File Coverage

blib/lib/Astro/Coord/ECI.pm

Criterion	Covered	Total	%
statement	775	914	84.7
branch	286	442	64.7
condition	70	137	51.0
subroutine	87	98	88.7
pod	43	43	100.0
total	1261	1634	77.1

=head1 NAME

Astro::Coord::ECI - Manipulate geocentric coordinates

=head1 SYNOPSIS

 use Astro::Coord::ECI;
 use Astro::Coord::ECI::Sun;
 use Astro::Coord::ECI::TLE;
 use Astro::Coord::ECI::Utils qw{rad2deg};

 # 1600 Pennsylvania Avenue, in radians, radians, and KM
 my ($lat, $lon, $elev) = (0.678911227503559,
     -1.34456123391096, 0.01668);

 # Record the time
 my $time = time ();

 # Set up observer's location
 my $loc = Astro::Coord::ECI->geodetic ($lat, $lon, $elev);

 # Uncomment to turn off atmospheric refraction if desired.
 # $loc->set( refraction => 0 );

 # Instantiate the Sun.
 my $sun = Astro::Coord::ECI::Sun->universal ($time);

 # Figure out if the Sun is up at the observer's location.
 my ($azimuth, $elevation, $range) = $loc->azel ($sun);
 print "The Sun is ", rad2deg ($elevation),
     " degrees above the horizon.\n";

See the L documentation
for an example involving satellite pass prediction.

=head1 DESCRIPTION

This module was written to provide a base class for a system to predict
satellite visibility. Its main task is to convert the
Earth-Centered Inertial (ECI) coordinates generated by the NORAD models
into other coordinate systems (e.g. latitude, longitude, and altitude
above mean sea level), but a other functions have accreted to it.
In addition, a few support routines have been exposed for testing, or
whatever.

All distances are in kilometers, and all angles are in radians
(including right ascension, which is usually measured in hours).

Times are normal Perl times, whether used as universal or dynamical
time. Universal time is what is usually meant, unless otherwise stated.

Known subclasses include B to predict the
position of the Moon, B to predict the
position of a star, or anything else that can be considered fixed on
the celestial sphere, B to predict the position
of the Sun, B to predict the position of a
satellite given the NORAD orbital parameters, and
B (a subclass of
Astro::Coord::ECI::TLE, and as of 0.099_01 moved to its own package) to
predict Iridium flares.

B that in version 0.022_01 the velocity code got a substantial
rework, which is still in progress. I am attempting give useful
velocities where they are available, and no velocities at all where they
are not.  Unfortunately I have yet to locate any worked problems, so the
velocity code is, in the most literal sense, untested.

In general, locations specified in Earth-fixed coordinates are
considered to share the rotational velocity of the Earth, and locations
specified in inertial coordinates are considered to have undefined
velocities unless a velocity was specified explicitly or produced by the
object's model. This involved a change in the behavior of the eci()
method when velocity was not specified but location was. See the next
paragraph for my excuse.

B This class and its subclasses should probably be
considered alpha code, meaning that the public interface may not be
completely stable. I will try not to change it, but given sufficient
reason I will do so. If I do so, I will draw attention to the change
in the documentation.

=head2 Methods

The following methods should be considered public.

=over 4

=cut

package Astro::Coord::ECI;

use strict;
use warnings;

our $VERSION = '0.134';

use Astro::Coord::ECI::Utils qw{ @CARP_NOT :mainstream };
use Carp;
use Clone ();
use POSIX qw{ floor };

use constant NO_CASCADING_STATIONS =>
    q{Cascading 'station' attributes are not supported};
use constant TWO_DEGREES => deg2rad( 2 );
use constant CIVIL_TWILIGHT	=> deg2rad( -6 );

my %mutator;	# Attribute mutators. We define these just after the
		# set() method, for convenience.
my %known_ellipsoid;	# Known reference ellipsoids. We define these
		# just before the reference_ellipsoid() method for
		# convenience.
my %static = (	# The geoid, etc. Geoid gets set later.
#   almanac_horizon	=> set when instantiated, so that the following
#   _almanac_horizon	=>     gets set automatically by the mutator.
    angularvelocity => 7.292114992e-5,	# Of surface of Earth, 1998. Meeus, p.83
    debug => 0,
    diameter => 0,
    edge_of_earths_shadow => 1,
    horizon => deg2rad (20),
    refraction => 1,
    twilight => CIVIL_TWILIGHT,
);
my %savatr;	# Attribs saved across "normal" operations. Set at end.
my @kilatr =	# Attributes to purge when setting coordinates.
    qw{_need_purge _ECI_cache inertial local_mean_time specified}; #?

=item $coord = Astro::Coord::ECI->new ();

This method instantiates a coordinate object. Any arguments are passed
to the set() method once the object has been instantiated.

=cut

sub new {
    my ( $class, @args ) = @_;
    my $self = bless { %static }, ref $class || $class;
    $self->{inertial} = $self->__initial_inertial();
    ref $static{sun}
	and $self->set( sun => $static{sun}->clone() );
    @args and $self->set( @args );
    exists $self->{almanac_horizon}
	or $self->set( almanac_horizon => 0 );
    $self->{debug} and do {
	require Data::Dumper;
	local $Data::Dumper::Terse = 1;
	print "Debug - Instantiated ", Data::Dumper::Dumper ($self);
    };
    return $self;
}

=item $angle = $coord->angle ($coord2, $coord3);

This method calculates the angle between the $coord2 and $coord3
objects, as seen from $coord. The calculation uses the law of
haversines, and does not take atmospheric refraction into account. The
return is a number of radians between 0 and pi.

The algorithm comes from "Ask Dr. Math" on the Math Forum,
C, which
attributes it to the Geographic Information Systems FAQ,
L, which in turn
attributes it to R. W. Sinnott, "Virtues of the Haversine," Sky and
Telescope, volume 68, number 2, 1984, page 159.

Unfortunately, as of early 2021 the National Council of Teachers of
Mathematics restricted the Dr. Math content to their members, but an
annotated and expanded version of the article on haversines is available
at
L.
If you want the original article, you can feed the URL
C to the Wayback
Machine.

Prior to version 0.011_03 the law of cosines was used, but this produced
errors on extremely small angles. The haversine law was selected based
on Jean Meeus, "Astronomical Algorithms", 2nd edition, chapter 17
"Angular Separation".

This method ignores the C attribute.

=cut

sub angle {
    my $self = shift;
    my $B = shift;
    my $C = shift;
    (ref $B && $B->represents (__PACKAGE__)
	&& ref $C && $C->represents (__PACKAGE__))
	or croak <
Error - Both arguments must represent @{[__PACKAGE__]} objects.
eod
    my ($ra1, $dec1) = $self->{inertial} ?
	$B->equatorial ($self) : $self->_angle_non_inertial ($B);
    my ($ra2, $dec2) = $self->{inertial} ?
	$C->equatorial ($self) : $self->_angle_non_inertial ($C);
    my $sindec = sin (($dec2 - $dec1)/2);
    my $sinra = sin (($ra2 - $ra1)/2);
    my $a = $sindec * $sindec +
	cos ($dec1) * cos ($dec2) * $sinra * $sinra;
    return 2 * atan2 (sqrt ($a), sqrt (1 - $a));

}

sub _angle_non_inertial {
    my $self = shift;
    my $other = shift;
    my ($x1, $y1, $z1) = $self->ecef ();
    my ($x2, $y2, $z2) = $other->ecef ();
    my $X = $x2 - $x1;
    my $Y = $y2 - $y1;
    my $Z = $z2 - $z1;
    my $lon = atan2 ($Y, $X);
    my $lat = mod2pi (atan2 ($Z, sqrt ($X * $X + $Y * $Y)));
    return ($lon, $lat);
}

=item $angle = $coord->angular_radius();

This method computes the angular radius of the C<$coord> body in
radians, as of the currently-set time. If the C attribute is
set, the range will be computed from it via C<< $self->azel() >>. If
not, the range from the center of the earth will be computed via
C<< $self->ecliptic() >>. If the C attribute is not set,
C is returned.

This method is not supported for terrestrial objects.

=cut

sub angular_radius {
    my ( $self ) = @_;
    defined( my $diameter = $self->get( 'diameter' ) )
	or return undef;	## no critic (ProhibitExplicitReturnUndef)
    my $range;
    if ( $self->get( 'station' ) ) {
	( undef, undef, $range ) = $self->azel();
    } else {
	( undef, undef, $range ) = $self->ecliptic();
    }
    return atan2 $diameter / 2, $range;
}


=item $which = $coord->attribute ($name);

This method returns the name of the class that implements the named
attribute, or undef if the attribute name is not valid.

=cut

sub attribute {return exists $mutator{$_[1]} ? __PACKAGE__ : undef}

=item ($azimuth, $elevation, $range) = $coord->azel( $coord2 );

This method takes another coordinate object, and computes its azimuth,
elevation, and range in reference to the object doing the computing.
The return is azimuth in radians measured clockwise from North (always
positive), elevation above the horizon in radians (negative if
below), and range in kilometers.

As a side effect, the time of the $coord object may be set from the
$coord2 object.

Atmospheric refraction is taken into account using the
C method if the C<$coord> object's
C attribute is true. The invocant's
C method is the one used; that is, if
C<$coord> and C<$coord2> have different C
methods, the C<$coord> object's method is used.

B that the C attribute defaults to true.
For cases where both invocant and argument are ground-based objects, you
should probably set the invocant's C false
before invoking this method.

This method is implemented in terms of azel_offset(). See that method's
documentation for further details.

=item ( $azimuth, $elevation, $range ) = $coord->azel();

This method computes the azimuth, elevation, and range if the C<$coord>
object as seen from the position stored in the C<$coord> object's
C attribute. An exception will be thrown if the C
attribute is not set.

=cut

sub azel {	## no critic (RequireArgUnpacking)
    @_ > 2
	and croak q{Too many arguments};
    @_ = _expand_args_default_station( @_ );
    $_[2] = $_[2] ? 1 : 0;
    goto &azel_offset;
}

=item ($azimuth, $elevation, $range) = $coord->azel_offset($coord2, $offset);

This method takes another coordinate object, and computes its azimuth,
elevation, and range in reference to the object doing the computing.
The return is azimuth in radians measured clockwise from North (always
positive), elevation above the horizon in radians (negative if
below), and range in kilometers.

If the optional C<$offset> argument is provided, the elevation is offset
upward by the given fraction of the radius of the C<$coord2> object.
Thus, an C<$offset> of C<1> specifies the upper limb of the object, C<0>
specifies the center of the object, and C<-1> specifies the lower limb.
This offset is applied before atmospheric refraction.

As a side effect, the time of the $coord object may be set from the
$coord2 object.

By default, atmospheric refraction is taken into account in the
calculation of elevation, using the C method.
This better represents the observed position in the sky when the object
is above the horizon, but produces a gap in the data when the object
passes below the horizon, since I have no refraction equations for rock.
See the C documentation for details.

The invocant's C method is the one used; that
is, if C<$coord> and C<$coord2> have different
C methods, the C<$coord> object's method is
used.

If you want to ignore atmospheric refraction (and not have a gap in your
data), set the C attribute of the $coord object
to any value Perl considers false (e.g. 0).

The algorithm for position is the author's, but he confesses to having
to refer to T. S. Kelso's "Computers and Satellites"
column in "Satellite Times", November/December 1995, titled "Orbital
Coordinate Systems, Part II" and available at
F to get the signs right.

If velocities are available for both bodies involved in the calculation,
they will be returned after the position (i.e. you get a six-element
array back instead of a three-element array). The velocity of a point
specified in Earth-fixed coordinates (e.g. geodetic latitude, longitude,
and altitude) is assumed to be the rotational velocity of the Earth at
that point. The returned velocities are azimuthal velocity in radians
per second (B radians of azimuth, which get smaller as you go
farther away from the Equator), elevational velocity in radians per
second, and velocity in recession in kilometers per second, in that
order.

If velocities are available for both bodies B the C<$coord2> object
has its C attribute set, the returned array will contain
seven elements, with the seventh being the Doppler shift.

The algorithm for recessional velocity comes from John A. Magliacane's
C program, available at
L. The transverse velocity
computations are the author's, but use the same basic approach: vector
dot product of the velocity vector with a unit vector in the appropriate
direction, the latter generated by the appropriate (I hope!) vector
cross products.

Velocity conversions to C appear to me to be mostly sane. See
L, below, for details.

If velocities are available I you have provided a non-zero value
for the C attribute, you will get the Doppler shift as the
seventh element of the returned array. The I about velocity in
recession apply to the Doppler shift as well. The frequency is from the
C<$coord2> object. Getting the frequency from the C<$coord> object used
to be supported as a fallback, but now results in an exception.

=item ( $azimuth, $elevation, $range ) = $coord->azel_offset( $offset );

This method computes the azimuth, elevation, and range if the C<$coord>
object as seen from the location stored in the C<$coord> object's
C attribute. The functionality is as above, except for the fact
that in the above version the station is the invocant, whereas in this
version the orbiting body is the invocant.

This version also returns velocities if they are available for both
bodies, and Doppler shift if in addition the C attribute of
the invocant is set.

=cut

sub azel_offset {
    my ( $self, $trn2, $offset ) = _expand_args_default_station( @_ );
    $self->{debug} and do {
	require Data::Dumper;
	local $Data::Dumper::Terse = 1;
	print "Debug azel_offset - ", Data::Dumper::Dumper ($self, $trn2, $offset);
    };

    # _local_cartesian() returns NEU coordinates. Converting these to
    # spherical gets Azimuth (clockwise from North), Elevation, and
    # Range.

    my ( $azimuth, $elevation, $range, @velocity ) =
	_convert_cartesian_to_spherical(
	    $self->_local_cartesian( $trn2 )
	);

    # If the velocity and frequency are defined, we provide the Doppler
    # shift as well.

    if ( defined $velocity[2] ) {
	my $freq = $trn2->get( 'frequency' );
	if ( not defined $freq ) {
	    $freq = $self->get( 'frequency' );
	    defined $freq
		and croak 'Calculation of Doppler shift based ',
		    'on the frequency attribute of the observing ',
		    'station is not allowed';
	}
	if ( defined $freq ) {
	    $velocity[3] = - $freq * $velocity[2] / SPEED_OF_LIGHT;
	}
    }

    # Adjust for upper limb and refraction if needed.

    $offset
	and $elevation += atan2(
	    $trn2->get( 'diameter' ) * $offset / 2,
	    $range,
	);

    $self->{refraction} and
	$elevation = $self->correct_for_refraction( $elevation );

    return ( $azimuth, $elevation, $range, @velocity );
}

=item $coord2 = $coord->clone ();

This method does a deep clone of an object, producing a different
but identical object.

At the moment, it's really just a wrapper for Clone::clone.

=cut

sub clone {
    my ( $self ) = @_;
    return Clone::clone( $self );
}

=item $elevation = $coord->correct_for_refraction ($elevation);

This method corrects the given angular elevation for atmospheric
refraction. This is done only if the elevation has a reasonable chance
of being visible; that is, if the elevation before correction is not
more than two degrees below either the 'horizon' attribute or zero,
whichever is lesser. Sorry for the discontinuity thus introduced, but I
did not wish to carry the refraction calculation too low because I am
uncertain about the behavior of the algorithm, and I do not have a
corresponding algorithm for refraction through magma.

This method can also be called as a class method, in which case the
correction is applied only if the uncorrected elevation is greater than
minus two degrees. It is really only exposed for testing purposes (hence
the cumbersome name). The azel() method calls it for you if the
C attribute is true.

The algorithm for atmospheric refraction comes from Thorfinn
Saemundsson's article in "Sky and Telescope", volume 72, page 70
(July 1986) as reported Jean Meeus' "Astronomical Algorithms",
2nd Edition, chapter 16, page 106, and includes the adjustment
suggested by Meeus.

=cut

sub correct_for_refraction {
    my $self = shift;
    my $elevation = shift;

    # If called as a static method, our effective horizon is zero. If
    # called as a normal method, our effective horizon is the lesser of
    # the 'horizon' setting or zero.

    my $horizon = ref $self ? min( 0, $self->get( 'horizon' ) ) : 0;

    # We exclude anything with an elevation <= 2 degrees below our
    # effective horizon; this is presumed to be not visible, since the
    # maximum deflection is about 35 minutes of arc. This is not
    # portable to (e.g.) Venus.

    if ( $elevation > $horizon - TWO_DEGREES ) {

#	Thorsteinn Saemundsson's algorithm for refraction, as reported
#	in Meeus, page 106, equation 16.4, and adjusted per the
#	suggestion in Meeus' following paragraph. Thorsteinn's
#	formula is in terms of angles in degrees and produces
#	a correction in minutes of arc. Meeus reports the original
#	publication as Sky and Telescope, volume 72 page 70, July 1986.

#	In deference to Thorsteinn I will point out:
#	* The Icelanders do not use family names. The "Saemundsson"
#	  simply means his father's name was Saemund.
#	* I have transcribed the names into 7-bit characters.
#	  "Thorsteinn" actually does not begin with "Th" but with
#	  the letter "Thorn." Similarly, the "ae" in "Saemund" is
#	  supposed to be a ligature (i.e. squished into one letter).

	my $deg = rad2deg ($elevation);
	my $correction = 1.02 / tan (deg2rad ($deg + 10.3/($deg + 5.11))) +
	    .0019279;
	$self->get ('debug') and print <
Debug correct_for_refraction
    input: $deg degrees of arc
    correction: $correction minutes of arc
eod

#	End of Thorsteinn's algorithm.

	$correction = deg2rad ($correction / 60);
	$elevation += $correction;
    }
    return $elevation;
}

=item $angle = $coord->dip ();

This method calculates the dip angle of the horizon due to the
altitude of the body, in radians. It will be negative for a location
above the surface of the reference ellipsoid, and positive for a
location below the surface.

The algorithm is simple enough to be the author's.

=cut

sub dip {
    my $self = shift;
    my (undef, undef, $h) = $self->geodetic ();
    my (undef, undef, $rho) = $self->geocentric ();
    return $h >= 0 ?
	- acos (($rho - $h) / $rho) :
	acos ($rho / ($rho - $h));
}

=item $coord = $coord->dynamical ($time);

This method sets the dynamical time represented by the object.

This method can also be called as a class method, in which case it
instantiates the desired object.

The algorithm for converting this to universal time comes from Jean
Meeus' "Astronomical Algorithms", 2nd Edition, Chapter 10, pages 78ff.

=item $time = $coord->dynamical ();

This method returns the dynamical time previously set, or the
universal time previously set, converted to dynamical.

The algorithm comes from Jean Meeus' "Astronomical Algorithms", 2nd
Edition, Chapter 10, pages 78ff.

=cut

sub dynamical {
    my ($self, @args) = @_;
    unless (@args) {
	ref $self or croak <
Error - The dynamical() method may not be called as a class method
        unless you specify arguments.
eod
	return ($self->{dynamical} ||= $self->{universal} +
	    dynamical_delta ($self->{universal} || croak <
Error - Universal time of object has not been set.
eod
    }

    if (@args == 1) {
	my $time = shift @args;
	$self = $self->new () unless ref $self;
	$self->{_no_set}++;	# Supress running the model if any
	$self->universal ($time - dynamical_delta ($time));
	$self->{dynamical} = $time;
	--$self->{_no_set};	# Undo supression of model
	$self->_call_time_set ();	# Run the model if any
    } else {
	croak <
Error - The dynamical() method must be called with either zero
        arguments (to retrieve the time) or one argument (to set the
        time).
eod
    }

    return $self;
}

=item $coord = $coord->ecef($x, $y, $z, $xdot, $ydot, $zdot)

This method sets the coordinates represented by the object in terms of
L in kilometers, with
the x axis being latitude 0 longitude 0, the y axis being latitude 0
longitude 90 degrees east, and the z axis being latitude 90 degrees
north. The velocities in kilometers per second are optional, and will
default to zero. ECEF velocities are considered to be relative to the
surface of the Earth; when converting to ECI, the rotational velocity of
the Earth will be added in.

B that prior to version 0.022_01, the documentation said that the
default velocity was the rotational velocity of the Earth. This was not
correct B The functionality of the code itself in
this respect did not change.

The object itself is returned.

This method can also be called as a class method, in which case it
instantiates the desired object.

=item ($x, $y, $z, $xdot, $ydot, $zdot) = $coord->ecef();

This method returns the object's L
coordinates>.

If the original coordinate setting was in an inertial system (e.g. eci,
equatorial, or ecliptic) B the absolute difference between the
current value of 'equinox_dynamical' and the current dynamical() setting
is greater than the value of $Astro::Coord::ECI::EQUINOX_TOLERANCE, the
coordinates will be precessed to the current dynamical time before
conversion.  Yes, this should be done any time the equinox is not the
current time, but for satellite prediction precession by a year or
so does not seem to make much difference in practice. The default value
of $Astro::Coord:ECI::EQUINOX_TOLERANCE is 365 days.  B that if
this behavior or the default value of $EQUINOX_TOLERANCE begins to look
like a bug, it will be changed, and noted in the documentation.

Some velocity conversions involving C appear to me to be mostly
sane. See L, below, for
details.

=cut

sub ecef {
    my ($self, @args) = @_;

    $self = $self->_check_coord (ecef => \@args);

    unless (@args) {
	my $cache = ($self->{_ECI_cache} ||= {});
	return @{$cache->{fixed}{ecef}} if $cache->{fixed}{ecef};
	return $self->_convert_eci_to_ecef () if $self->{inertial};
	croak <
Error - Object has not been initialized.
eod
    }

    @args == 3 and push @args, 0, 0, 0;

    if (@args == 6) {
	foreach my $key (@kilatr) {delete $self->{$key}}
##	$self->{_ECI_cache}{fixed}{ecef} = \@args;
	$self->{_ECI_cache} = {fixed => {ecef => \@args}};
	$self->{specified} = 'ecef';
	$self->{inertial} = 0;
    } else {
	croak <
Error - The ecef() method must be called with either zero arguments (to
        retrieve coordinates), three arguments (to set coordinates,
        with velocity defaulting to zero) or six arguments.
eod
    }

    return $self;
}

=item $coord = $coord->eci ($x, $y, $z, $xdot, $ydot, $zdot, $time)

This method sets the coordinates represented by the object in terms of
L in kilometers, time being
universal time, the x axis being 0 hours L 0 degrees
L, y being 6 hours L 0 degrees
L, and z being 90 degrees north L. The
velocities in kilometers per second are optional. If omitted, the object
will be considered not to have a velocity. B
with version 0.022_01.> Before that, the velocity defaulted to 0.

The time argument is optional if the time represented by the object
has already been set (e.g. by the universal() or dynamical() methods).

The object itself is returned.

This method can also be called as a class method, in which case it
instantiates the desired object. In this case the time is not optional.

The algorithm for converting from ECI to geocentric coordinates and
back is based on the description of ECI coordinates in T. S. Kelso's
"Computers and Satellites" column in "Satellite Times",
September/October 1995, titled "Orbital Coordinate Systems, Part I"
and available at F.

=item ($x, $y, $z, $xdot, $ydot, $zdot) = $coord->eci($time);

This method returns the L
of the object at the given time. The time argument is actually
optional if the time represented by the object has already been set.

If you specify a time, the time represented by the object will be set
to that time. The net effect of specifying a time is equivalent to

 ($x, $y, $z, $xdot, $ydot, $zdot) = $coord->universal($time)->eci()

If the original coordinate setting was in a non-inertial system (e.g.
ECEF or geodetic), the equinox_dynamical attribute will be set to the
object's dynamical time.

Velocities will be returned if they were originally provided. B
a change introduced in version 0.022_01.> Prior to that version,
velocities were always returned.

Some velocity conversions involving C appear to me to be mostly
sane. See L, below, for
details.

=cut

sub eci {
    my ($self, @args) = @_;

    $self = $self->_check_coord (eci => \@args);

    unless (@args) {
	my $cache = ($self->{_ECI_cache} ||= {});
	return @{$cache->{inertial}{eci}} if $cache->{inertial}{eci};
	return $self->_convert_ecef_to_eci () if $self->{specified};
	croak <
Error - Object has not been initialized.
eod

    }

##    @args == 3 and push @args, 0, 0, 0;

    if (@args == 3 || @args == 6) {
	foreach my $key (@kilatr) {delete $self->{$key}}
	$self->{_ECI_cache} = {inertial => {eci => \@args}};
	$self->{specified} = 'eci';
	$self->{inertial} = 1;
    } else {
	croak <
Error - The eci() method must be called with either zero or one
        arguments (to retrieve coordinates), three or four arguments
        (to set coordinates, with velocity defaulting to zero), or
        six or seven arguments.
eod
    }
    return $self;
}

=item $coord = $coord->ecliptic_cartesian( $X, $Y, $Z, $time );

This method sets the Cartesian L coordinates represented by
the object in terms of C (kilometers toward the vernal equinox), C
(90 degrees along the ecliptic from C), and C (toward ecliptic
North).  The time is universal time.  The object itself is returned.

The time argument is optional if the time represented by the object has
already been set (e.g. by the universal() or dynamical() methods).

This method can also be called as a class method, in which case it
instantiates the desired object. In this case the time is not optional.

B you can pass in velocities (before the C<$time>) but they are
unsupported, meaning that I can not at this point say whether they will
be transformed sanely, much less correctly. B.

=item ( $X, $Y, $Z) = $coord->ecliptic_cartesian( $time );

This method returns the Cartesian ecliptic coordinates of the object at
the given time. The time is optional if the time represented by the
object has already been set (e.g. by the universal() or dynamical()
methods).

B velocities are unsupported by this method. That means you may
(or may not, depending on the coordinates originally set) get them back,
but I have not looked into whether they are actually correct.

=cut

sub ecliptic_cartesian {
    my ( $self, @args ) = @_;

    $self = $self->_check_coord( ecliptic_cartesian => \@args );

    if ( @args ) {
	@args % 3
	    and croak <<'EOD';
Error - The ecliptic_cartesian() method must be called with either zero
        or one arguments (to retrieve coordinates), or three or four
        arguments (to set coordinates). There is currently no six or
        seven argument version.
EOD
	my $epsilon = $self->obliquity();
	my $sin_epsilon = sin $epsilon;
	my $cos_epsilon = cos $epsilon;
	my @eci;
	for ( my $inx = 0; $inx < @args; $inx += 3 ) {
	    push @eci,   $args[ $inx ];
	    push @eci, - $args[ $inx + 2 ] * $sin_epsilon +
		$args[ $inx + 1 ] * $cos_epsilon;
	    push @eci,   $args[ $inx + 2 ] * $cos_epsilon +
		$args[ $inx + 1 ] * $sin_epsilon;
	}
	$self->eci( @eci );
	$self->{_ECI_cache}{inertial}{ecliptic_cartesian} = \@args;
	$self->{inertial} = 1;
	$self->{specified} = 'ecliptic_cartesian';
	return $self;
    } else {
	$self->{_ECI_cache}{inertial}{ecliptic_cartesian}
	    and return @{
		$self->{_ECI_cache}{inertial}{ecliptic_cartesian}
	    };
	my @eci = $self->eci();
	my $epsilon = $self->obliquity();
	my $sin_epsilon = sin $epsilon;
	my $cos_epsilon = cos $epsilon;
	my @ecliptic_cartesian;
	for ( my $inx = 0; $inx < @eci; $inx += 3 ) {
	    push @ecliptic_cartesian,   $eci[ $inx ];
	    push @ecliptic_cartesian,   $eci[ $inx + 2 ] * $sin_epsilon +
		$eci[ $inx + 1 ] * $cos_epsilon;
	    push @ecliptic_cartesian,   $eci[ $inx + 2 ] * $cos_epsilon -
		$eci[ $inx + 1 ] * $sin_epsilon;
	}
	return @{
		$self->{_ECI_cache}{inertial}{ecliptic_cartesian} =
		    \@ecliptic_cartesian
	    };
    }

}

=item $coord = $coord->ecliptic ($latitude, $longitude, $range, $time);

This method sets the L coordinates represented by the object
in terms of L and L in radians,
and the range to the object in kilometers, time being universal time.
The object itself is returned.

The time argument is optional if the time represented by the object has
already been set (e.g. by the universal() or dynamical() methods).

The latitude should be a number between -PI/2 and PI/2 radians
inclusive. The longitude should be a number between -2*PI and 2*PI
radians inclusive.  The increased range (one would expect -PI to PI) is
because in some astronomical usages latitudes outside the range + or -
180 degrees are employed. A warning will be generated if either of these
range checks fails.

This method can also be called as a class method, in which case it
instantiates the desired object. In this case the time is not optional.

The algorithm for converting from ecliptic latitude and longitude to
right ascension and declination comes from Jean Meeus'
"Astronomical Algorithms", 2nd Edition, Chapter 13, page 93.

=item ($latitude, $longitude, $range) = $coord->ecliptic ($time);

This method returns the ecliptic latitude and longitude of the
object at the given time. The time is optional if the time represented
by the object has already been set (e.g. by the universal() or
dynamical() methods).

B velocities are not returned by this method.

=cut

sub ecliptic {
    my ($self, @args) = @_;

    $self = $self->_check_coord( ecliptic => \@args );

    if ( @args ) {

	@args % 3
	    and croak 'Arguments are in threes, plus an optional time';
	$self->{_ECI_cache}{inertial}{ecliptic} = \@args;
	$self->ecliptic_cartesian(
	    _convert_spherical_x_to_cartesian( @args ) );
	$self->{specified} = 'ecliptic';
	$self->{inertial} = 1;
	return $self;

    } else {

	return @{ $self->{_ECI_cache}{inertial}{ecliptic} ||= [
	    _convert_cartesian_to_spherical_x(
		$self->ecliptic_cartesian() )
	    ]
	};

    }
}

=item $longitude = $coord->ecliptic_longitude();

This method returns the ecliptic longitude of the body at its current
time setting, in radians. It is really just a convenience method, since
it is equivalent to C<< ( $coord->ecliptic() )[1] >>, and in fact that
is how it is implemented.

=cut

sub ecliptic_longitude {
    my ( $self ) = @_;
    return ( $self->ecliptic() )[1];
}

=item $seconds = $self->equation_of_time( $time );

This method returns the equation of time at the given B
time. If the time is C, the invocant's dynamical time is used.

The algorithm is from W. S. Smart's "Text-Book on Spherical Astronomy",
as reported in Jean Meeus' "Astronomical Algorithms", 2nd Edition,
Chapter 28, page 185.

=cut

sub equation_of_time {
    my ( $self, $time ) = @_;

    if ( looks_like_number( $self ) ) {
	( $self, $time ) = ( __PACKAGE__, $self );
    }
    defined $time
	or $time = $self->dynamical();

    my $epsilon = $self->obliquity( $time );
    my $y = tan($epsilon / 2);
    $y *= $y;

#	The following algorithm is from Meeus, chapter 25, page, 163 ff.

    my $T = jcent2000( $time );				# Meeus (25.1)
    my $L0 = mod2pi( deg2rad( (.0003032 * $T + 36000.76983 ) * $T
	    + 280.46646 ) );	# Meeus (25.2)
    my $M = mod2pi( deg2rad( ( ( -.0001537 ) * $T + 35999.05029 )
	    * $T + 357.52911 ) );	# Meeus (25.3)
    my $e = ( -0.0000001267 * $T - 0.000042037 ) * $T
	    + 0.016708634;	# Meeus (25.4)

    my $E = $y * sin( 2 * $L0 ) - 2 * $e * sin( $M ) +
	4 * $e * $y * sin( $M ) * cos( 2 * $L0 ) -
	$y * $y * .5 * sin( 4 * $L0 ) -
	1.25 * $e * $e * sin( 2 * $M );				# Meeus (28.3)

    return $E * SECSPERDAY / TWOPI;	# The formula gives radians.
}

=item $coord->equatorial ($rightasc, $declin, $range, $time);

This method sets the L coordinates represented by the
object (relative to the center of the Earth) in terms of
L and L in radians, and the range to the
object in kilometers, time being universal time. The object itself is
returned.

If C is called in this way, the C attribute will
be ignored, for historical reasons.

The right ascension should be a number between 0 and 2*PI radians
inclusive. The declination should be a number between -PI/2 and PI/2
radians inclusive. A warning will be generated if either of these range
checks fails.

The time argument is optional if the time represented by the object
has already been set (e.g. by the universal() or dynamical() methods).

This method can also be called as a class method, in which case it
instantiates the desired object. In this case the time is not optional.

You may optionally pass velocity information in addition to position
information. If you do this, the velocity in right ascension (in radians
per second), declination (ditto), and range (in kilometers per second in
recession) are passed after the position arguments, and before the $time
argument if any.

=item ( $rightasc, $declin, $range ) = $coord->equatorial( $time );

This method returns the L coordinates of the object at the
relative to the center of the Earth. The C attribute is
ignored.  The time argument is optional if the time represented by the
object has already been set (e.g. by the universal() or dynamical()
methods).

If velocities are available from the object (i.e. if it is an instance
of Astro::Coord::ECI::TLE) the return will contain velocity in right
ascension, declination, and range, the first two being in radians per
second and the third in kilometers per second in recession.

B these velocities are believed by me to be sane B they are
derived from ECI coordinates. This method does not support setting
velocities. See L, below,
for details.

=item ($rightasc, $declin, $range) = $coord->equatorial( $coord2 );

This method is retained (for the moment) for historical reasons, but
C is preferred.

This method returns the apparent equatorial coordinates of the object
represented by $coord2, as seen from the location represented by
$coord.

As a side effect, the time of the $coord object may be set from the
$coord2 object.

If the C attribute of the $coord object is
true, the coordinates will be corrected for atmospheric refraction using
the correct_for_refraction() method.

If velocities are available from both objects (i.e. if both objects are
Astro::Coord::ECI::TLE objects) the return will contain velocity in
right ascension, declination, and range, the first two being in radians
per second and the third in kilometers per second in recession.

B these velocities are believed by me B to be correct.

=cut

sub equatorial {
    my ( $self, @args ) = @_;

    my $body;
    @args
	and embodies( $args[0], __PACKAGE__ )
	and $body = shift @args;

    $self = $self->_check_coord( equatorial => \@args );
    my $time;
    $body or $time = $self->universal;

    if ( @args ) {
	@args % 3
	    and croak 'Arguments must be in threes, with an optional time';
	$body
	    and croak 'You may not set the equatorial coordinates ',
	'relative to an observer';

##	my ($ra, $dec, $range, @eqvel) = @args;
	$args[0] = _check_right_ascension( 'right ascension' => $args[0] );
	$args[1] = _check_latitude( declination => $args[1] );
	foreach my $key (@kilatr) {delete $self->{$key}}
	$self->{_ECI_cache}{inertial}{equatorial} = \@args;
	$self->eci(
	    _convert_spherical_to_cartesian( @args ) );
	$self->{specified} = 'equatorial';
	$self->{inertial} = 1;
	return $self;

    } elsif ( $body ) {

	return $self->_equatorial_reduced( $body );

    } else {

	return @{ $self->{_ECI_cache}{inertial}{equatorial} ||= [
	    _convert_cartesian_to_spherical( $self->eci() )
	] };

    }
}

=item ($ra, $decl, $rng) = $coord->equatorial_apparent( $sta );

This method returns the apparent equatorial coordinates of the C<$coord>
object as seen from the object specified by the C<$sta> argument, or by
the object in the C attribute of the C<$coord> object if no
argument is specified.

This method will return velocities if available, but I have no idea
whether they are correct.

=cut

sub equatorial_apparent {
    my ( $self, $station ) = @_;
    ( $station ||= $self->get( 'station' ) )
	or croak 'Station attribute is required';
    return $station->_equatorial_reduced( $self );
}

=item my ($rasc, $decl, $range, $v_rasc, $v_decl, $v_r) = $coord->equatorial_unreduced($body);

This method computes the unreduced equatorial position of the second ECI
object as seen from the first. If the second argument is undefined,
computes the equatorial position of the first object as seen from the
center of the Earth. Unlike the equatorial() method itself, this method
is an accessor only. This method would probably not be exposed except
for the anticipation of the usefulness of $range and $v_r in satellite
conjunction computations, and the desirability of not taking the time to
make the two atan2() calls that are unneeded in this application.

The 'unreduced' in the name of the method is intended to refer to the
fact that the $rasc and $decl are not the right ascension and
declination themselves, but the arguments to atan2() needed to compute
them, and $v_rasc and $v_decl are in km/sec, rather than being divided
by the range to get radians per second.

This method ignores the C<'station'> attribute.

The returned data are:

$rasc is an array reference, which can be converted to the right
ascension in radians by mod2pi(atan2($rasc->[0], $rasc->[1])).

$decl is an array reference, which can be converted to the declination
in radians by atan2($decl->[0], $decl->[1]).

$range is the range in kilometers.

$v_rasc is the velocity in the right ascensional direction in kilometers
per second. It can be converted to radians per second by dividing by
$range.

$v_decl is the velocity in the declinational direction in kilometers per
second. It can be converted to radians per second by dividing by $range.

$v_r is the velocity in recession in kilometers per second. Negative
values indicate that the objects are approaching.

The velocities are only returned if they are available from the input
objects.

B these velocities are believed by me B to be correct.

=cut

sub equatorial_unreduced {

    # Unpack the two objects.

    my ($self, $body) = @_;

    # Compute the relative positions if there are in fact two objects;
    # otherwise just get the absolute position.

    my @pos;
    if ($body) {
	@pos = $body->eci();
	my @base = $self->eci($body->universal());
	my $limit = @pos < @base ? @pos : @base;
	foreach my $inx (0 .. $limit - 1) {
	    $pos[$inx] -= $base[$inx];
	}
	splice @pos, $limit;
    } else {
	$self->{_ECI_cache}{inertial}{equatorial_unreduced} and return
	    @{$self->{_ECI_cache}{inertial}{equatorial_unreduced}};
	@pos = $self->eci();
    }

    # Rotate the coordinate system so that the second body lies in the
    # X-Z plane. The matrix is
    # +-                     -+
    # | cos rasc -sin rasc  0 |
    # | sin rasc  cos rasc  0 |
    # |     0         0     1 |
    # +-                     -+
    # where rasc is -atan2(y,x). This means sin rasc = -y/h and
    # cos rasc = x/h where h = sqrt(x*x + y*y). You postmultiply
    # the matrix by the vector to get the new vector.

    my $hypot_rasc = sqrt($pos[0] * $pos[0] + $pos[1] * $pos[1]);

    # Now we need to rotate the coordinates in the new X-Z plane until
    # the second body lies along the X axis. The matrix is
    # +-                      -+
    # | cos decl  0  -sin decl |
    # |     0     1       0    |
    # | sin decl  0   cos decl |
    # +-                      -+
    # where decl is -atan2(z,x') (in the new coordinate system), or
    # -atan2(z,h) in the old coordinate system. This means that sin decl
    # = z/h' and cos decl = x/h' where h' = sqrt(h*h + z*z). Again you
    # postmultiply the matrix by the vector to get the result.

    my $hypot_decl = sqrt($hypot_rasc * $hypot_rasc + $pos[2] * $pos[2]);

    # But at this point we have the equatorial coordinates themselves in
    # terms of the original coordinates and the various intermediate
    # quantities needed to compute the matrix. So we only need the
    # matrix if we need to deal with the velocities. For this, the
    # velocity rotation matrix is
    # +-                      -+   +-                     -+
    # | cos decl  0  -sin decl |   | cos rasc -sin rasc  0 |
    # |     0     1       0    | x | sin rasc  cos rasc  0 | =
    # | sin decl  0   cos decl |   |     0         0     1 |
    # +-                      -+   +-                     -+
    #
    # +-                                                -+
    # | cos decl cos rasc  -cos decl sin rasc  -sin decl |
    # | sin rasc            cos rasc               0     |
    # | sin decl cos rasc  -sin decl sin rasc   cos decl |
    # +-                                                 +

    my @rslt = ([$pos[1], $pos[0]], [$pos[2], $hypot_rasc], $hypot_decl);
    if (@pos >= 6) {
	my $cos_rasc =  $pos[0]/$hypot_rasc;
	my $sin_rasc = -$pos[1]/$hypot_rasc;
	my $cos_decl =  $hypot_rasc/$hypot_decl;
	my $sin_decl = -$pos[2]/$hypot_decl;
	push @rslt,
	    $sin_rasc * $pos[3] + $cos_rasc * $pos[4],
	    $sin_decl * $cos_rasc * $pos[3]
		- $sin_decl * $sin_rasc * $pos[4] + $cos_decl * $pos[5],
	    # Computationally the following is the top row of the
	    # matrix, but it is swapped to the bottom in the output for
	    # consistency of returned data sequence.
	    $cos_decl * $cos_rasc * $pos[3]
		- $cos_decl * $sin_rasc * $pos[4] - $sin_decl * $pos[5];
    }
    $body
	or $self->{_ECI_cache}{inertial}{equatorial_unreduced} = \@rslt;
    return @rslt;

}

sub _equatorial_reduced {
    my ( $self, $body ) = @_;
    my @rslt = $self->equatorial_unreduced( $body );
    $rslt[0] = mod2pi( atan2( $rslt[0][0], $rslt[0][1] ) );
    $rslt[1] = atan2( $rslt[1][0], $rslt[1][1] );
    if (@rslt >= 6) {
	$rslt[3] /= $rslt[2];
	$rslt[4] /= $rslt[2];
    }
    return @rslt;
}

=item $coord = $coord->equinox_dynamical ($value);

This method sets the value of the equinox_dynamical attribute, and
returns the modified object. If called without an argument, it returns
the current value of the equinox_dynamical attribute.

Yes, it is equivalent to $coord->set (equinox_dynamical => $value) and
$coord->get ('equinox_dynamical'). But there seems to be a significant
performance penalty in the $self->SUPER::set () needed to get this
attribute set from a subclass. It is possible that more methods like
this will be added, but I do not plan to eliminate the set() interface.

=cut

sub equinox_dynamical {
    if (@_ > 1) {
	$_[0]{equinox_dynamical} = $_[1];
	return $_[0];
    } else {
	return $_[0]{equinox_dynamical};
    }
}

=item $coord = $coord->geocentric($psiprime, $lambda, $rho);

This method sets the L coordinates represented by the
object in terms of L psiprime and L
lambda in radians, and distance from the center of the Earth rho in
kilometers.

The latitude should be a number between -PI/2 and PI/2 radians
inclusive. The longitude should be a number between -2*PI and 2*PI
radians inclusive.  The increased range (one would expect -PI to PI) is
because in some astronomical usages latitudes outside the range + or -
180 degrees are employed. A warning will be generated if either of these
range checks fails.

This method can also be called as a class method, in which case it
instantiates the desired object.

B because map
latitude is L, measured in terms of the tangent of
the reference ellipsoid, whereas geocentric coordinates are,
essentially, spherical coordinates.

The algorithm for conversion between geocentric and ECEF is the
author's.

=item ($psiprime, $lambda, $rho) = $coord->geocentric();

This method returns the L, L, and
distance to the center of the Earth.

=cut

sub geocentric {
    my ($self, @args) = @_;

    $self = $self->_check_coord (geocentric => \@args);

    unless (@args) {

	if ( ! $self->{_ECI_cache}{fixed}{geocentric} ) {

##	    my ($x, $y, $z, $xdot, $ydot, $zdot) = $self->ecef;
	    my ($x, $y, $z) = $self->ecef;
	    my $rsq = $x * $x + $y * $y;
	    my $rho = sqrt ($z * $z + $rsq);
	    my $lambda = atan2 ($y, $x);
	    my $psiprime = atan2 ($z, sqrt ($rsq));
	    $self->get ('debug') and print <
Debug geocentric () - ecef -> geocentric
    inputs:
        x = $x
        y = $y
        z = $z
    outputs:
        psiprime = $psiprime
        lambda = $lambda
        rho = $rho
eod
	    $self->{_ECI_cache}{fixed}{geocentric} =
		[ $psiprime, $lambda, $rho ];
	}

	return @{ $self->{_ECI_cache}{fixed}{geocentric} };
    }

    if (@args == 3) {
	my ($psiprime, $lambda, $rho) = @args;
	$psiprime = _check_latitude(latitude => $psiprime);
	$lambda = _check_longitude(longitude => $lambda);
	my $z = $rho * sin ($psiprime);
	my $r = $rho * cos ($psiprime);
	my $x = $r * cos ($lambda);
	my $y = $r * sin ($lambda);
	$self->get ('debug') and print <
Debug geocentric () - geocentric -> ecef
    inputs:
        psiprime = $psiprime
        lambda = $lambda
        rho = $rho
    outputs:
        x = $x
        y = $y
        z = $z
eod
	$self->ecef ($x, $y, $z);
	$self->{_ECI_cache}{fixed}{geocentric} = \@args;
	$self->{specified} = 'geocentric';
	$self->{inertial} = 0;
    } else {
	croak <
Error - Method geocentric() must be called with either zero arguments
        (to retrieve coordinates) or three arguments (to set
        coordinates). There is currently no six argument version.
eod
    }
    return $self;
}

=item $coord = $coord->geodetic($psi, $lambda, $h, $ellipsoid);

This method sets the L coordinates represented by the object
in terms of its L psi and L lambda in
radians, and its height h above mean sea level in kilometers.

The latitude should be a number between -PI/2 and PI/2 radians
inclusive.  The longitude should be a number between -2*PI and 2*PI
radians inclusive. The increased range (one would expect -PI to PI) is
because in some astronomical usages latitudes outside the range + or -
180 degrees are employed. A warning will be generated if either of these
range checks fails.

The ellipsoid argument is the name of a L known
to the class, and is optional. If passed, it will set the ellipsoid
to be used for calculations with this object. If not passed, the
default ellipsoid is used.

This method can also be called as a class method, in which case it
instantiates the desired object.

The conversion from geodetic to geocentric comes from Jean Meeus'
"Astronomical Algorithms", 2nd Edition, Chapter 11, page 82.

B

=item ($psi, $lambda, $h) = $coord->geodetic($ellipsoid);

This method returns the geodetic latitude, longitude, and height
above mean sea level.

The ellipsoid argument is the name of a L known
to the class, and is optional. If not specified, the most-recently-set
ellipsoid will be used.

The conversion from geocentric to geodetic comes from Kazimierz
Borkowski's "Accurate Algorithms to Transform Geocentric to Geodetic
Coordinates", at F.
This is best viewed with Internet Explorer because of its use of Microsoft's
Symbol font.

=cut

sub geodetic {
    my ($self, @args) = @_;

#	Detect and acquire the optional ellipsoid name argument. We do
#	this before the check, since the check expects the extra
#	argument to be a time.

    my $elps = (@args == 1 || @args == 4) ? pop @args : undef;

    $self = $self->_check_coord (geodetic => \@args, $elps ? $elps : ());

#	The following is just a sleazy way to get a consistent
#	error message if the ellipsoid name is unknown.

    $elps && $self->reference_ellipsoid ($elps);

#	If we're fetching the geodetic coordinates

    unless (@args) {

#	Return cached coordinates if they exist and we did not
#	override the default ellipsoid.

	return @{$self->{_ECI_cache}{fixed}{geodetic}}
	    if $self->{_ECI_cache}{fixed}{geodetic} && !$elps;
	$self->{debug} and do {
	    require Data::Dumper;
	    local $Data::Dumper::Terse = 1;
	    print "Debug geodetic - explicit ellipsoid ",
		Data::Dumper::Dumper( $elps );
	};

#	Get a reference to the ellipsoid data to use.

	$elps = $elps ? $known_ellipsoid{$elps} : $self;
	$self->{debug} and do {
	    require Data::Dumper;
	    local $Data::Dumper::Terse = 1;
	    print "Debug geodetic - ellipsoid ", Data::Dumper::Dumper( $elps );
	};

#	Calculate geodetic coordinates.

	my ($phiprime, $lambda, $rho) = $self->geocentric;
	my $r = $rho * cos ($phiprime);
	my $b = $elps->{semimajor} * (1- $elps->{flattening});
	my $a = $elps->{semimajor};
	my $z = $rho * sin ($phiprime);
	# The $b is _not_ a magic variable, since we are not inside
	# any of the specialized blocks that make $b magic. Perl::Critic
	# is simply confused.
	$b = - $b	## no critic (RequireLocalizedPunctuationVars)
	    if $z < 0;	# Per Borkowski, for southern hemisphere.

#	The following algorithm is due to Kazimierz Borkowski's
#	paper "Accurate Algorithms to Transform Geocentric to Geodetic
#	Coordinates", from
#	http://www.astro.uni.torun.pl/~kb/Papers/geod/Geod-BG.htm

	my $bz = $b * $z;
	my $asq_bsq = $a * $a - $b * $b;
	my $ar = $a * $r;
	my $E = ($bz - $asq_bsq) / $ar;		# Borkowski (10)
	my $F = ($bz + $asq_bsq) / $ar;		# Borkowski (11)
	my $Q = ($E * $E - $F * $F) * 2;	# Borkowski (17)
	my $P = ($E * $F + 1) * 4 / 3;		# Borkowski (16)
	my $D = $P * $P * $P + $Q * $Q;		# Borkowski (15)
	my $v = $D >= 0 ? do {
	    my $d = sqrt $D;
	    my $onethird = 1 / 3;
	    my $vp = ($d - $Q) ** $onethird -	# Borkowski (14a)
		($d + $Q) ** $onethird;
	    $vp * $vp >= abs ($P) ? $vp :
		- ($vp * $vp * $vp + 2 * $Q) /	# Borkowski (20)
		    (3 * $P);
	} : do {
	    my $p = - $P;
	    sqrt (cos (acos ($Q /			# Borkowski (14b)
		sqrt ($p * $p * $p)) / 3) * $p) * 2;
	};
	my $G = (sqrt ($E * $E + $v)		# Borkowski (13)
		+ $E) / 2;
	my $t = sqrt ($G * $G + ($F - $v * $G)	# Borkowski (12)
		/ (2 * $G - $E)) - $G;

	# Borkowski (18)
	# equivalent to atan (arg1)
	my $phi = do {
	    my $Y = $a * ( 1 - $t * $t );
	    my $X = 2 * $b * $t;
	    ( $Y, $X ) = ( -$Y, -$X ) if $X < 0;
	    atan2 $Y, $X;
	};

	my $h = ($r - $a * $t) * cos ($phi) +	# Borkowski (19)
	    ($z - $b) * sin ($phi);

#	End of Borkowski's algorthm.

#	Cache the results of the calculation if they were done using
#	the default ellipsoid.

	$self->{_ECI_cache}{fixed}{geodetic} = [$phi, $lambda, $h] unless $elps;

#	Return the results in any event.

	$self->get ('debug') and print <
Debug geodetic: geocentric to geodetic
    inputs:
        phiprime = $phiprime
        lambda = $lambda
        rho = $rho
    intermediates:
        z = $z
        r = $r
        E = $E
        F = $F
        P = $P
        Q = $Q
        D = $D
        v = $v
        G = $G
        t = $t
    outputs:
        phi = atan2 (a * (1 - t * t), 2 * b * t)
            = atan2 (@{[$a * (1 - $t * $t)]}, @{[2 * $b * $t]})
            = $phi (radians)
        h = (r - a * t) * cos (phi) + (z - b) * sin (phi)
          = @{[$r - $a * $t]} * cos (phi) + @{[$z - $b]} * sin (phi)
          = $h (kilometers)
eod
	return ($phi, $lambda, $h);
    }

#	If we're setting the geodetic coordinates.

    if (@args == 3) {

#	Set the ellipsoid for the object if one was specified.

	$self->set (ellipsoid => $elps) if $elps;

#	Calculate the geocentric data.

	my ($phi, $lambda, $h) = @args;
	$phi = _check_latitude(latitude => $phi);
	$lambda = _check_longitude(longitude => $lambda);
	my $bovera = 1 - $self->{flattening};

#	The following algorithm appears on page 82 of the second
#	edition of Jean Meeus' "Astronomical Algorithms."

	my $u = atan2 ($bovera * tan ($phi), 1);
	my $rhosinlatprime = $bovera * sin ($u) +
	    $h / $self->{semimajor} * sin ($phi);
	my $rhocoslatprime = cos ($u) +
	    $h / $self->{semimajor} * cos ($phi);
	my $phiprime = atan2 ($rhosinlatprime, $rhocoslatprime);
	my $rho = $self->{semimajor} * ($rhocoslatprime ?
	    $rhocoslatprime / cos ($phiprime) :
	    $rhosinlatprime / sin ($phiprime));

#	End of Meeus' algorithm.

#	Set the geocentric data as the coordinates.

	$self->geocentric ($phiprime, $lambda, $rho);


 #	Cache the geodetic coordinates.

	$self->{_ECI_cache}{fixed}{geodetic} = [$phi, $lambda, $h];
	$self->{specified} = 'geodetic';
	$self->{inertial} = 0;

#	Else if the number of coordinates is bogus, croak.

    } else {
	croak <
Error - Method geodetic() must be called with either zero arguments
        (to retrieve coordinates) or three arguments (to set
        coordinates). There is currently no six argument version.
eod
    }

#	Return the object, wherever it came from.

    return $self;

}

=item $value = $coord->get ($attrib);

This method returns the named attributes of the object. If called in
list context, you can give more than one attribute name, and it will
return all their values.

If called as a class method, it returns the current default values.

See L for a list of the attributes you can get.

=cut

{	# Begin local symbol block.

    my %accessor = (
	sun	=> sub {
##	    my ( $self, $name ) = @_;
	    my ( $self ) = @_;		# Name unused
	    $self->{sun}
		and return $self->{sun};
	    my $sun_class = $self->SUN_CLASS();
	    ( my $path = "$sun_class.pm" ) =~ s< :: >smxg;
	    require $path;
	    return( $self->{sun} ||= $sun_class->new() );
	},
    );

    sub get {
	my ($self, @args) = @_;
	ref $self or $self = \%static;
	my @rslt;
	foreach my $name (@args) {
	    exists $mutator{$name}
		or croak " Error - Attribute '$name' does not exist";
	    if ($accessor{$name}) {
		push @rslt, $accessor{$name}->($self, $name);
	    } else {
		push @rslt, $self->{$name};
	    }
	}
	return wantarray ? @rslt : $rslt[0];
    }

}	# End local symbol block

=item @coords = $coord->enu();

This method reports the coordinates of C<$coord> in the East-North-Up
coordinate system, as seen from the position stored in the C
attribute of C<$coord>.

Velocity conversions to C appear to me to be mostly sane. See
L, below, for details.

=cut

sub enu {				# East, North, Up
    my ( $self ) = @_;
    my @vector = $self->neu();
    @vector[ 0, 1 ] =  @vector[ 1, 0 ];	# Swap North and East
    @vector > 3				# If we have velocity,
	and @vector[ 3, 4 ] = @vector[ 4, 3 ];	# Ditto
    return @vector;
}

=item @coords = $coord->neu();

This method reports the coordinates of C<$coord2> in the North-East-Up
coordinate system, as seen from the position stored in the C
attribute of C<$coord>. This is a B coordinate system.

Velocity conversions to C appear to me to be mostly sane. See
L, below, for details.

=cut

sub neu {				# North, East, Up
    my ( $self ) = @_;
    my $station = $self->get( 'station' )
	or croak 'Station attribute required';
    return $station->_local_cartesian( $self );
}

=item @coords = $coord->heliocentric_ecliptic_cartesian()

This method returns the Heliocentric ecliptic Cartesian position of the
object. You can optionally pass time as an argument; this is equivalent
to

 @coords = $coord->universal( $time )
     ->heliocentric_ecliptic_cartesian();

At this time this object is not able to derive Heliocentric ecliptic
Cartesian coordinates from other coordinates (say, ECI).

=item $coord->heliocentric_ecliptic_cartesian( $X, $Y, $Z )

This method sets the object's position in Heliocentric ecliptic
Cartesian coordinates. Velocities may not be set at the moment, though
this is planned. You may also pass an optional time as the last
argument.

The invocant is returned.

=cut

sub heliocentric_ecliptic_cartesian {
    my ( $self, @args ) = @_;

    $self = $self->_check_coord( heliocentric_ecliptic_cartesian => \@args );

    if ( @args == 3 ) {
	$self->{_ECI_cache} = {
	    inertial => {
		heliocentric_ecliptic_cartesian => \@args,
	    },
	};
	my $sun = $self->get( 'sun' )
	    or croak q{Attribute 'sun' not set};
	my ( $X_sun, $Y_sun, $Z_sun ) = $sun->universal(
	    $self->universal() )->ecliptic_cartesian();
	my ( $X, $Y, $Z ) = ( $args[0] + $X_sun, $args[1] + $Y_sun,
	    $args[2] + $Z_sun );
	$self->ecliptic_cartesian( $X, $Y, $Z );
	$self->{specified} = 'heliocentric_ecliptic_cartesian';
	$self->{inertial} = 1;
    } elsif ( @args ) {
	croak 'heliocentric_ecliptic_cartesian() wants 0, 1, 3 or 4 arguments';
    } else {
	$self->{_ECI_cache}{inertial}{heliocentric_ecliptic_cartesian}
	    and return @{
		$self->{_ECI_cache}{inertial}{heliocentric_ecliptic_cartesian}
	    };
	my $sun = $self->get( 'sun' )
	    or croak q{Attribute 'sun' not set};
	my ( $X_self, $Y_self, $Z_self ) = $self->ecliptic_cartesian();
	my ( $X_sun, $Y_sun, $Z_sun ) = $sun->universal(
	    $self->universal() )->ecliptic_cartesian();

	my @hec = ( $X_self - $X_sun, $Y_self - $Y_sun, $Z_self - $Z_sun );

	$self->set( equinox_dynamical => $self->dynamical() );
	$self->{_ECI_cache}{inertial}{heliocentric_ecliptic_cartesian} = \@hec;
	return @hec;
    }
    return $self;
}

=item @coords = $coord->heliocentric_ecliptic();

This method returns the Heliocentric ecliptic coordinates of the object.

=cut

sub heliocentric_ecliptic {
    my ( $self, @args ) = @_;

    $self = $self->_check_coord( heliocentric_ecliptic => \@args );

    if ( @args ) {
	@args % 3
	    and croak 'Arguments must be in threes, plus an optional time';
	$self->{_ECI_cache}{inertial}{heliocentric_ecliptic} = \@args;
	$self->heliocentric_ecliptic_cartesian(
	    _convert_spherical_x_to_cartesian( @args ) );
	$self->{specified} = 'heliocentric_ecliptic';
	return $self;
    } else {
	return @{
	    $self->{_ECI_cache}{inertial}{heliocentric_ecliptic} ||= [
		_convert_cartesian_to_spherical_x(
		    $self->heliocentric_ecliptic_cartesian() ) ] }
    }
    return $self;
}

# ( $temp, $X, $Y, $Z, $Xprime, $Yprime, $Zprime ) =
#     $self->_local_cartesian( $trn2 );
# This method computes local Cartesian coordinates of $trn2 as seen from
# $self. The first return is intermediate results useful for the azimuth
# and elevation. The subsequent results are X, Y, and Z coordinates (and
# velocities if available), in the North, East, Up coordinate system.
# This is a left-handed coordinate system, but so is the azel() system,
# which it serves.

sub _local_cartesian {
    my ( $self, $trn2 ) = @_;
    $self->{debug} and do {
	require Data::Dumper;
	local $Data::Dumper::Terse = 1;
	print "Debug local_cartesian - ", Data::Dumper::Dumper( $self, $trn2 );
    };

    my $time = $trn2->universal();
    $self->universal( $time );

    # Pick up the ECEF of the body and the station

    my @tgt = $trn2->ecef();
    my @obs = $self->ecef();

    # Translate the ECEF coordinates to the station.

    my $limit = int( min( scalar @tgt, scalar @obs ) / 3 ) * 3 - 1;
    foreach my $inx ( 0 .. $limit ) {
	$tgt[$inx] -= $obs[$inx];
    }
    splice @tgt, $limit + 1;

    # Pick up the latitude and longitude.

    my ( $lat, $lon ) = $self->geodetic();
    my $sinlat = sin $lat;
    my $coslat = cos $lat;
    my $sinlon = sin $lon;
    my $coslon = cos $lon;

    # Rotate X->Y to longitude of station,
    # then Z->X to latitude of station

    # NOTE to Flat Earthers: For the next two statements to produce a
    # position in a local Cartesian system with the X-Y plane coincident
    # with the horizon, the Earth must be the oblate spheroid specified
    # by the currently-set reference elllipsoid.

    @tgt[ 0, 1 ] = (
	  $tgt[0] * $coslon + $tgt[1] * $sinlon,
	- $tgt[0] * $sinlon + $tgt[1] * $coslon,
    );
    @tgt[ 0, 2 ] = (
	$tgt[0] * $sinlat - $tgt[2] * $coslat,
	$tgt[0] * $coslat + $tgt[2] * $sinlat,
    );

    $tgt[0] = - $tgt[0];	# Convert Southing to Northing

    if ( @tgt > 5 ) {
	@tgt[ 3, 4 ] = (
	      $tgt[3] * $coslon + $tgt[4] * $sinlon,
	    - $tgt[3] * $sinlon + $tgt[4] * $coslon,
	);
	@tgt[ 3, 5 ] = (
	    $tgt[3] * $sinlat - $tgt[5] * $coslat,
	    $tgt[3] * $coslat + $tgt[5] * $sinlat,
	);

	$tgt[3] = - $tgt[3];	# Convert South velocity to North
    }

    return @tgt;
}

=item $coord = $coord->local_mean_time ($time);

This method sets the local mean time of the object. B
local standard time,> but the universal time plus the longitude
of the object expressed in seconds. Another definition is mean
solar time plus 12 hours (since the solar day begins at noon).
You will get an exception of some sort if the position of the
object has not been set, or if the object represents inertial
coordinates, or on any subclass whose position depends on the time.

=item $time = $coord->local_mean_time ()

This method returns the local mean time of the object. It will raise
an exception if the time has not been set.

Note that this returns the actual local time, not the GMT equivalent.
This means that in formatting for output, you call

 strftime $format, gmtime $coord->local_mean_time ();

=cut

sub local_mean_time {
    my ($self, @args) = @_;

    ref $self or croak <
Error - The local_mean_time() method may not be called as a class method.
eod

    unless (@args) {
	$self->{universal} || croak <
Error - Object's time has not been set.
eod
	$self->{local_mean_time} = $self->universal () +
	    _local_mean_delta ($self)
	    unless defined $self->{local_mean_time};
	return $self->{local_mean_time};
    }

    if (@args == 1) {
	my $time = shift @args;
	$self->{specified} or croak <
Error - Object's coordinates have not been set.
eod
	$self->{inertial} and croak <
Error - You can not set the local time of an object that represents
       a position in an inertial coordinate system, because this
       causes the earth-fixed position to change, invalidating the
       local time.
	$self->can ('time_set') and croak <
Error - You can not set the local time on an @{[ref $self]}
        object, because when you do the time_set() method will just
	move the object, invalidating the local time.
eod
	$self->universal($time - _local_mean_delta($self));
	$self->{local_mean_time} = $time;
    } else {
	croak <
Error - The local_mean_time() method must be called with either zero
        arguments (to retrieve the time) or one argument (to set
        the time).
eod
    }

    return $self;
}

=item $time = $coord->local_time ();

This method returns the local time (defined as solar time plus 12 hours)
of the given object. There is no way to set the local time.

This is really just a convenience routine - the calculation is simply
the local mean time plus the equation of time at the given time.

Note that this returns the actual local time, not the GMT equivalent.
This means that in formatting for output, you call

 strftime $format, gmtime $coord->local_time ();

=cut

sub local_time {
    my $self = shift;
    return $self->local_mean_time() + $self->equation_of_time();
}

=item ( $maidenhead_loc, $height ) = $coord->maidenhead( $precision );

This method returns the location of the object in the Maidenhead Locator
System.  Height above the reference ellipsoid is not part of the system,
but is returned anyway, in kilometers. The $precision is optional, and
is an integer greater than 0.

The default precision is 4, but this can be modified by setting
C<$Astro::Coord::ECI::DEFAULT_MAIDENHEAD_PRECISION> to the desired
value. For example, if you want the default precision to be 3 (which it
probably should have been in the first place), you can use

 no warnings qw{ once };
 local $Astro::Coord::ECI::DEFAULT_MAIDENHEAD_PRECISION = 3;

Note that precisions greater than 4 are not defined by the standard.
This method extends the system by alternating letters (base 24) with
digits (base 10), but this is unsupported since the results will change,
possibly without notice, if the standard is extended in a manner
incompatible with this implementation.

Conversion of latitudes and longitudes to Maidenhead Grid is subject to
truncation error, perhaps more so since latitude and longitude are
specified in radians. An attempt has been made to minimize this by using
Perl's stringification of numbers to ensure that something that looks
like C<42> is not handled as C<41.999999999385>. This probably amounts
to shifting some grid squares very slightly to the north-west, but in
practice it seems to give better results for points on the boundaries of
the grid squares.

=item $coord->maidenhead( $maidenhead_loc, $height );

This method sets the geodetic location in the Maidenhead Locator System.
Height above the reference ellipsoid is not part of the system, but is
accepted anyway, in kilometers, defaulting to 0.

The actual location set is the center of the specified grid square.

Locations longer than 8 characters are not defined by the standard. This
method extends precision by assuming alternate letters (base 24) and
digits (base 10), but this will change, possibly without notice, if the
standard is extended in a manner incompatible with this implementation.

The object itself is returned, to allow call chaining.

=cut
{

    our $DEFAULT_MAIDENHEAD_PRECISION = 4;

    my $alpha = 'abcdefghijklmnopqrstuvwxyz';

    sub maidenhead {
	my ( $self, @args ) = @_;
	if ( @args > 1 || defined $args[0] && $args[0] =~ m/ [^0-9] /smx ) {
	    my ( $loc, $alt ) = @args;
	    defined $alt or $alt = 0;

	    my $precision = length $loc;
	    $precision % 2
		and croak "Invalid Maidenhead locator; length must be even";
	    $precision /= 2;

	    my $lat = 0.5;
	    my $lon = 0.5;
	    my @chars = split qr{}smx, $loc;
	    foreach my $base ( reverse _maidenhead_bases( $precision ) ) {
		foreach ( $lat, $lon ) {
		    my $chr = pop @chars;
		    if ( $base > 10 ) {
			( my $inx = index $alpha, lc $chr ) < 0
			    and croak 'Invalid Maidenhead locator; ',
				"'$chr' is not a letter";
			my $limit = @chars > 1 ? $base - 1 : $base;
			$inx > $limit
			    and croak 'Invalid Maidenhead locator; ',
				"'$chr' is greater than '",
				substr( $alpha, $limit, 1 ), "'";
			$_ += $inx;
			$_ /= $base;
		    } else {
			$chr =~ m/ [^0-9] /smx
			    and croak 'Invalid Maidenhead locator; ',
				"'$chr' is not a digit";
			$_ += $chr;
			$_ /= $base;
		    }
		}
	    }

	    $self->geodetic(
		deg2rad( $lat * 180 - 90 ),
		deg2rad( $lon * 360 - 180 ),
		$alt
	    );
	    return $self;

	} else {
	    my ( $precision ) = @args;
	    defined $precision
		and $precision > 0
		or $precision = $DEFAULT_MAIDENHEAD_PRECISION;

	    my @bases = reverse _maidenhead_bases( $precision );

	    # In order to minimize truncation errors we multiply
	    # everything out, and then work right-to-left using the mod
	    # operator. We stringify to take advantage of Perl's smarts
	    # about when something is 42 versus 41.999999999583.

	    my $multiplier = 1;
	    foreach ( @bases ) {
		$multiplier *= $_;
	    }

	    my ( $lat, $lon, $alt ) = $self->geodetic();
	    $lat = ( $lat + PIOVER2 ) * $multiplier / PI;
	    $lon = ( $lon + PI ) * $multiplier / TWOPI;
	    $lat = floor( "$lat" );
	    $lon = floor( "$lon" );

	    my @rslt;
	    foreach my $base ( @bases ) {
		foreach ( $lat, $lon ) {
		    my $inx = $_ % $base;
		    push @rslt, $base > 10 ?
			substr( $alpha, $inx, 1 ) : $inx;
		    $_ = floor( $_ / $base );
		}
	    }

	    @rslt
		and $rslt[-1] = uc $rslt[-1];
	    @rslt > 1
		and $rslt[-2] = uc $rslt[-2];

	    return (
		join( '', reverse @rslt ),
		$alt,
	    );
	}
    }

    sub _maidenhead_bases {
	my ( $precision ) = @_;
	my @bases;
	foreach my $inx ( 1 .. $precision ) {
	    push @bases, $inx % 2 ? 24 : 10;
	}
	$bases[0] = 18;
	return @bases;
    }
}

=item $value = $coord->mean_angular_velocity();

This method returns the mean angular velocity of the body in radians
per second. If the $coord object has a period() method, this method
just returns two pi divided by the period. Otherwise it returns the
contents of the angularvelocity attribute.

=cut

sub mean_angular_velocity {
    my $self = shift;
    return $self->can ('period') ?
	TWOPI / $self->period :
	$self->{angularvelocity};
}

=item $time = $coord->next_azimuth( $body, $azimuth );

This method returns the next time the given C<$body> passes the given
C<$azimuth> as seen from the given C<$coord>, calculated to the nearest
second. The start time is the current time setting of the C<$body>
object.

=item $time = $coord->next_azimuth( $azimuth );

This method returns the next time the C<$coord> object passes the given
C<$azimuth> as seen from the location in the C<$coord> object's
C attribute, calculated to the nearest second. The start time
is the current time setting of the C<$coord> object.

=cut

sub next_azimuth {
    my ( $self, $body, $azimuth ) = _expand_args_default_station( @_ );
    ref $self or croak <<'EOD';
Error - The next_azimuth() method may not be called as a class method.
EOD

    $body->represents(__PACKAGE__) or croak <<"EOD";
Error - The argument to next_azimuth() must be a subclass of
        @{[__PACKAGE__]}.
EOD

    my $want = shift;
    defined $want and $want = $want ? 1 : 0;

    my $denom = $body->mean_angular_velocity -
	$self->mean_angular_velocity;
##  my $retro = $denom >= 0 ? 0 : 1;
    ($denom = abs ($denom)) < 1e-11 and croak <
Error - The next_azimuth() method will not work for geosynchronous
        bodies.
eod

    my $apparent = TWOPI / $denom;
    my $begin = $self->universal;
    my $delta = floor( $apparent / 8 );
    my $end = $begin + $delta;

    my $begin_angle = mod2pi(
	( $self->azel( $body->universal( $begin ) ) )[0] - $azimuth );
    my $end_angle = mod2pi(
	( $self->azel( $body->universal( $end ) ) )[0] - $azimuth );
    while ( $begin_angle < PI || $end_angle >= PI ) {
	$begin_angle = $end_angle;
	$begin = $end;
	$end = $end + $delta;
	$end_angle = mod2pi(
	    ( $self->azel( $body->universal( $end ) ) )[0] - $azimuth );
    }

    while ($end - $begin > 1) {
	my $mid = floor (($begin + $end) / 2);
	my $mid_angle = mod2pi(
	    ( $self->azel( $body->universal( $mid ) ) )[0] - $azimuth );
	( $begin, $end ) = ( $mid_angle >= PI ) ?
	    ( $mid, $end ) : ( $begin, $mid );
    }

    $body->universal ($end);
    $self->universal ($end);
    return $end;
}

=item ($time, $rise) = $coord->next_elevation ($body, $elev, $upper)

This method calculates the next time the given body passes above or
below the given elevation (in radians) as seen from C<$coord>. The
C<$elev> argument may be omitted (or passed as undef), and will default
to 0. If the C<$upper> argument is true, the calculation will be based
on the upper limb of the body (as determined from its C
attribute); if false, the calculation will be based on the center of the
body. The C<$upper> argument defaults to true if the C<$elev> argument
is zero or positive, and false if the C<$elev> argument is negative.

In list context, it returns the time, and an indicator which is true if
the body passes above the given elevation at that time, and false if it
passes below the given elevation. In scalar context it just returns the
time.

The algorithm is successive approximation, and assumes that the
body will be at its highest at meridian passage. It also assumes
that if the body hasn't passed the given elevation in 183 days it
never will. In this case it returns undef in scalar context, or
an empty list in list context.

=item ($time, $rise) = $coord->next_elevation( $elev, $upper );

This method calculates the next time the C<$coord> object passes above
or below the given elevation (in radians) as seen from the position
found in the C<$coord> object's C attribute. The C<$elev>
argument may be omitted (or passed as undef), and will default to 0. If
the C<$upper> argument is true, the calculation will be based on the
upper limb of the body (as determined from its C
attribute); if false, the calculation will be based on the center of the
body. The C<$upper> argument defaults to true if the C<$elev> argument
is zero or positive, and false if the C<$elev> argument is negative.

In list context, it returns the time, and an indicator which is true if
the body passes above the given elevation at that time, and false if it
passes below the given elevation. In scalar context it just returns the
time.

The algorithm is successive approximation, and assumes that the
body will be at its highest at meridian passage. It also assumes
that if the body hasn't passed the given elevation in 183 days it
never will. In this case it returns undef in scalar context, or
an empty list in list context.

=cut

use constant NEVER_PASS_ELEV => 183 * SECSPERDAY;

sub next_elevation {
    my ( $self, $body, $angle, $upper ) = _expand_args_default_station( @_ );
    ref $self or croak <<'EOD';
Error - The next_elevation() method may not be called as a class method.
EOD

    $body->represents(__PACKAGE__) or croak <<"EOD";
Error - The first argument to next_elevation() must be a subclass of
        @{[__PACKAGE__]}.
EOD

    $angle ||= 0;
    defined $upper or $upper = $angle >= 0;
    $upper = $upper ? 1 : 0;

    my $begin = $self->universal;
    my $original = $begin;
    my ( undef, $elev ) = $self->azel_offset(
	$body->universal( $begin ), $upper );
    my $rise = ( $elev < $angle ) || 0;

    my ($end, $above) = $self->next_meridian ($body, $rise);

    my $give_up = $body->NEVER_PASS_ELEV ();

    while ( 1 ) {
	my ( undef, $elev ) = $self->azel_offset( $body, $upper );
	( ( $elev < $angle ) || 0 ) == $rise
	    or last;
	return if $end - $original > $give_up;
	$begin = $end;
	($end, $above) = $self->next_meridian ($body, $rise);
    }

    while ($end - $begin > 1) {
	my $mid = floor (($begin + $end) / 2);
	my ( undef, $elev ) = $self->universal( $mid )->
	    azel_offset( $body->universal( $mid ), $upper );
	( $begin, $end ) =
	    ($elev < $angle || 0) == $rise ? ($mid, $end) : ($begin, $mid);
    }

    $self->universal ($end);	# Ensure consistent time.
    $body->universal ($end);
    return wantarray ? ($end, $rise) : $end;
}

=item ( $time, $type ) = $coord->next_elevation_extreme_tod( $body, $elev, $ipper );

B

This variation on C returns the earliest (or latest)
time the given body passes above (or below) the given elevation (in
radians) as seen from the given coordinates. If called in list context
it returns the time of the event and the type of the event, encoded as
follows:

 0 - earliest pass below the given elevation
 1 - earliest pass above the given elevation
 2 - latest pass below the given elevation
 3 - latest pass above the given elevation

If called in scalar context this method just returns the time.

If no extreme can be computed, this method returns nothing. It will
always return nothing unless otherwise documented for the class of the
C<$body> object.

The motivation for this method was a desire to compute the earliest and
latest sunrise and sunset at a given location. It appears here in the
object hierarchy out of a perhaps-foolish desire to be consistent with
C.

Use this method with caution. I have serious doubts about what will
happen when the invocant is north of the Arctic Circle or south of the
Antarctic Circle, and I am a little unsure of its behaviour near the
Equator. In addition, in an early implementation consecutive calls
returned the same extreme over and over. I believe this to be fixed, but
the fix raises the possibility of skipped extrema. B

=item ( $time, $type ) = $body->next_elevation_extreme_tod( $elev, $ipper );

This way to call C is equivalent to the
above, but uses the body's C attribute for the C<$coord>
object.

=cut

sub next_elevation_extreme_tod {
    my ( $self, $body, $angle, $upper ) = _expand_args_default_station( @_ );

    ref $self or croak <<'EOD';
Error - The next_elevation_extreme_tod() method may not be called as a
        class method.
EOD

    $body->represents(__PACKAGE__) or croak <<"EOD";
Error - The first argument to next_elevation_extreme_tod() must be a
        subclass of @{[__PACKAGE__]}.
EOD

    return $body->__next_elevation_extreme_tod( $self, $angle, $upper );
}

sub __next_elevation_extreme_tod {
    return;
}

=item ( $time, $above ) = $coord->next_meridian( $body, $want )

This method calculates the next meridian passage of the given C<$body>
over (or under) the location specified by the C<$coord> object. The
C<$body> object must be a subclass of Astro::Coord::ECI.

The optional C<$want> argument should be specified as true (e.g. 1) if
you want the next passage above the observer, or as false (e.g. 0) if
you want the next passage below the observer. If this argument is
omitted or undefined, you get whichever passage is next.

The start time of the search is the current time setting of the
C<$coord> object.

The returns are the time of the meridian passage, and an indicator which
is true if the passage is above the observer (i.e. local noon if the
C<$body> represents the Sun), or false if below (i.e. local midnight if
the C<$body> represents the Sun). If called in scalar context, you get
the time only.

The current time of both C<$coord> and C<$body> objects are left at the
returned time.

The algorithm is by successive approximation. It will croak if the
period of the C<$body> is close to synchronous, and will probably not
work well for bodies in highly eccentric orbits. The calculation is to
the nearest second, and the time returned is the first even second after
the body crosses the meridian.

=item ( $time, $above ) = $coord->next_meridian( $want )

This method calculates the next meridian passage of the C<$coord> object
over (or under) the location specified by the C<$coord> object's
C attribute.

The optional C<$want> argument should be specified as true (e.g. 1) if
you want the next passage above the observer, or as false (e.g. 0) if
you want the next passage below the observer. If this argument is
omitted or undefined, you get whichever passage is next.

The start time of the search is the current time setting of the
C<$coord> object.

The returns are the time of the meridian passage, and an indicator which
is true if the passage is above the observer (i.e. local noon if the
C<$coord> object represents the Sun), or false if below (i.e. local
midnight if the C<$coord> object represents the Sun). If called in
scalar context, you get the time only.

The current time of both C<$coord> and its C are left at the
returned time.

The algorithm is by successive approximation. It will croak if the
period of the C<$body> is close to synchronous, and will probably not
work well for bodies in highly eccentric orbits. The calculation is to
the nearest second, and the time returned is the first even second after
the body crosses the meridian.

=cut

sub next_meridian {
    my ( $self, $body, $want ) = _expand_args_default_station( @_ );
    ref $self or croak <<'EOD';
Error - The next_meridian() method may not be called as a class method.
EOD

    $body->represents(__PACKAGE__) or croak <<"EOD";
Error - The argument to next_meridian() must be a subclass of
        @{[__PACKAGE__]}.
EOD

    defined $want and $want = $want ? 1 : 0;

    my $denom = $body->mean_angular_velocity -
	$self->mean_angular_velocity;
    my $retro = $denom >= 0 ? 0 : 1;
    ($denom = abs ($denom)) < 1e-11 and croak <<'EOD';
Error - The next_meridian() method will not work for geosynchronous
        bodies.
EOD

    my $apparent = TWOPI / $denom;
    my $begin = $self->universal;
    my $delta = floor ($apparent / 16);
    my $end = $begin + $delta;

    my ($above, $opposite) =
	mod2pi (($body->universal($begin)->geocentric)[1]
	    - ($self->universal($begin)->geocentric)[1]) >= PI ?
	(1 - $retro, PI) : ($retro, 0);

    ($begin, $end) = ($end, $end + $delta)
	while mod2pi (($body->universal($end)->geocentric)[1] -
	    ($self->universal($end)->geocentric)[1] + $opposite) < PI;

    if (defined $want && $want != $above) {
	$above = $want;
	$opposite = $opposite ? 0 : PI;
	($begin, $end) = ($end, $end + $delta)
	    while mod2pi (($body->universal($end)->geocentric)[1] -
		($self->universal($end)->geocentric)[1] + $opposite) < PI;
    }

    while ($end - $begin > 1) {
	my $mid = floor (($begin + $end) / 2);
	my $long = ($body->universal($mid)->geocentric)[1];
	my $merid = ($self->universal($mid)->geocentric)[1];
	($begin, $end) =
	    mod2pi ($long - $merid + $opposite) < PI ?
	    ($mid, $end) : ($begin, $mid);
    }

    $body->universal ($end);
    $self->universal ($end);
    return wantarray ? ($end, $above) : $end;
}

=item ( $delta_psi, $delta_epsilon ) = $self->nutation( $time )

This method calculates the nutation in longitude (delta psi) and
obliquity (delta epsilon) for the given B time. If the time
is unspecified or specified as C, the current B time
of the object is used.

The algorithm comes from Jean Meeus' "Astronomical Algorithms", 2nd
Edition, Chapter 22, pages 143ff. Meeus states that it is good to
0.5 seconds of arc.

=cut

sub nutation {
    my ( $self, $time ) = @_;
    defined $time
	or $time = $self->dynamical();

    my $T = jcent2000( $time );	# Meeus (22.1)

    my $omega = mod2pi( deg2rad( ( ( $T / 450000 + .0020708 ) * $T -
	    1934.136261 ) * $T + 125.04452 ) );

    my $L = mod2pi( deg2rad( 36000.7698 * $T + 280.4665 ) );
    my $Lprime = mod2pi( deg2rad( 481267.8813 * $T + 218.3165 ) );

    my $delta_psi = deg2rad( ( -17.20 * sin( $omega )
	    - 1.32 * sin( 2 * $L )
	    - 0.23 * sin( 2 * $Lprime )
	    + 0.21 * sin( 2 * $omega ) ) / 3600 );
    my $delta_epsilon = deg2rad( ( 9.20 * cos( $omega )
	    + 0.57 * cos( 2 * $L )
	    + 0.10 * cos( 2 * $Lprime )
	    - 0.09 * cos( 2 * $omega ) ) / 3600 );

    return ( $delta_psi, $delta_epsilon );
}

=item $epsilon = $self->obliquity( $time )

This method calculates the obliquity of the ecliptic in radians at the
given B time. If the time is unspecified or specified as
C, the current B time of the object is used.

The algorithm comes from Jean Meeus' "Astronomical Algorithms", 2nd
Edition, Chapter 22, pages 143ff. The conversion from universal to
dynamical time comes from chapter 10, equation 10.2  on page 78.

=cut

use constant E0BASE => (21.446 / 60 + 26) / 60 + 23;

sub obliquity {
    my ( $self, $time ) = @_;

    if ( looks_like_number( $self ) ) {
	( $self, $time ) = ( __PACKAGE__, $self );
    }
    defined $time
	or $time = $self->dynamical();

    my $T = jcent2000 ($time);	# Meeus (22.1)

    my ( undef, $delta_epsilon ) = $self->nutation( $time );

    my $epsilon0 = deg2rad( ( ( 0.001813 * $T - 0.00059 ) * $T - 46.8150 )
	    * $T / 3600 + E0BASE);
    return $epsilon0 + $delta_epsilon;
}

=item $coord = $coord->precess ($time);

This method is a convenience wrapper for precess_dynamical(). The
functionality is the same except that B
universal time.>

=cut

sub precess {
    $_[1]
	and $_[1] += dynamical_delta( $_[1] );
    goto &precess_dynamical;
}

=item $coord = $coord->precess_dynamical ($time);

This method precesses the coordinates of the object to the given
equinox, B The starting equinox is the
value of the 'equinox_dynamical' attribute, or the current time setting
if the 'equinox_dynamical' attribute is any false value (i.e. undef, 0,
or ''). A warning will be issued if the value of 'equinox_dynamical' is
undef (which is the default setting) and the object represents inertial
coordinates. As of version 0.013_02, B
unaffected by this operation.>

As a side effect, the value of the 'equinox_dynamical' attribute will be
set to the dynamical time corresponding to the argument.

As of version 0.061_01, this does nothing to non-inertial
objects -- that is, those whose position was set in Earth-fixed
coordinates.

If the object's 'station' attribute is set, the station is also
precessed.

The object itself is returned.

The algorithm comes from Jean Meeus, "Astronomical Algorithms", 2nd
Edition, Chapter 21, pages 134ff (a.k.a. "the rigorous method").

=cut

sub precess_dynamical {
    my ( $self, $end ) = @_;

    $end
	or croak "No equinox time specified";

    # Non-inertial coordinate systems are not referred to the equinox,
    # and so do not get precessed.
    $self->get( 'inertial' )
	or return $self;

    defined ( my $start = $self->get( 'equinox_dynamical' ) )
	or carp "Warning - Precess called with equinox_dynamical ",
	    "attribute undefined";
    $start ||= $self->dynamical ();

    my $sta;
    if ( $sta = $self->get( 'station' ) and $sta->get( 'inertial' )
    ) {
	$sta->get( 'station' )
	    and croak NO_CASCADING_STATIONS;
	$sta->universal( $self->universal() );
	$sta->precess_dynamical( $end );
    }

    $start == $end
	and return $self;

    my ($alpha0, $delta0, $rho0) = $self->equatorial ();

    my $T = jcent2000($start);
    my $t = ($end - $start) / (36525 * SECSPERDAY);

    #	The following four assignments correspond to Meeus' (21.2).
    my $zterm = (- 0.000139 * $T + 1.39656) * $T + 2306.2181;
    my $zeta = deg2rad((((0.017998 * $t + (- 0.000344 * $T + 0.30188)) *
	    $t + $zterm) * $t) / 3600);
    my $z = deg2rad((((0.018203 * $t + (0.000066 * $T + 1.09468)) * $t +
	    $zterm) * $t) / 3600);
    my $theta = deg2rad(((( - 0.041833 * $t - (0.000217 * $T + 0.42665))
	    * $t + (- 0.000217 * $T - 0.85330) * $T + 2004.3109) * $t) /
	    3600);

    #	The following assignments correspond to Meeus' (21.4).
    my $sindelta0 = sin ($delta0);
    my $cosdelta0 = cos ($delta0);
    my $sintheta = sin ($theta);
    my $costheta = cos ($theta);
    my $cosdelta0cosalpha0 = cos ($alpha0 + $zeta) * $cosdelta0;
    my $A = $cosdelta0 * sin ($alpha0 + $zeta);
    my $B = $costheta * $cosdelta0cosalpha0 - $sintheta * $sindelta0;
    my $C = $sintheta * $cosdelta0cosalpha0 + $costheta * $sindelta0;

    my $alpha = mod2pi(atan2 ($A , $B) + $z);
    my $delta = asin($C);

    $self->equatorial ($alpha, $delta, $rho0);
    $self->set (equinox_dynamical => $end);

    return $self;
}

=item Astro::Coord::ECI->reference_ellipsoid($semi, $flat, $name);

This class method can be used to define or redefine reference
ellipsoids.

Nothing bad will happen if you call this as an object method, but
it still just creates a reference ellipsoid definition -- the object
is unaffected.

It is not an error to redefine an existing ellipsoid.

=item ($semi, $flat, $name) = Astro::Coord::ECI->reference_ellipsoid($name)

This class method returns the definition of the named reference
ellipsoid. It croaks if there is no such ellipsoid.

You can also call this as an object method, but the functionality is
the same.

The following reference ellipsoids are known to the class initially:

 CLARKE-1866 - 1866.
   semimajor => 6378.2064 km, flattening => 1/294.98.
   Source: http://www.colorado.edu/geography/gcraft/notes/datum/elist.html

 GRS67 - Geodetic Reference System 1967.
   semimajor => 6378.160 km, flattening => 1/298.247.
   Source: http://www.colorado.edu/geography/gcraft/notes/datum/elist.html

 GRS80 - Geodetic Reference System 1980.
   semimajor => 6378.137 km, flattening => 1/298.25722210088
     (flattening per U.S. Department of Commerce 1989).
   Source: http://biology.usgs.gov/fgdc.metadata/version2/spref/horiz/geodet/faq.htm

 IAU68 - International Astronomical Union, 1968.
   semimajor => 6378.160 km, flattening => 1/298.25.
   Source: http://maic.jmu.edu/sic/standards/ellipsoid.htm

 IAU76 - International Astronomical Union, 1976.
   semimajor => 6378.14 km, flattening => 1 / 298.257.
   Source: Jean Meeus, "Astronomical Algorithms", 2nd Edition

 NAD83 - North American Datum, 1983.
    semimajor => 6378.137 km, flattening => 1/298.257.
    Source:  http://biology.usgs.gov/fgdc.metadata/version2/spref/horiz/geodet/faq.htm
    (NAD83 uses the GRS80 ellipsoid)

 sphere - Just in case you were wondering how much difference it
   makes (a max of 11 minutes 32.73 seconds of arc, per Jean
   Meeus).
   semimajor => 6378.137 km (from GRS80), flattening => 0.

 WGS72 - World Geodetic System 1972.
   semimajor => 6378.135 km, flattening=> 1/298.26.
   Source: http://biology.usgs.gov/fgdc.metadata/version2/spref/horiz/geodet/faq.htm

 WGS84 - World Geodetic System 1984.
   semimajor => 6378.137 km, flattening => 1/1/298.257223563.
   Source: http://www.colorado.edu/geography/gcraft/notes/datum/elist.html

Reference ellipsoid names are case-sensitive.

The default model is WGS84.

=cut

# Wish I had:
# Maling, D.H., 1989, Measurements from Maps: Principles and methods of cartometry, Pergamon Press, Oxford, England.
# Maling, D.H., 1993, Coordinate Systems and Map Projections, Pergamon Press, Oxford, England.

# http://www.gsi.go.jp/PCGIAP/95wg/wg3/geodinf.htm has a partial list of who uses
#	what in the Asia/Pacific.

%known_ellipsoid = (	# Reference Ellipsoids
    'CLARKE-1866' => {	# http://www.colorado.edu/geography/gcraft/notes/datum/elist.html
	semimajor => 6378.2064,
	flattening => 1/294.9786982,
    },
    GRS67 => {		# http://www.colorado.edu/geography/gcraft/notes/datum/elist.html
	semimajor => 6378.160,
	flattening => 1/298.247167427,
    },
    GRS80 => {		# http://biology.usgs.gov/fgdc.metadata/version2/spref/horiz/geodet/faq.htm
	semimajor => 6378.137,		# km
	flattening => 1/298.25722210088, # U.S. Dept of Commerce 1989 (else 1/298.26)
    },
    IAU68 => {		# http://maic.jmu.edu/sic/standards/ellipsoid.htm
	semimajor => 6378.160,
	flattening => 1/298.25,
    },
    IAU76 => {		# Meeus, p. 82.
	semimajor => 6378.14,
	flattening => 1/298.257,
    },
    NAD83 => {		# http://biology.usgs.gov/fgdc.metadata/version2/spref/horiz/geodet/faq.htm
	semimajor => 6378.137,		# km
	flattening => 1/298.257,
    },
    sphere => {		# Defined by me for grins, with semimajor from GRS80.
	semimajor => 6378.137,		# km, from GRS80
	flattening => 0,		# It's a sphere!
    },
    WGS72 => {		# http://biology.usgs.gov/fgdc.metadata/version2/spref/horiz/geodet/faq.htm
	semimajor => 6378.135,		# km
	flattening => 1/298.26,
    },
    WGS84 => {		# http://www.colorado.edu/geography/gcraft/notes/datum/elist.html
	semimajor => 6378.137,
	flattening => 1/298.257223563,
    },
);
foreach my $name (keys %known_ellipsoid) {
    $known_ellipsoid{$name}{name} = $name;
}

sub reference_ellipsoid {
    my ( undef, @args ) = @_;	# Invocant unused
    my $name = pop @args or croak <
Error - You must specify the name of a reference ellipsoid.
eod
    if (@args == 0) {
	$known_ellipsoid{$name} or croak <
Error - Reference ellipsoid $name is unknown to this software. Known
        ellipsoids are:
        @{[join ', ', sort keys %known_ellipsoid]}.
eod
    } elsif (@args == 2) {
	$known_ellipsoid{$name} = {
	    semimajor => $args[0],
	    flattening => $args[1],
	    name => $name,
	};
    } else {
	croak <
Error - You must call the reference_ellipsoid class method with either one
        argument (to fetch the definition of a known ellipsoid) or three
        arguments (to define a new ellipsoid or redefine an old one).
eod
    }
    return @{$known_ellipsoid{$name}}{qw{semimajor flattening name}}
}

=item $coord->represents ($class);

This method returns true if the $coord object represents the given class.
It is pretty much like isa (), but if called on a container class (i.e.
Astro::Coord::ECI::TLE::Set), it returns true based on the class of
the members of the set, and dies if the set has no members.

The $class argument is optional. If not specified (or undef), it is
pretty much like ref $coord || $coord (i.e. it returns the class
name), but with the delegation behavior described in the previous
paragraph if the $coord object is a container.

There. This took many more words to explain than it did to implement.

=cut

sub represents {
    return defined ($_[1]) ?
##	$_[0]->represents()->isa($_[1]) :
	__classisa($_[0]->represents(), $_[1]) ? 1 : 0 :
	(ref $_[0] || $_[0]);
}

=item $coord->set (name => value ...);

This method sets various attributes of the object. If called as a class
method, it changes the defaults.

For reasons that seemed good at the time, this method returns the
object it was called on (i.e. $coord in the above example).

See L for a list of the attributes you can set.

=cut

use constant SET_ACTION_NONE => 0;	# Do nothing.
use constant SET_ACTION_RESET => 1;	# Reset the coordinates based on the initial setting.

sub set {
    my ($self, @args) = @_;
    my $original = $self;
    ref $self
	or $self = \%static;
    @args % 2
	and croak 'Error - The set() method requires an even ',
	    'number of arguments.';
    my $action = 0;
    while (@args) {
	my $name = shift @args;
	exists $mutator{$name}
	    or croak "Error - Attribute '$name' does not exist.";
	CODE_REF eq ref $mutator{$name}
	    or croak "Error - Attribute '$name' is read-only";
	$action |= $mutator{$name}->($self, $name, shift @args);
    }

    $self->{_need_purge} = 1
	if ref $self && $self->{specified} && $action & SET_ACTION_RESET;

    return $original;
}

#	The following are the mutators for the attributes. All are
#	passed three arguments: a reference to the hash to be set,
#	the hash key to be set, and the value. They must return the
#	bitwise 'or' of the desired action masks, defined above.

%mutator = (
    almanac_horizon	=> \&_set_almanac_horizon,
    angularvelocity => \&_set_value,
    debug => \&_set_value,
    diameter => \&_set_value,
    edge_of_earths_shadow => \&_set_value,
    ellipsoid => \&_set_reference_ellipsoid,
    equinox_dynamical => \&_set_value,	# CAVEAT: _convert_eci_to_ecef
					# accesses this directly for
					# speed.
    flattening => \&_set_custom_ellipsoid,
    frequency => \&_set_value,
    horizon => \&_set_elevation,
    id => \&_set_id,
    inertial => undef,
    name => \&_set_value,
    refraction => \&_set_value,
    specified => undef,
    semimajor => \&_set_custom_ellipsoid,
    station => \&_set_station,
    sun	=> \&_set_sun,
    twilight => \&_set_elevation,
);

{
    # TODO this will all be nicer if I had state variables.

    my %special = (
##	horizon	=> sub { return $_[0]->get( 'horizon' ); },
	height	=> sub { return $_[0]->dip(); },
    );

    sub _set_almanac_horizon {
	my ( $hash, $attr, $value ) = @_;
	defined $value
	    or $value = 0;	# Default
	if ( $special{$value} ) {
	    $hash->{"_$attr"} = $special{$value};
	} elsif ( looks_like_number( $value )
	    && $value >= - PIOVER2 # Not -PIOVER2 to avoid warning under 5.10.1.
	    && $value <= PIOVER2
	) {
	    $hash->{"_$attr"} = sub { return $_[0]->get( $attr ) };
	} else {
	    croak "'$value' is an invalid value for '$attr'";
	}
	$hash->{$attr} = $value;
	return SET_ACTION_NONE;
    }
}

#	Get the actual value of the almanac horizon, from whatever
#	source.

sub __get_almanac_horizon {
    goto $_[0]{_almanac_horizon};
}

#	If you set semimajor or flattening, the ellipsoid name becomes
#	undefined. Also clear any cached geodetic coordinates.

sub _set_custom_ellipsoid {
    $_[0]->{ellipsoid} = undef;
    $_[0]->{$_[1]} = $_[2];
    return SET_ACTION_RESET;
}

{
    my %dflt = (
	twilight	=> CIVIL_TWILIGHT,
    );

    # Any attribute specified as angle above or below the horizon.
    sub _set_elevation {
	my ( $self, $name, $value ) = @_;
	defined $value
	    or $value = ( $dflt{$name} || 0 );	# Default
	looks_like_number( $value )
	    and $value >= - PIOVER2 # Not -PIOVER2 to avoid warning under 5.10.1.
	    and $value <= PIOVER2
	    or croak "'$value' is an invalid value for '$name'";
	$self->{$name} = $value;
	return SET_ACTION_NONE;
    }
}

sub _set_station {
    my ( $hash, $attr, $value ) = @_;
    if ( defined $value ) {
	embodies( $value, 'Astro::Coord::ECI' )
	    or croak "Attribute $attr must be undef or an ",
		'Astro::Coord::ECI object';
	$value->get( 'station' )
	    and croak NO_CASCADING_STATIONS;
    }
    $hash->{$attr} = $value;
    return SET_ACTION_RESET;
}

use constant SUN_CLASS => 'Astro::Coord::ECI::Sun';

sub _set_sun {
    my ( $self, $name, $value ) = @_;
    embodies( $value, $self->SUN_CLASS() )
	or croak sprintf 'The sun attribute must be a %s',
	    $self->SUN_CLASS();
    ref $value
	or $value = $value->new();
    $self->{$name} = $value;
    return SET_ACTION_NONE;
}

#	Unfortunately, the TLE subclass may need objects reblessed if
#	the ID changes. So much for factoring. Sigh.

sub _set_id {
    $_[0]{$_[1]} = $_[2];
    $_[0]->rebless () if $_[0]->can ('rebless');
    return SET_ACTION_NONE;
}

#	If this is a reference ellipsoid name, check it, and if it's
#	OK set semimajor and flattening also. Also clear any cached
#	geodetic coordinates.

sub _set_reference_ellipsoid {
    my ($self, $name, $value) = @_;
    defined $value or croak <
Error - Can not set reference ellipsoid to undefined.
eod
    exists $known_ellipsoid{$value} or croak <
Error - Reference ellipsoid '$value' is unknown.
eod

    $self->{semimajor} = $known_ellipsoid{$value}{semimajor};
    $self->{flattening} = $known_ellipsoid{$value}{flattening};
    $self->{$name} = $value;
    return SET_ACTION_RESET;
}

#	If this is a vanilla setting, just do it.

sub _set_value {
    $_[0]->{$_[1]} = $_[2];
    return SET_ACTION_NONE;
}

=item $coord->universal ($time)

This method sets the time represented by the object, in universal time
(a.k.a. CUT, a.k.a. Zulu, a.k.a. Greenwich).

This method can also be called as a class method, in which case it
instantiates the desired object.

=item $time = $coord->universal ();

This method returns the universal time previously set.

=cut

sub universal {
    my ( $self, $time, @args ) = @_;

    @args
	and croak <<'EOD';
Error - The universal() method must be called with either zero
        arguments (to retrieve the time) or one argument (to set the
        time).
EOD

    if ( defined $time ) {
	return $self->__universal( $time, 1 );
    } else {
	ref $self or croak <<'EOD';
Error - The universal() method may not be called as a class method
        unless you specify arguments.
EOD
	defined $self->{universal}
	    or croak <<'EOD';
Error - Object's time has not been set.
EOD
	return $self->{universal};
    }
}

# Set universal time without running model

sub __universal {
    my ( $self, $time, $run_model ) = @_;
    defined $time
	or confess 'Programming error - __universal() requires a defined time';
    $self = $self->new () unless ref $self;
    defined $self->{universal}
	and $time == $self->{universal}
	and return $self;
    $self->__clear_time();
    $self->{universal} = $time;
    $run_model
	and $self->_call_time_set();
    return $self;
}

sub __clear_time {
    my ( $self ) = @_;
    delete $self->{universal};
    delete $self->{local_mean_time};
    delete $self->{dynamical};
    if ($self->{specified}) {
	if ($self->{inertial}) {
	    delete $self->{_ECI_cache}{fixed};
	} else {
	    delete $self->{_ECI_cache}{inertial};
	}
    }
    return;
}

sub __event_name {
    my ( $self, $event, $tplt ) = @_;
    ARRAY_REF eq ref $tplt
	or confess 'Programming error - $tplt must be array ref';
    return __sprintf( $tplt->[$event], $self->__object_name() );
}

sub __horizon_name {
    my ( $self, $event, $tplt ) = @_;
    return $self->__event_name(
	$event,
	$tplt || $self->__horizon_name_tplt(),
    );
}

sub __horizon_name_tplt {
    return [ '%s sets', '%s rises' ];
}

sub __horizon_extreme_tod_name {
    my ( $self, $event, $tplt ) = @_;
    return $self->__event_name(
	$event,
	$tplt || $self->__horizon_extreme_tod_name_tplt(),
    );
}

sub __horizon_extreme_tod_name_tplt {
    return [
	'earliest %sset', 'earliest %srise',
	'latest %sset', 'latest %srise',
    ];
}

sub __transit_name {
    my ( $self, $event, $tplt ) = @_;
    return $self->__event_name(
	$event,
	$tplt || $self->__transit_name_tplt(),
    );
}

sub __transit_name_tplt {
    return [ undef, '%s transits meridian' ];
}

sub __object_name {
    my ( $self ) = @_;
    return $self->get( 'name' ) || $self->get( 'id' ) || (
	$self->{_name} ||= $self->__object_name_from_class_name()
    );
}

sub __object_name_from_class_name {
    my ( $self ) = @_;
    ( my $name = ref $self || $self ) =~ s/ (?: :: | _ ) XS \z //smx;
    $name =~ s/ .* :: //smx;
    return $name;
}

sub __object_is_self_named {
    my ( $self ) = @_;
    $self->{_name_re} ||= do {
	my $re = $self->__object_name_from_class_name();
	qr< \A \Q$re\E \z >smxi;
    };
    return $self->__object_name() =~ $self->{_name_re};
}

#######################################################################
#
#	Internal
#

#	$coord->_call_time_set ()

#	This method calls the time_set method if it exists and if we are
#	not already in it. It is a way to avoid endless recursion if the
#	time_set method should happen to set the time.

sub _call_time_set {
    my $self = shift;
    $self->can ('time_set') or return;
    my $exception;
    unless ($self->{_no_set}++) {
	eval {$self->time_set (); 1;}
	    or $exception = $@;
    }
    --$self->{_no_set} or delete $self->{_no_set};
    if ($exception) {
	$self->__clear_time();
	# Re-raise the exception now that we have cleaned up.
	die $exception;	## no critic (RequireCarping)
    }
    return;
}

#	$coord->_check_coord (method => \@_)

#	This is designed to be called "up front" for any of the methods
#	that retrieve or set coordinates, to be sure the object is in
#	a consistent state.
#	* If $self is not a reference, it creates a new object if there
#	  are arguments, or croaks if there are none.
#	* If the number arguments passed (after removing self and the
#	  method name) is one more than a multiple of three, the last
#	  argument is removed, and used to set the universal time of
#	  the object.
#	* If there are arguments, all coordinate caches are cleared;
#	  otherwise the coordinates are reset if needed.
#	The object itself is returned.

sub _check_coord {
    my ($self, $method, $args, @extra) = @_;

    unless (ref $self) {
	@$args or croak <
Error - The $method() method may not be called as a class method
        unless you specify arguments.
eod
	$self = $self->new ();
    }

    $self->{debug} and do {
	require Data::Dumper;
	local $Data::Dumper::Terse = 1;
	print "Debug $method (", Data::Dumper::Dumper (@$args, @extra), ")\n";
    };

    {
	my $inx = 0;
	foreach (@$args) {
	    defined $_
		and next;
	    require Data::Dumper;
	    local $Data::Dumper::Terse = 1;
	    croak <
Error - @{[ (caller (1))[3] ]} argument $inx is undefined.
        Arguments are (@{[ join ', ',
	    map {my $x = Data::Dumper::Dumper $_; chomp $x; $x} @$args ]})
eod
	    $inx++;
	}
    }

    $self->universal (pop @$args) if @$args % 3 == 1;

    if ($self->{specified}) {
	if (@$args) {
#	    Cached coordinate deletion moved to ecef(), so it's only done once.
	} elsif ($self->{_need_purge}) {
	    delete $self->{_need_purge};
	    my $spec = $self->{specified};
	    my $data = [$self->$spec ()];
	    foreach my $key (@kilatr) {delete $self->{$key}}
	    $self->$spec (@$data);
	}
    }

    return $self;
}

#	_check_latitude($arg_name => $value)
#
#	This subroutine range-checks the given latitude value,
#	generating a warning if it is outside the range -PIOVER2 <=
#	$value <= PIOVER2. The $arg_name is used in the exception, if
#	any. The value is normalized to the range -PI to PI, and
#	returned. Not that a value outside the validation range makes
#	any sense.

sub _check_latitude {
    (-&PIOVER2 <= $_[1] && $_[1] <= &PIOVER2)
	or carp (ucfirst $_[0],
	' must be in radians, between -PI/2 and PI/2');
    return mod2pi($_[1] + PI) - PI;
}

#	$value = _check_longitude($arg_name => $value)
#
#	This subroutine range-checks the given longitude value,
#	generating a warning if it is outside the range -TWOPI <= $value
#	<= TWOPI. The $arg_name is used in the exception, if any. The
#	value is normalized to the range -PI to PI, and returned.

sub _check_longitude {
    (-&TWOPI <= $_[1] && $_[1] <= &TWOPI)
	or carp (ucfirst $_[0],
	' must be in radians, between -2*PI and 2*PI');
    return mod2pi($_[1] + PI) - PI;
}

#	_check_right_ascension($arg_name => $value)
#
#	This subroutine range-checks the given right ascension value,
#	generating a warning if it is outside the range 0 <= $value <=
#	TWOPI. The $arg_name is used in the exception, if any. The value
#	is normalized to the range 0 to TWOPI, and returned.

sub _check_right_ascension {
    (0 <= $_[1] && $_[1] <= &TWOPI)
	or carp (ucfirst $_[0],
	' must be in radians, between 0 and 2*PI');
    return mod2pi($_[1]);
}

# @cartesian = _convert_spherical_to_cartesian( @spherical )
#
# This subroutine converts spherical coordinates to Cartesian
# coordinates of the same handedness.
#
# The first three arguments are Phi (in the X-Y plane, e.g. azimuth or
# longitude, in radians), Theta (perpendicular to the X-Y plane, e.g.
# elevation or latitude, in radians) and Rho (range). Subsequent
# triplets of arguments are optional, and can represent derivitaves of
# position (velocity, acceleration ... ) at that point.
#
# The return is the corresponding X, Y, and Z coordinates. If more than
# three triplets of arguments are specified, they will be converted to
# spherical coordinates as though measured at that point, and returned
# in the same order. That is, if you supplied Phi, Theta, and Rho
# velocities, you will get back X, Y, and Z velocities.  velocities, you
# will get back Phi, Theta, and Rho velocities, in that order.

sub _convert_spherical_to_cartesian {
    my @sph_data = @_;
    @sph_data
	and not @sph_data % 3
	or confess( 'Programming error - Want 3 or 6 arguments' );

    # The first triplet is position. We extract it into its own array,
    # then compute the corresponding Cartesian coordinates using the
    # definition of the coordinate system.

    my @sph_pos = splice @sph_data, 0, 3;
    my $range = $sph_pos[2];
    my $sin_theta = sin $sph_pos[1];
    my $cos_theta = cos $sph_pos[1];
    my $sin_phi = sin $sph_pos[0];
    my $cos_phi = cos $sph_pos[0];
    my $diag = $range * $cos_theta;

    my $X = $diag * $cos_phi;
    my $Y = $diag * $sin_phi;
    my $Z = $range * $sin_theta;

    # Accumulate results.

    my @rslt = ( $X, $Y, $Z );

    # If we have velocity (accelelation, etc) components

    if ( @sph_data ) {

	# We compute unit vectors in the Cartesian coordinate system.

	# x = cos Theta cos Phi r - sin Theta cos Phi theta - sin Phi phi
	# y = cos Theta sin Phi r - sin Theta sin Phi theta - cos Phi phi
	# z = sin Theta r + cos Theta theta

	my $X_hat = [ - $sin_phi, - $sin_theta * $cos_phi,
	    $cos_theta * $cos_phi ];
	my $Y_hat = [   $cos_phi, - $sin_theta * $sin_phi,
	    $cos_theta * $sin_phi ];
	my $Z_hat = [ 0, $cos_theta, $sin_theta ];

	while ( @sph_data ) {

	    # Each triplet is then converted by projecting the Cartesian
	    # vector onto the appropriate unit vector. Azimuth and
	    # elevation are also converted to length by multiplying by
	    # the range. NOTE that this is the small-angle
	    # approximation, but should be OK since we assume we're
	    # dealing with derivatives of position.

	    my @sph_info = splice @sph_data, 0, 3;
	    $sph_info[0] *= $range;
	    $sph_info[1] *= $range;
	    push @rslt, vector_dot_product( $X_hat, \@sph_info );
	    push @rslt, vector_dot_product( $Y_hat, \@sph_info );
	    push @rslt, vector_dot_product( $Z_hat, \@sph_info );
	}

    }

    return @rslt;
}

# @cartesian = _convert_spherical_x_to_cartesian( @spherical )
#
# This subroutine is a convenience wrapper for
# _convert_spherical_to_cartesian(). The only difference is that the
# arguments are Theta (perpendicular to the X-Y plane), Phi (in the X-Y
# plane) and Rho, rather than Phi, Theta, Rho. This subroutine is my
# penance for not having all the methods that involve spherical
# coordinates return their arguments in the same order.

sub _convert_spherical_x_to_cartesian {
    my @args = @_;
    for ( my $inx = 0; $inx < @args; $inx += 3 ) {
	@args[ $inx, $inx + 1 ] = @args[ $inx + 1, $inx ];
    }
    return _convert_spherical_to_cartesian( @args );
}

# @spherical = _convert_cartesian_to_spherical( @cartesian )
#
# This subroutine converts three-dimensional Cartesian coordinates to
# spherical coordinates of the same handedness.
#
# The first three arguments are the X, Y, and Z coordinates, and are
# required. Subsequent triplets af arguments are optional, and can
# represent anything (velocity, acceleration ... ) at that point.
#
# The return is the spherical coordinates Phi (in the X-Y plane, e.g.
# azimuth or longitude, in radians), Theta (perpendicular to the X-Y
# plane, e.g.  elevation or latitude, in radians), and Rho (range). If
# more than three triplets of arguments are specified, they will be
# converted to spherical coordinates as though measured at that point,
# and returned in the same order. That is, if you supplied X, Y, and Z
# velocities, you will get back Phi, Theta, and Rho velocities, in that
# order.

sub _convert_cartesian_to_spherical {
    my @cart_data = @_;
    @cart_data
	and not @cart_data % 3
	or confess( 'Programming error - Want 3 or 6 arguments' );

    # The first triplet is position. We extract it into its own array,
    # then compute the corresponding spherical coordinates using the
    # definition of the coordinate system.

    my @cart_pos = splice @cart_data, 0, 3;
    my $range = vector_magnitude( \@cart_pos );
    my $azimuth = ( $cart_pos[0] || $cart_pos[1] ) ?
	mod2pi( atan2 $cart_pos[1], $cart_pos[0] ) :
	0;
    my $elevation = $range ? asin( $cart_pos[2] / $range ) : 0;

    # Accumulate results.

    my @rslt = ( $azimuth, $elevation, $range );

    # If we have velocity (accelelation, etc) components

    if ( @cart_data ) {

	# We compute unit vectors in the spherical coordinate system.
	#
	# The "Relationships among Unit Vectors" at
	# http://plaza.obu.edu/corneliusk/mp/rauv.pdf (and retained in
	# the ref/ directory) gives the transformation both ways. With
	# x, y, and z being the Cartesian unit vectors, Theta and Phi
	# being the elevation (in the range 0 to pi, 0 being along the +
	# Z axis) and azimuth (X toward Y, i.e. right-handed), and r,
	# theta, and phi being the corresponding unit vectors:
	#
	# r = sin Theta cos Phi x + sin Theta sin Phi y + cos Theta z
	# theta = cos Theta cos Phi x + cos Theta sin Phi y - sin Theta z
	# phi = - sin Phi x + cos phi y
	#
	# and
	#
	# x = sin Theta cos Phi r + cos Theta cos Phi theta - sin Phi phi
	# y = sin Theta sin Phi r + cos Theta sin Phi theta - cos Phi phi
	# z = cos Theta r - sin Theta theta
	#
	# It looks to me like I get the Theta convention I'm using by
	# replacing sin Theta with cos Theta and cos Theta by sin Theta
	# (because Dr. Cornelius takes 0 as the positive Z axis whereas
	# I take zero as the X-Y plane) and changing the sign of theta
	# (since Dr. Cornelius' Theta increases in the negative Z
	# direction, whereas mine increases in the positive Z
	# direction).
	#
	# The document was found at http://plaza.obu.edu/corneliusk/
	# which is the page for Dr. Kevin Cornelius' Mathematical
	# Physics (PHYS 4053) course at Ouachita Baptist University in
	# Arkadelphia AR.

	my $diag = sqrt( $cart_pos[0] * $cart_pos[0] +
	    $cart_pos[1] * $cart_pos[1] );
	my ( $sin_theta, $cos_theta, $sin_phi, $cos_phi );
	if ( $range > 0 ) {
	    $sin_theta = $cart_pos[2] / $range;
	    $cos_theta = $diag / $range;
	    if ( $diag > 0 ) {
		$sin_phi = $cart_pos[1] / $diag;
		$cos_phi = $cart_pos[0] / $diag;
	    } else {
		$sin_phi = 0;
		$cos_phi = 1;
	    }

	    # phi = - sin Phi x + cos phi y
	    # theta = - sin Theta cos Phi x - sin Theta sin Phi y + cos Theta z
	    # r = cos Theta cos Phi x + cos Theta sin Phi y + sin Theta z

	    my $az_hat = [ - $sin_phi, $cos_phi, 0 ];
	    my $el_hat = [ - $sin_theta * $cos_phi, - $sin_theta * $sin_phi,
		$cos_theta ];
	    my $rng_hat = [ $cos_theta * $cos_phi, $cos_theta * $sin_phi,
		$sin_theta ];

	    while ( @cart_data ) {

		# Each triplet is then converted by projecting the
		# Cartesian vector onto the appropriate unit vector.
		# Azimuth and elevation are also converted to radians by
		# dividing by the range. NOTE that this is the
		# small-angle approximation, but since we assume we're
		# dealing with derivitaves, it's OK.

		my @cart_info = splice @cart_data, 0, 3;
		push @rslt, vector_dot_product( $az_hat, \@cart_info ) / $range;
		push @rslt, vector_dot_product( $el_hat, \@cart_info ) / $range;
		push @rslt, vector_dot_product( $rng_hat, \@cart_info );
	    }
	} else {
	    # $sin_theta = $sin_phi = 0;
	    # $cos_theta = $cos_phi = 1;
	    # We used to do the above and then drop through and do the
	    # velocity calculation if needed. But in this case the
	    # velocity calulation blows up. So for the moment we are
	    # just going to punt.
	}

    }

    return @rslt;

}

# @spherical = _convert_cartesian_to_spherical_x( @cartesian )
#
# This subroutine is a convenience wrapper for
# _convert_cartesian_to_spherical(). The only difference is that the
# return is Theta (perpendicular to the X-Y plane), Phi (in the X-Y
# plane) and Rho, rather than Phi, Theta, Rho. This subroutine is my
# penance for not having all the methods that involve spherical
# coordinates return their arguments in the same order.

sub _convert_cartesian_to_spherical_x {
    my @sph_data = _convert_cartesian_to_spherical( @_ );
    for ( my $inx = 0; $inx < @sph_data; $inx += 3 ) {
	@sph_data[ $inx, $inx + 1 ] = @sph_data[ $inx + 1, $inx ];
    }
    return @sph_data;
}

#	@eci = $self->_convert_ecef_to_eci ()

#	This subroutine converts the object's ECEF setting to ECI, and
#	both caches and returns the result.

sub _convert_ecef_to_eci {
    my $self = shift;
    my $thetag = thetag ($self->universal);
    my @data = $self->ecef ();
    $self->{debug} and print <
Debug eci - thetag = $thetag
    (x, y) = (@data[0, 1])
eod
    my $costh = cos ($thetag);
    my $sinth = sin ($thetag);
    @data[0, 1] = ($data[0] * $costh - $data[1] * $sinth,
	    $data[0] * $sinth + $data[1] * $costh);
    $self->{debug} and print <
Debug eci - after rotation,
    (x, y) = (@data[0, 1])
eod
    if ( @data > 5 ) {

	@data[3, 4] = ($data[3] * $costh - $data[4] * $sinth,
		$data[3] * $sinth + $data[4] * $costh);
	$data[3] -= $data[1] * $self->{angularvelocity};
	$data[4] += $data[0] * $self->{angularvelocity};
    }

    # A bunch of digging says this was added by Subversion commit 697,
    # which was first released with Astro::Coord::ECI version 0.014. The
    # comment says "Handle equinox in conversion between eci and ecef.
    # Correctly, I hope." I'm leaving it in for now, but ...
    $self->set (equinox_dynamical => $self->dynamical);
    return @{$self->{_ECI_cache}{inertial}{eci} = \@data};
}

#	This subroutine converts the object's ECI setting to ECEF, and
#	both caches and returns the result.

our $EQUINOX_TOLERANCE = 365 * SECSPERDAY;

sub _convert_eci_to_ecef {
    my $self = shift;
    my $thetag = thetag ($self->universal);

    my $dyn = $self->dynamical;
    ## my $equi = $self->get ('equinox_dynamical') || do {
    ##     $self->set (equinox_dynamical => $dyn); $dyn};
    my $equi = $self->{equinox_dynamical} ||= $dyn;
    if (abs ($equi - $dyn) > $EQUINOX_TOLERANCE) {
	$self->precess_dynamical ($dyn);
    }

    my @ecef = $self->eci ();
    my $costh = cos (- $thetag);
    my $sinth = sin (- $thetag);
    @ecef[0, 1] = ($ecef[0] * $costh - $ecef[1] * $sinth,
	    $ecef[0] * $sinth + $ecef[1] * $costh);

    if ( @ecef > 3 ) {
	@ecef[ 3, 4 ] = ( $ecef[3] * $costh - $ecef[4] * $sinth,
	    $ecef[3] * $sinth + $ecef[4] * $costh );
	$ecef[3] += $ecef[1] * $self->{angularvelocity};
	$ecef[4] -= $ecef[0] * $self->{angularvelocity};
    }

    $self->{_ECI_cache}{fixed}{ecef} = \@ecef;

    defined wantarray
	or return;
    return @ecef;
}

#	my @args = _expand_args_default_station( @_ )
#
#	This subroutine handles the placing of the contents of the
#	'station' attribute into the argument list of methods that,
#	prior to the introduction of the 'station' attribute, formerly
#	took two Astro::Coord::ECI objects and computed the position of
#	the second as seen from the first.

sub _expand_args_default_station {
    my @args = @_;
    if ( ! embodies( $args[1], 'Astro::Coord::ECI' ) ) {
	unshift @args, $args[0]->get( 'station' );
	embodies( $args[0], 'Astro::Coord::ECI' )
	    or croak 'Station not set';
	defined $args[1]->{universal}
	    and $args[0]->universal( $args[1]->universal() );
    }
    return @args;
}

#	$value = $self->__initial_inertial
#
#	Return the initial setting of the inertial attribute. At this
#	level we are assumed not inertial until we acquire a position.
#	This is not part of the public interface, but may be used by
#	subclasses to set an initial value for this read-only attribute.
#	Setting the coordinates explicitly will still set the {inertial}
#	attribute appropriately.

sub __initial_inertial{ return };

#	$value = _local_mean_delta ($coord)

#	Calculate the delta from universal to civil time for the object.
#	An exception is raised if the coordinates of the object have not
#	been set.

sub _local_mean_delta {
    return ($_[0]->geodetic ())[1] * SECSPERDAY / TWOPI;
}

=begin comment

#	$string = _rad2dms ($angle)

#	Convert radians to degrees, minutes, and seconds of arc.
#	Used for debugging.

sub _rad2dms {
    my $angle = rad2deg (shift);
    my $deg = floor ($angle);
    $angle = ($angle - $deg) * 60;
    my $min = floor ($angle);
    $angle = ($angle - $min) * 60;
    return "$deg degrees $min minutes $angle seconds of arc";
}

=end comment

=cut

#######################################################################
#
#	Package initialization
#

__PACKAGE__->set (ellipsoid => 'WGS84');

%savatr = map {$_ => 1} (keys %static, qw{dynamical id name universal});
#	Note that local_mean_time does not get preserved, because it
#	changes if the coordinates change.

1;

__END__