trig.pm

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#
# Trigonometric functions, mostly inherited from Math::Complex.
# -- Jarkko Hietaniemi, since April 1997
# -- Raphael Manfredi, September 1996 (indirectly: because of Math::Complex)
#

require Exporter;
package Math::Trig;

use 5.005;
use strict;

use Math::Complex 1.36;
use Math::Complex qw(:trig :pi);

use vars qw($VERSION $PACKAGE @ISA @EXPORT @EXPORT_OK %EXPORT_TAGS);

@ISA = qw(Exporter);

$VERSION = 1.04;

my @angcnv = qw(rad2deg rad2grad
		deg2rad deg2grad
		grad2rad grad2deg);

@EXPORT = (@{$Math::Complex::EXPORT_TAGS{'trig'}},
	   @angcnv);

my @rdlcnv = qw(cartesian_to_cylindrical
		cartesian_to_spherical
		cylindrical_to_cartesian
		cylindrical_to_spherical
		spherical_to_cartesian
		spherical_to_cylindrical);

my @greatcircle = qw(
		     great_circle_distance
		     great_circle_direction
		     great_circle_bearing
		     great_circle_waypoint
		     great_circle_midpoint
		     great_circle_destination
		    );

my @pi = qw(pi pi2 pi4 pip2 pip4);

@EXPORT_OK = (@rdlcnv, @greatcircle, @pi);

# See e.g. the following pages:
# http://www.movable-type.co.uk/scripts/LatLong.html
# http://williams.best.vwh.net/avform.htm

%EXPORT_TAGS = ('radial' => [ @rdlcnv ],
	        'great_circle' => [ @greatcircle ],
	        'pi'     => [ @pi ]);

sub _DR  () { pi2/360 }
sub _RD  () { 360/pi2 }
sub _DG  () { 400/360 }
sub _GD  () { 360/400 }
sub _RG  () { 400/pi2 }
sub _GR  () { pi2/400 }

#
# Truncating remainder.
#

sub _remt ($$) {
    # Oh yes, POSIX::fmod() would be faster. Possibly. If it is available.
    $_[0] - $_[1] * int($_[0] / $_[1]);
}

#
# Angle conversions.
#

sub rad2rad($)     { _remt($_[0], pi2) }

sub deg2deg($)     { _remt($_[0], 360) }

sub grad2grad($)   { _remt($_[0], 400) }

sub rad2deg ($;$)  { my $d = _RD * $_[0]; $_[1] ? $d : deg2deg($d) }

sub deg2rad ($;$)  { my $d = _DR * $_[0]; $_[1] ? $d : rad2rad($d) }

sub grad2deg ($;$) { my $d = _GD * $_[0]; $_[1] ? $d : deg2deg($d) }

sub deg2grad ($;$) { my $d = _DG * $_[0]; $_[1] ? $d : grad2grad($d) }

sub rad2grad ($;$) { my $d = _RG * $_[0]; $_[1] ? $d : grad2grad($d) }

sub grad2rad ($;$) { my $d = _GR * $_[0]; $_[1] ? $d : rad2rad($d) }

sub cartesian_to_spherical {
    my ( $x, $y, $z ) = @_;

    my $rho = sqrt( $x * $x + $y * $y + $z * $z );

    return ( $rho,
             atan2( $y, $x ),
             $rho ? acos( $z / $rho ) : 0 );
}

sub spherical_to_cartesian {
    my ( $rho, $theta, $phi ) = @_;

    return ( $rho * cos( $theta ) * sin( $phi ),
             $rho * sin( $theta ) * sin( $phi ),
             $rho * cos( $phi   ) );
}

sub spherical_to_cylindrical {
    my ( $x, $y, $z ) = spherical_to_cartesian( @_ );

    return ( sqrt( $x * $x + $y * $y ), $_[1], $z );
}

sub cartesian_to_cylindrical {
    my ( $x, $y, $z ) = @_;

    return ( sqrt( $x * $x + $y * $y ), atan2( $y, $x ), $z );
}

sub cylindrical_to_cartesian {
    my ( $rho, $theta, $z ) = @_;

    return ( $rho * cos( $theta ), $rho * sin( $theta ), $z );
}

sub cylindrical_to_spherical {
    return ( cartesian_to_spherical( cylindrical_to_cartesian( @_ ) ) );
}

sub great_circle_distance {
    my ( $theta0, $phi0, $theta1, $phi1, $rho ) = @_;

    $rho = 1 unless defined $rho; # Default to the unit sphere.

    my $lat0 = pip2 - $phi0;
    my $lat1 = pip2 - $phi1;

    return $rho *
        acos(cos( $lat0 ) * cos( $lat1 ) * cos( $theta0 - $theta1 ) +
             sin( $lat0 ) * sin( $lat1 ) );
}

sub great_circle_direction {
    my ( $theta0, $phi0, $theta1, $phi1 ) = @_;

    my $distance = &great_circle_distance;

    my $lat0 = pip2 - $phi0;
    my $lat1 = pip2 - $phi1;

    my $direction =
	acos((sin($lat1) - sin($lat0) * cos($distance)) /
	     (cos($lat0) * sin($distance)));

    $direction = pi2 - $direction
	if sin($theta1 - $theta0) < 0;

    return rad2rad($direction);
}

*great_circle_bearing = \&great_circle_direction;

sub great_circle_waypoint {
    my ( $theta0, $phi0, $theta1, $phi1, $point ) = @_;

    $point = 0.5 unless defined $point;

    my $d = great_circle_distance( $theta0, $phi0, $theta1, $phi1 );

    return undef if $d == pi;

    my $sd = sin($d);

    return ($theta0, $phi0) if $sd == 0;

    my $A = sin((1 - $point) * $d) / $sd;
    my $B = sin(     $point  * $d) / $sd;

    my $lat0 = pip2 - $phi0;
    my $lat1 = pip2 - $phi1;

    my $x = $A * cos($lat0) * cos($theta0) + $B * cos($lat1) * cos($theta1);
    my $y = $A * cos($lat0) * sin($theta0) + $B * cos($lat1) * sin($theta1);
    my $z = $A * sin($lat0)                + $B * sin($lat1);

    my $theta = atan2($y, $x);
    my $phi   = acos($z);
    
    return ($theta, $phi);
}

sub great_circle_midpoint {
    great_circle_waypoint(@_[0..3], 0.5);
}

sub great_circle_destination {
    my ( $theta0, $phi0, $dir0, $dst ) = @_;

    my $lat0 = pip2 - $phi0;

    my $phi1   = asin(sin($lat0)*cos($dst)+cos($lat0)*sin($dst)*cos($dir0));
    my $theta1 = $theta0 + atan2(sin($dir0)*sin($dst)*cos($lat0),
				 cos($dst)-sin($lat0)*sin($phi1));

    my $dir1 = great_circle_bearing($theta1, $phi1, $theta0, $phi0) + pi;

    $dir1 -= pi2 if $dir1 > pi2;

    return ($theta1, $phi1, $dir1);
}

1;

__END__
=pod

=head1 NAME

Math::Trig - trigonometric functions

=head1 SYNOPSIS

    use Math::Trig;

    $x = tan(0.9);
    $y = acos(3.7);
    $z = asin(2.4);

    $halfpi = pi/2;

    $rad = deg2rad(120);

    # Import constants pi2, pip2, pip4 (2*pi, pi/2, pi/4).
    use Math::Trig ':pi';

    # Import the conversions between cartesian/spherical/cylindrical.
    use Math::Trig ':radial';

        # Import the great circle formulas.
    use Math::Trig ':great_circle';

=head1 DESCRIPTION

C<Math::Trig> defines many trigonometric functions not defined by the
core Perl which defines only the C<sin()> and C<cos()>.  The constant
B<pi> is also defined as are a few convenience functions for angle
conversions, and I<great circle formulas> for spherical movement.

=head1 TRIGONOMETRIC FUNCTIONS

The tangent

=over 4

=item B<tan>

=back

The cofunctions of the sine, cosine, and tangent (cosec/csc and cotan/cot
are aliases)

B<csc>, B<cosec>, B<sec>, B<sec>, B<cot>, B<cotan>

The arcus (also known as the inverse) functions of the sine, cosine,
and tangent

B<asin>, B<acos>, B<atan>

The principal value of the arc tangent of y/x

B<atan2>(y, x)

The arcus cofunctions of the sine, cosine, and tangent (acosec/acsc
and acotan/acot are aliases).  Note that atan2(0, 0) is not well-defined.

B<acsc>, B<acosec>, B<asec>, B<acot>, B<acotan>

The hyperbolic sine, cosine, and tangent

B<sinh>, B<cosh>, B<tanh>

The cofunctions of the hyperbolic sine, cosine, and tangent (cosech/csch
and cotanh/coth are aliases)

B<csch>, B<cosech>, B<sech>, B<coth>, B<cotanh>

The arcus (also known as the inverse) functions of the hyperbolic
sine, cosine, and tangent

B<asinh>, B<acosh>, B<atanh>

The arcus cofunctions of the hyperbolic sine, cosine, and tangent
(acsch/acosech and acoth/acotanh are aliases)

B<acsch>, B<acosech>, B<asech>, B<acoth>, B<acotanh>

The trigonometric constant B<pi> and some of handy multiples
of it are also defined.

B<pi, pi2, pi4, pip2, pip4>

=head2 ERRORS DUE TO DIVISION BY ZERO

The following functions

    acoth
    acsc
    acsch
    asec
    asech
    atanh
    cot
    coth
    csc
    csch
    sec
    sech
    tan
    tanh

cannot be computed for all arguments because that would mean dividing
by zero or taking logarithm of zero. These situations cause fatal
runtime errors looking like this

    cot(0): Division by zero.
    (Because in the definition of cot(0), the divisor sin(0) is 0)
    Died at ...

or

    atanh(-1): Logarithm of zero.
    Died at...

For the C<csc>, C<cot>, C<asec>, C<acsc>, C<acot>, C<csch>, C<coth>,
C<asech>, C<acsch>, the argument cannot be C<0> (zero).  For the