trig.pm

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C<atanh>, C<acoth>, the argument cannot be C<1> (one).  For the
C<atanh>, C<acoth>, the argument cannot be C<-1> (minus one).  For the
C<tan>, C<sec>, C<tanh>, C<sech>, the argument cannot be I<pi/2 + k *
pi>, where I<k> is any integer.

Note that atan2(0, 0) is not well-defined.

=head2 SIMPLE (REAL) ARGUMENTS, COMPLEX RESULTS

Please note that some of the trigonometric functions can break out
from the B<real axis> into the B<complex plane>. For example
C<asin(2)> has no definition for plain real numbers but it has
definition for complex numbers.

In Perl terms this means that supplying the usual Perl numbers (also
known as scalars, please see L<perldata>) as input for the
trigonometric functions might produce as output results that no more
are simple real numbers: instead they are complex numbers.

The C<Math::Trig> handles this by using the C<Math::Complex> package
which knows how to handle complex numbers, please see L<Math::Complex>
for more information. In practice you need not to worry about getting
complex numbers as results because the C<Math::Complex> takes care of
details like for example how to display complex numbers. For example:

    print asin(2), "\n";

should produce something like this (take or leave few last decimals):

    1.5707963267949-1.31695789692482i

That is, a complex number with the real part of approximately C<1.571>
and the imaginary part of approximately C<-1.317>.

=head1 PLANE ANGLE CONVERSIONS

(Plane, 2-dimensional) angles may be converted with the following functions.

=over

=item deg2rad

    $radians  = deg2rad($degrees);

=item grad2rad

    $radians  = grad2rad($gradians);

=item rad2deg

    $degrees  = rad2deg($radians);

=item grad2deg

    $degrees  = grad2deg($gradians);

=item deg2grad

    $gradians = deg2grad($degrees);

=item rad2grad

    $gradians = rad2grad($radians);

=back

The full circle is 2 I<pi> radians or I<360> degrees or I<400> gradians.
The result is by default wrapped to be inside the [0, {2pi,360,400}[ circle.
If you don't want this, supply a true second argument:

    $zillions_of_radians  = deg2rad($zillions_of_degrees, 1);
    $negative_degrees     = rad2deg($negative_radians, 1);

You can also do the wrapping explicitly by rad2rad(), deg2deg(), and
grad2grad().

=over 4

=item rad2rad

    $radians_wrapped_by_2pi = rad2rad($radians);

=item deg2deg

    $degrees_wrapped_by_360 = deg2deg($degrees);

=item grad2grad

    $gradians_wrapped_by_400 = grad2grad($gradians);

=back

=head1 RADIAL COORDINATE CONVERSIONS

B<Radial coordinate systems> are the B<spherical> and the B<cylindrical>
systems, explained shortly in more detail.

You can import radial coordinate conversion functions by using the
C<:radial> tag:

    use Math::Trig ':radial';

    ($rho, $theta, $z)     = cartesian_to_cylindrical($x, $y, $z);
    ($rho, $theta, $phi)   = cartesian_to_spherical($x, $y, $z);
    ($x, $y, $z)           = cylindrical_to_cartesian($rho, $theta, $z);
    ($rho_s, $theta, $phi) = cylindrical_to_spherical($rho_c, $theta, $z);
    ($x, $y, $z)           = spherical_to_cartesian($rho, $theta, $phi);
    ($rho_c, $theta, $z)   = spherical_to_cylindrical($rho_s, $theta, $phi);

B<All angles are in radians>.

=head2 COORDINATE SYSTEMS

B<Cartesian> coordinates are the usual rectangular I<(x, y, z)>-coordinates.

Spherical coordinates, I<(rho, theta, pi)>, are three-dimensional
coordinates which define a point in three-dimensional space.  They are
based on a sphere surface.  The radius of the sphere is B<rho>, also
known as the I<radial> coordinate.  The angle in the I<xy>-plane
(around the I<z>-axis) is B<theta>, also known as the I<azimuthal>
coordinate.  The angle from the I<z>-axis is B<phi>, also known as the
I<polar> coordinate.  The North Pole is therefore I<0, 0, rho>, and
the Gulf of Guinea (think of the missing big chunk of Africa) I<0,
pi/2, rho>.  In geographical terms I<phi> is latitude (northward
positive, southward negative) and I<theta> is longitude (eastward
positive, westward negative).

B<BEWARE>: some texts define I<theta> and I<phi> the other way round,
some texts define the I<phi> to start from the horizontal plane, some
texts use I<r> in place of I<rho>.

Cylindrical coordinates, I<(rho, theta, z)>, are three-dimensional
coordinates which define a point in three-dimensional space.  They are
based on a cylinder surface.  The radius of the cylinder is B<rho>,
also known as the I<radial> coordinate.  The angle in the I<xy>-plane
(around the I<z>-axis) is B<theta>, also known as the I<azimuthal>
coordinate.  The third coordinate is the I<z>, pointing up from the
B<theta>-plane.

=head2 3-D ANGLE CONVERSIONS

Conversions to and from spherical and cylindrical coordinates are
available.  Please notice that the conversions are not necessarily
reversible because of the equalities like I<pi> angles being equal to
I<-pi> angles.

=over 4

=item cartesian_to_cylindrical

    ($rho, $theta, $z) = cartesian_to_cylindrical($x, $y, $z);

=item cartesian_to_spherical

    ($rho, $theta, $phi) = cartesian_to_spherical($x, $y, $z);

=item cylindrical_to_cartesian

    ($x, $y, $z) = cylindrical_to_cartesian($rho, $theta, $z);

=item cylindrical_to_spherical

    ($rho_s, $theta, $phi) = cylindrical_to_spherical($rho_c, $theta, $z);

Notice that when C<$z> is not 0 C<$rho_s> is not equal to C<$rho_c>.

=item spherical_to_cartesian

    ($x, $y, $z) = spherical_to_cartesian($rho, $theta, $phi);

=item spherical_to_cylindrical

    ($rho_c, $theta, $z) = spherical_to_cylindrical($rho_s, $theta, $phi);

Notice that when C<$z> is not 0 C<$rho_c> is not equal to C<$rho_s>.

=back

=head1 GREAT CIRCLE DISTANCES AND DIRECTIONS

A great circle is section of a circle that contains the circle
diameter: the shortest distance between two (non-antipodal) points on
the spherical surface goes along the great circle connecting those two
points.

=head2 great_circle_distance

You can compute spherical distances, called B<great circle distances>,
by importing the great_circle_distance() function:

  use Math::Trig 'great_circle_distance';

  $distance = great_circle_distance($theta0, $phi0, $theta1, $phi1, [, $rho]);

The I<great circle distance> is the shortest distance between two
points on a sphere.  The distance is in C<$rho> units.  The C<$rho> is
optional, it defaults to 1 (the unit sphere), therefore the distance
defaults to radians.

If you think geographically the I<theta> are longitudes: zero at the
Greenwhich meridian, eastward positive, westward negative--and the
I<phi> are latitudes: zero at the North Pole, northward positive,
southward negative.  B<NOTE>: this formula thinks in mathematics, not
geographically: the I<phi> zero is at the North Pole, not at the
Equator on the west coast of Africa (Bay of Guinea).  You need to
subtract your geographical coordinates from I<pi/2> (also known as 90
degrees).

  $distance = great_circle_distance($lon0, pi/2 - $lat0,
                                    $lon1, pi/2 - $lat1, $rho);

=head2 great_circle_direction

The direction you must follow the great circle (also known as I<bearing>)
can be computed by the great_circle_direction() function:

  use Math::Trig 'great_circle_direction';

  $direction = great_circle_direction($theta0, $phi0, $theta1, $phi1);

=head2 great_circle_bearing

Alias 'great_circle_bearing' for 'great_circle_direction' is also available.

  use Math::Trig 'great_circle_bearing';

  $direction = great_circle_bearing($theta0, $phi0, $theta1, $phi1);

The result of great_circle_direction is in radians, zero indicating
straight north, pi or -pi straight south, pi/2 straight west, and
-pi/2 straight east.

You can inversely compute the destination if you know the
starting point, direction, and distance:

=head2 great_circle_destination

  use Math::Trig 'great_circle_destination';

  # thetad and phid are the destination coordinates,
  # dird is the final direction at the destination.

  ($thetad, $phid, $dird) =
    great_circle_destination($theta, $phi, $direction, $distance);

or the midpoint if you know the end points:

=head2 great_circle_midpoint

  use Math::Trig 'great_circle_midpoint';

  ($thetam, $phim) =
    great_circle_midpoint($theta0, $phi0, $theta1, $phi1);

The great_circle_midpoint() is just a special case of

=head2 great_circle_waypoint

  use Math::Trig 'great_circle_waypoint';

  ($thetai, $phii) =
    great_circle_waypoint($theta0, $phi0, $theta1, $phi1, $way);

Where the $way is a value from zero ($theta0, $phi0) to one ($theta1,
$phi1).  Note that antipodal points (where their distance is I<pi>
radians) do not have waypoints between them (they would have an an
"equator" between them), and therefore C<undef> is returned for
antipodal points.  If the points are the same and the distance
therefore zero and all waypoints therefore identical, the first point
(either point) is returned.

The thetas, phis, direction, and distance in the above are all in radians.

You can import all the great circle formulas by

  use Math::Trig ':great_circle';

Notice that the resulting directions might be somewhat surprising if
you are looking at a flat worldmap: in such map projections the great
circles quite often do not look like the shortest routes-- but for
example the shortest possible routes from Europe or North America to
Asia do often cross the polar regions.

=head1 EXAMPLES

To calculate the distance between London (51.3N 0.5W) and Tokyo
(35.7N 139.8E) in kilometers:

    use Math::Trig qw(great_circle_distance deg2rad);

    # Notice the 90 - latitude: phi zero is at the North Pole.
    sub NESW { deg2rad($_[0]), deg2rad(90 - $_[1]) }
    my @L = NESW( -0.5, 51.3);
    my @T = NESW(139.8, 35.7);
    my $km = great_circle_distance(@L, @T, 6378); # About 9600 km.

The direction you would have to go from London to Tokyo (in radians,
straight north being zero, straight east being pi/2).

    use Math::Trig qw(great_circle_direction);

    my $rad = great_circle_direction(@L, @T); # About 0.547 or 0.174 pi.

The midpoint between London and Tokyo being

    use Math::Trig qw(great_circle_midpoint);

    my @M = great_circle_midpoint(@L, @T);

or about 89.16N 68.93E, practically at the North Pole.

=head2 CAVEAT FOR GREAT CIRCLE FORMULAS

The answers may be off by few percentages because of the irregular
(slightly aspherical) form of the Earth.  The errors are at worst
about 0.55%, but generally below 0.3%.

=head1 BUGS

Saying C<use Math::Trig;> exports many mathematical routines in the
caller environment and even overrides some (C<sin>, C<cos>).  This is
construed as a feature by the Authors, actually... ;-)

The code is not optimized for speed, especially because we use
C<Math::Complex> and thus go quite near complex numbers while doing
the computations even when the arguments are not. This, however,
cannot be completely avoided if we want things like C<asin(2)> to give
an answer instead of giving a fatal runtime error.

Do not attempt navigation using these formulas.

=head1 AUTHORS

Jarkko Hietaniemi <F<jhi!at!iki.fi>> and 
Raphael Manfredi <F<Raphael_Manfredi!at!pobox.com>>.

=cut

# eof