#
# Trigonometric functions, mostly inherited from Math::Complex.
-# -- Jarkko Hietaniemi, April 1997
+# -- Jarkko Hietaniemi, since April 1997
# -- Raphael Manfredi, September 1996 (indirectly: because of Math::Complex)
#
require Exporter;
package Math::Trig;
+use 5.005_64;
use strict;
use Math::Complex qw(:trig);
-use vars qw($VERSION $PACKAGE
- @ISA
- @EXPORT);
+our($VERSION, $PACKAGE, @ISA, @EXPORT, @EXPORT_OK, %EXPORT_TAGS);
@ISA = qw(Exporter);
@EXPORT = (@{$Math::Complex::EXPORT_TAGS{'trig'}},
@angcnv);
-use constant pi2 => 2 * pi;
-use constant DR => pi2/360;
-use constant RD => 360/pi2;
-use constant DG => 400/360;
-use constant GD => 360/400;
-use constant RG => 400/pi2;
-use constant GR => pi2/400;
+my @rdlcnv = qw(cartesian_to_cylindrical
+ cartesian_to_spherical
+ cylindrical_to_cartesian
+ cylindrical_to_spherical
+ spherical_to_cartesian
+ spherical_to_cylindrical);
+
+@EXPORT_OK = (@rdlcnv, 'great_circle_distance');
+
+%EXPORT_TAGS = ('radial' => [ @rdlcnv ]);
+
+sub pi2 () { 2 * pi } # use constant generates warning
+sub pip2 () { pi / 2 } # use constant generates warning
+use constant DR => pi2/360;
+use constant RD => 360/pi2;
+use constant DG => 400/360;
+use constant GD => 360/400;
+use constant RG => 400/pi2;
+use constant GR => pi2/400;
#
# Truncating remainder.
sub grad2rad ($) { remt(GR * $_[0], pi2) }
+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 ) );
+}
+
+=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);
The tangent
- tan
+=over 4
+
+=item B<tan>
+
+=back
The cofunctions of the sine, cosine, and tangent (cosec/csc and cotan/cot
are aliases)
- csc cosec sec cot cotan
+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
- asin acos atan
+B<asin>, B<acos>, B<atan>
The principal value of the arc tangent of y/x
- atan2(y, x)
+B<atan2>(y, x)
The arcus cofunctions of the sine, cosine, and tangent (acosec/acsc
and acotan/acot are aliases)
- acsc acosec asec acot acotan
+B<acsc>, B<acosec>, B<asec>, B<acot>, B<acotan>
The hyperbolic sine, cosine, and tangent
- sinh cosh tanh
+B<sinh>, B<cosh>, B<tanh>
The cofunctions of the hyperbolic sine, cosine, and tangent (cosech/csch
and cotanh/coth are aliases)
- csch cosech sech coth cotanh
+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
- asinh acosh atanh
+B<asinh>, B<acosh>, B<atanh>
The arcus cofunctions of the hyperbolic sine, cosine, and tangent
(acsch/acosech and acoth/acotanh are aliases)
- acsch acosech asech acoth acotanh
+B<acsch>, B<acosech>, B<asech>, B<acoth>, B<acotanh>
The trigonometric constant B<pi> is also defined.
- $pi2 = 2 * pi;
+$pi2 = 2 * B<pi>;
=head2 ERRORS DUE TO DIVISION BY ZERO
The following functions
- tan
- sec
- csc
- cot
- asec
+ acoth
acsc
- tanh
- sech
- csch
- coth
- atanh
- asech
acsch
- acoth
+ 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
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 ANGLE CONVERSIONS
+=head1 PLANE ANGLE CONVERSIONS
(Plane, 2-dimensional) angles may be converted with the following functions.
$radians = deg2rad($degrees);
$radians = grad2rad($gradians);
-
+
$degrees = rad2deg($radians);
$degrees = grad2deg($gradians);
-
+
$gradians = deg2grad($degrees);
$gradians = rad2grad($radians);
The full circle is 2 I<pi> radians or I<360> degrees or I<400> gradians.
+=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, $phi);
+
+B<All angles are in radians>.
+
+=head2 COORDINATE SYSTEMS
+
+B<Cartesian> coordinates are the usual rectangular I<(x, y,
+z)>-coordinates.
+
+Spherical coordinates, I<(rho, theta, pi)>, are three-dimensional
+coordinates which define a point in three-dimensional space. They are
+based on a sphere surface. The radius of the sphere is B<rho>, also
+known as the I<radial> coordinate. The angle in the I<xy>-plane
+(around the I<z>-axis) is B<theta>, also known as the I<azimuthal>
+coordinate. The angle from the I<z>-axis is B<phi>, also known as the
+I<polar> coordinate. The `North Pole' is therefore I<0, 0, rho>, and
+the `Bay of Guinea' (think of the missing big chunk of Africa) I<0,
+pi/2, rho>. In geographical terms I<phi> is latitude (northward
+positive, southward negative) and I<theta> is longitude (eastward
+positive, westward negative).
+
+B<BEWARE>: some texts define I<theta> and I<phi> the other way round,
+some texts define the I<phi> to start from the horizontal plane, some
+texts use I<r> in place of I<rho>.
+
+Cylindrical coordinates, I<(rho, theta, z)>, are three-dimensional
+coordinates which define a point in three-dimensional space. They are
+based on a cylinder surface. The radius of the cylinder is B<rho>,
+also known as the I<radial> coordinate. The angle in the I<xy>-plane
+(around the I<z>-axis) is B<theta>, also known as the I<azimuthal>
+coordinate. The third coordinate is the I<z>, pointing up from the
+B<theta>-plane.
+
+=head2 3-D ANGLE CONVERSIONS
+
+Conversions to and from spherical and cylindrical coordinates are
+available. Please notice that the conversions are not necessarily
+reversible because of the equalities like I<pi> angles being equal to
+I<-pi> angles.
+
+=over 4
+
+=item cartesian_to_cylindrical
+
+ ($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
+
+You can compute spherical distances, called B<great circle distances>,
+by importing the C<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);
+
+=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.
+ @L = (deg2rad(-0.5), deg2rad(90 - 51.3));
+ @T = (deg2rad(139.8),deg2rad(90 - 35.7));
+
+ $km = great_circle_distance(@L, @T, 6378);
+
+The answer may be off by few percentages because of the irregular
+(slightly aspherical) form of the Earth. The used formula
+
+ lat0 = 90 degrees - phi0
+ lat1 = 90 degrees - phi1
+ d = R * arccos(cos(lat0) * cos(lat1) * cos(lon1 - lon01) +
+ sin(lat0) * sin(lat1))
+
+is also somewhat unreliable for small distances (for locations
+separated less than about five degrees) because it uses arc cosine
+which is rather ill-conditioned for values close to zero.
+
=head1 BUGS
Saying C<use Math::Trig;> exports many mathematical routines in the
=head1 AUTHORS
Jarkko Hietaniemi <F<jhi@iki.fi>> and
-Raphael Manfredi <F<Raphael_Manfredi@grenoble.hp.com>>.
+Raphael Manfredi <F<Raphael_Manfredi@pobox.com>>.
=cut