#
# 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 5.006;
use strict;
use Math::Complex qw(:trig);
-use vars qw($VERSION $PACKAGE
- @ISA
- @EXPORT
- $pi2 $DR $RD $DG $GD $RG $GR);
+our($VERSION, $PACKAGE, @ISA, @EXPORT, @EXPORT_OK, %EXPORT_TAGS);
@ISA = qw(Exporter);
-$VERSION = 1.00;
+$VERSION = 1.01;
-my @angcnv = qw(rad_to_deg rad_to_grad
- deg_to_rad deg_to_grad
- grad_to_rad grad_to_dec);
+my @angcnv = qw(rad2deg rad2grad
+ deg2rad deg2grad
+ grad2rad grad2deg);
@EXPORT = (@{$Math::Complex::EXPORT_TAGS{'trig'}},
@angcnv);
-sub pi2 () {
- $pi2 = 2 * pi unless ($pi2);
- $pi2;
-}
+my @rdlcnv = qw(cartesian_to_cylindrical
+ cartesian_to_spherical
+ cylindrical_to_cartesian
+ cylindrical_to_spherical
+ spherical_to_cartesian
+ spherical_to_cylindrical);
-sub DR () {
- $DR = pi2/360 unless ($DR);
- $DR;
-}
+@EXPORT_OK = (@rdlcnv, 'great_circle_distance', 'great_circle_direction');
-sub RD () {
- $RD = 360/pi2 unless ($RD);
- $RD;
-}
+%EXPORT_TAGS = ('radial' => [ @rdlcnv ]);
-sub DG () {
- $DG = 400/360 unless ($DG);
- $DG;
-}
+sub pi2 () { 2 * pi }
+sub pip2 () { pi / 2 }
-sub GD () {
- $GD = 360/400 unless ($GD);
- $GD;
-}
-
-sub RG () {
- $RG = 400/pi2 unless ($RG);
- $RG;
-}
-
-sub GR () {
- $GR = pi2/400 unless ($GR);
- $GR;
-}
+sub DR () { pi2/360 }
+sub RD () { 360/pi2 }
+sub DG () { 400/360 }
+sub GD () { 360/400 }
+sub RG () { 400/pi2 }
+sub GR () { pi2/400 }
#
# Truncating remainder.
# Angle conversions.
#
-sub rad_to_deg ($) {
- remt(RD * $_[0], 360);
+sub rad2rad($) { 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 / $rho ) : 0 );
}
-sub deg_to_rad ($) {
- remt(DR * $_[0], pi2);
+sub spherical_to_cartesian {
+ my ( $rho, $theta, $phi ) = @_;
+
+ return ( $rho * cos( $theta ) * sin( $phi ),
+ $rho * sin( $theta ) * sin( $phi ),
+ $rho * cos( $phi ) );
}
-sub grad_to_deg ($) {
- remt(GD * $_[0], 360);
+sub spherical_to_cylindrical {
+ my ( $x, $y, $z ) = spherical_to_cartesian( @_ );
+
+ return ( sqrt( $x * $x + $y * $y ), $_[1], $z );
}
-sub deg_to_grad ($) {
- remt(DG * $_[0], 400);
+sub cartesian_to_cylindrical {
+ my ( $x, $y, $z ) = @_;
+
+ return ( sqrt( $x * $x + $y * $y ), atan2( $y, $x ), $z );
}
-sub rad_to_grad ($) {
- remt(RG * $_[0], 400);
+sub cylindrical_to_cartesian {
+ my ( $rho, $theta, $z ) = @_;
+
+ return ( $rho * cos( $theta ), $rho * sin( $theta ), $z );
}
-sub grad_to_rad ($) {
- remt(GR * $_[0], pi2);
+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 $lat0 = pip2 - $phi0;
+ my $lat1 = pip2 - $phi1;
+
+ my $direction =
+ atan2(sin($theta0 - $theta1) * cos($lat1),
+ cos($lat0) * sin($lat1) -
+ sin($lat0) * cos($lat1) * cos($theta0 - $theta1));
+
+ return rad2rad($direction);
+}
+
+=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 = deg_to_rad(120);
+ $rad = deg2rad(120);
=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
+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.
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
+
+ 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 ...
-=head2 SIMPLE ARGUMENTS, COMPLEX RESULTS
+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
+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.
+
+=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
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 E<1.571>
-and the imaginary part of approximately E<-1.317>.
+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.
- $radians = deg_to_rad($degrees);
- $radians = grad_to_rad($gradians);
-
- $degrees = rad_to_deg($radians);
- $degrees = grad_to_deg($gradians);
-
- $gradians = deg_to_grad($degrees);
- $gradians = rad_to_grad($radians);
+ $radians = deg2rad($degrees);
+ $radians = grad2rad($gradians);
-The full circle is 2 B<pi> radians or E<360> degrees or E<400> gradians.
+ $degrees = rad2deg($radians);
+ $degrees = grad2deg($gradians);
-=head1
+ $gradians = deg2grad($degrees);
+ $gradians = rad2grad($radians);
-The following functions
+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:
- tan
- sec
- csc
- cot
- atan
- acot
- tanh
- sech
- csch
- coth
- atanh
- asech
- acsch
- acoth
+ $zillions_of_radians = deg2rad($zillions_of_degrees, 1);
+ $negative_degrees = rad2deg($negative_radians, 1);
-cannot be computed for all arguments because that would mean dividing
-by zero. These situations cause fatal runtime errors looking like this
+You can also do the wrapping explicitly by rad2rad(), deg2deg(), and
+grad2grad().
- cot(0): Division by zero.
- (Because in the definition of cot(0), sin(0) is 0)
- Died at ...
+=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 AND DIRECTIONS
+
+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);
+
+The direction you must follow the great circle can be computed by the
+great_circle_direction() function:
+
+ use Math::Trig 'great_circle_direction';
+
+ $direction = great_circle_direction($theta0, $phi0, $theta1, $phi1);
+
+The result is in radians, zero indicating straight north, pi or -pi
+straight south, pi/2 straight west, and -pi/2 straight east.
+
+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.
+ @L = (deg2rad(-0.5), deg2rad(90 - 51.3));
+ @T = (deg2rad(139.8),deg2rad(90 - 35.7));
+
+ $km = great_circle_distance(@L, @T, 6378);
+
+The direction you would have to go from London to Tokyo
+
+ use Math::Trig qw(great_circle_direction);
+
+ $rad = great_circle_direction(@L, @T);
+
+=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 formula used for
+grear circle distances
+
+ 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
+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.
+
+=head1 AUTHORS
+
+Jarkko Hietaniemi <F<jhi@iki.fi>> and
+Raphael Manfredi <F<Raphael_Manfredi@pobox.com>>.
=cut
projects / p5sagit/p5-mst-13.2.git / blobdiff