434 lines
12 KiB
Perl
434 lines
12 KiB
Perl
|
#
|
||
|
|
# 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 strict;
|
||
|
|
|
||
|
|
use Math::Complex qw(:trig);
|
||
|
|
|
||
|
|
use vars qw($VERSION $PACKAGE
|
||
|
|
@ISA
|
||
|
|
@EXPORT @EXPORT_OK %EXPORT_TAGS);
|
||
|
|
|
||
|
|
@ISA = qw(Exporter);
|
||
|
|
|
||
|
|
$VERSION = 1.00;
|
||
|
|
|
||
|
|
my @angcnv = qw(rad2deg rad2grad
|
||
|
|
deg2rad deg2grad
|
||
|
|
grad2rad grad2deg);
|
||
|
|
|
||
|
|
@EXPORT = (@{$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);
|
||
|
|
|
||
|
|
@EXPORT_OK = (@rdlcnv, 'great_circle_distance');
|
||
|
|
|
||
|
|
%EXPORT_TAGS = ('radial' => [ @rdlcnv ]);
|
||
|
|
|
||
|
|
use constant pi2 => 2 * pi;
|
||
|
|
use constant pip2 => pi / 2;
|
||
|
|
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 remt ($$) {
|
||
|
|
# Oh yes, POSIX::fmod() would be faster. Possibly. If it is available.
|
||
|
|
$_[0] - $_[1] * int($_[0] / $_[1]);
|
||
|
|
}
|
||
|
|
|
||
|
|
#
|
||
|
|
# Angle conversions.
|
||
|
|
#
|
||
|
|
|
||
|
|
sub rad2deg ($) { remt(RD * $_[0], 360) }
|
||
|
|
|
||
|
|
sub deg2rad ($) { remt(DR * $_[0], pi2) }
|
||
|
|
|
||
|
|
sub grad2deg ($) { remt(GD * $_[0], 360) }
|
||
|
|
|
||
|
|
sub deg2grad ($) { remt(DG * $_[0], 400) }
|
||
|
|
|
||
|
|
sub rad2grad ($) { remt(RG * $_[0], 400) }
|
||
|
|
|
||
|
|
sub grad2rad ($) { 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);
|
||
|
|
|
||
|
|
=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.
|
||
|
|
|
||
|
|
=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)
|
||
|
|
|
||
|
|
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> is also defined.
|
||
|
|
|
||
|
|
$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 ...
|
||
|
|
|
||
|
|
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
|
||
|
|
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.
|
||
|
|
|
||
|
|
$radians = deg2rad($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.
|
||
|
|
|
||
|
|
=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@grenoble.hp.com>>.
|
||
|
|
|
||
|
|
=cut
|
||
|
|
|
||
|
|
# eof
|