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Diffstat (limited to 'contrib/perl5/lib/Math/Trig.pm')
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diff --git a/contrib/perl5/lib/Math/Trig.pm b/contrib/perl5/lib/Math/Trig.pm deleted file mode 100644 index b28f150..0000000 --- a/contrib/perl5/lib/Math/Trig.pm +++ /dev/null @@ -1,456 +0,0 @@ -# -# 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_64; -use strict; - -use Math::Complex qw(:trig); - -our($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 ]); - -sub pi2 () { 2 * pi } -sub pip2 () { pi / 2 } - -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 ) ); -} - -=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. -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(). - -=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 -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 - -# eof |
