## 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 NAMEMath::Trig - trigonometric functions=head1 SYNOPSIS	use Math::Trig;	$x = tan(0.9);	$y = acos(3.7);	$z = asin(2.4);	$halfpi = pi/2;	$rad = deg2rad(120);=head1 DESCRIPTIONC<Math::Trig> defines many trigonometric functions not defined by thecore Perl which defines only the C<sin()> and C<cos()>.  The constantB<pi> is also defined as are a few convenience functions for angleconversions.=head1 TRIGONOMETRIC FUNCTIONSThe tangent=over 4=item B<tan>=backThe cofunctions of the sine, cosine, and tangent (cosec/csc and cotan/cotare aliases)B<csc>, B<cosec>, B<sec>, B<sec>, B<cot>, B<cotan>The arcus (also known as the inverse) functions of the sine, cosine,and tangentB<asin>, B<acos>, B<atan>The principal value of the arc tangent of y/xB<atan2>(y, x)The arcus cofunctions of the sine, cosine, and tangent (acosec/acscand acotan/acot are aliases)B<acsc>, B<acosec>, B<asec>, B<acot>, B<acotan>The hyperbolic sine, cosine, and tangentB<sinh>, B<cosh>, B<tanh>The cofunctions of the hyperbolic sine, cosine, and tangent (cosech/cschand cotanh/coth are aliases)B<csch>, B<cosech>, B<sech>, B<coth>, B<cotanh>The arcus (also known as the inverse) functions of the hyperbolicsine, cosine, and tangentB<asinh>, B<acosh>, B<atanh>The arcus cofunctions of the hyperbolic sine, cosine, and tangent(acsch/acosech and acoth/acotanh are aliases)B<acsch>, B<acosech>, B<asech>, B<acoth>, B<acotanh>The trigonometric constant B<pi> is also defined.$pi2 = 2 * B<pi>;=head2 ERRORS DUE TO DIVISION BY ZEROThe following functions	acoth	acsc	acsch	asec	asech	atanh	cot	coth	csc	csch	sec	sech	tan	tanhcannot be computed for all arguments because that would mean dividingby zero or taking logarithm of zero. These situations cause fatalruntime 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 theC<atanh>, C<acoth>, the argument cannot be C<1> (one).  For theC<atanh>, C<acoth>, the argument cannot be C<-1> (minus one).  For theC<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 RESULTSPlease note that some of the trigonometric functions can break outfrom the B<real axis> into the B<complex plane>. For exampleC<asin(2)> has no definition for plain real numbers but it hasdefinition for complex numbers.In Perl terms this means that supplying the usual Perl numbers (alsoknown as scalars, please see L<perldata>) as input for thetrigonometric functions might produce as output results that no moreare simple real numbers: instead they are complex numbers.The C<Math::Trig> handles this by using the C<Math::Complex> packagewhich knows how to handle complex numbers, please see L<Math::Complex>for more information. In practice you need not to worry about gettingcomplex numbers as results because the C<Math::Complex> takes care ofdetails 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):	1.5707963267949-1.31695789692482iThat 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(), andgrad2grad().=head1 RADIAL COORDINATE CONVERSIONSB<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 theC<: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, $phi);B<All angles are in radians>.=head2 COORDINATE SYSTEMSB<Cartesian> coordinates are the usual rectangular I<(x, y,z)>-coordinates.Spherical coordinates, I<(rho, theta, pi)>, are three-dimensionalcoordinates which define a point in three-dimensional space.  They arebased on a sphere surface.  The radius of the sphere is B<rho>, alsoknown as the I<radial> coordinate.  The angle in the I<xy>-plane(around the I<z>-axis) is B<theta>, also known as the I<azimuthal>coordinate.  The angle from the I<z>-axis is B<phi>, also known as theI<polar> coordinate.  The `North Pole' is therefore I<0, 0, rho>, andthe `Bay of Guinea' (think of the missing big chunk of Africa) I<0,pi/2, rho>.  In geographical terms I<phi> is latitude (northwardpositive, southward negative) and I<theta> is longitude (eastwardpositive, westward negative).B<BEWARE>: some texts define I<theta> and I<phi> the other way round,some texts define the I<phi> to start from the horizontal plane, sometexts use I<r> in place of I<rho>.Cylindrical coordinates, I<(rho, theta, z)>, are three-dimensionalcoordinates which define a point in three-dimensional space.  They arebased on a cylinder surface.  The radius of the cylinder is B<rho>,also known as the I<radial> coordinate.  The angle in the I<xy>-plane(around the I<z>-axis) is B<theta>, also known as the I<azimuthal>coordinate.  The third coordinate is the I<z>, pointing up from theB<theta>-plane.=head2 3-D ANGLE CONVERSIONSConversions to and from spherical and cylindrical coordinates areavailable.  Please notice that the conversions are not necessarilyreversible because of the equalities like I<pi> angles being equal toI<-pi> angles.=over 4=item cartesian_to_cylindrical        ($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 DISTANCESYou 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, [, $rho]);The I<great circle distance> is the shortest distance between twopoints on a sphere.  The distance is in C<$rho> units.  The C<$rho> isoptional, it defaults to 1 (the unit sphere), therefore the distancedefaults to radians.If you think geographically the I<theta> are longitudes: zero at theGreenwhich meridian, eastward positive, westward negative--and theI<phi> are latitudes: zero at the North Pole, northward positive,southward negative.  B<NOTE>: this formula thinks in mathematics, notgeographically: the I<phi> zero is at the North Pole, not at theEquator on the west coast of Africa (Bay of Guinea).  You need tosubtract your geographical coordinates from I<pi/2> (also known as 90degrees).  $distance = great_circle_distance($lon0, pi/2 - $lat0,                                    $lon1, pi/2 - $lat1, $rho);=head1 EXAMPLESTo calculate the distance between London (51.3N 0.5W) and Tokyo (35.7N139.8E) 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 locationsseparated less than about five degrees) because it uses arc cosinewhich is rather ill-conditioned for values close to zero.=head1 BUGSSaying C<use Math::Trig;> exports many mathematical routines in thecaller environment and even overrides some (C<sin>, C<cos>).  This isconstrued as a feature by the Authors, actually... ;-)The code is not optimized for speed, especially because we useC<Math::Complex> and thus go quite near complex numbers while doingthe computations even when the arguments are not. This, however,cannot be completely avoided if we want things like C<asin(2)> to givean answer instead of giving a fatal runtime error.=head1 AUTHORSJarkko Hietaniemi <F<jhi@iki.fi>> and Raphael Manfredi <F<Raphael_Manfredi@pobox.com>>.=cut# eof