# # 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 5.006; use strict; use Math::Complex qw(:trig); our($VERSION, $PACKAGE, @ISA, @EXPORT, @EXPORT_OK, %EXPORT_TAGS); @ISA = qw(Exporter); $VERSION = 1.01; my @angcnv = qw(rad2deg rad2grad deg2rad deg2grad grad2rad grad2deg); @EXPORT = (@{$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', 'great_circle_direction'); %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 ) ); } 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 = deg2rad(120); =head1 DESCRIPTION C defines many trigonometric functions not defined by the core Perl which defines only the C and C. The constant B is also defined as are a few convenience functions for angle conversions. =head1 TRIGONOMETRIC FUNCTIONS The tangent =over 4 =item B =back The cofunctions of the sine, cosine, and tangent (cosec/csc and cotan/cot are aliases) B, B, B, B, B, B The arcus (also known as the inverse) functions of the sine, cosine, and tangent B, B, B The principal value of the arc tangent of y/x B(y, x) The arcus cofunctions of the sine, cosine, and tangent (acosec/acsc and acotan/acot are aliases) B, B, B, B, B The hyperbolic sine, cosine, and tangent B, B, B The cofunctions of the hyperbolic sine, cosine, and tangent (cosech/csch and cotanh/coth are aliases) B, B, B, B, B The arcus (also known as the inverse) functions of the hyperbolic sine, cosine, and tangent B, B, B The arcus cofunctions of the hyperbolic sine, cosine, and tangent (acsch/acosech and acoth/acotanh are aliases) B, B, B, B, B The trigonometric constant B is also defined. $pi2 = 2 * B; =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, C, C, C, C, C, C, C, C, the argument cannot be C<0> (zero). For the C, C, the argument cannot be C<1> (one). For the C, C, the argument cannot be C<-1> (minus one). For the C, C, C, C, the argument cannot be I, where I is any integer. =head2 SIMPLE (REAL) ARGUMENTS, COMPLEX RESULTS Please note that some of the trigonometric functions can break out from the B into the B. For example C 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) 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 handles this by using the C package which knows how to handle complex numbers, please see L for more information. In practice you need not to worry about getting complex numbers as results because the C 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 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 are the B and the B 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, $phi); B. =head2 COORDINATE SYSTEMS B coordinates are the usual rectangular I<(x, y, z)>-coordinates. Spherical coordinates, I<(rho, theta, pi)>, are three-dimensional coordinates which define a point in three-dimensional space. They are based on a sphere surface. The radius of the sphere is B, also known as the I coordinate. The angle in the I-plane (around the I-axis) is B, also known as the I coordinate. The angle from the I-axis is B, also known as the I coordinate. The `North Pole' is therefore I<0, 0, rho>, and the `Bay of Guinea' (think of the missing big chunk of Africa) I<0, pi/2, rho>. In geographical terms I is latitude (northward positive, southward negative) and I is longitude (eastward positive, westward negative). B: some texts define I and I the other way round, some texts define the I to start from the horizontal plane, some texts use I in place of I. Cylindrical coordinates, I<(rho, theta, z)>, are three-dimensional coordinates which define a point in three-dimensional space. They are based on a cylinder surface. The radius of the cylinder is B, also known as the I coordinate. The angle in the I-plane (around the I-axis) is B, also known as the I coordinate. The third coordinate is the I, pointing up from the B-plane. =head2 3-D ANGLE CONVERSIONS Conversions to and from spherical and cylindrical coordinates are available. Please notice that the conversions are not necessarily reversible because of the equalities like I angles being equal to I<-pi> angles. =over 4 =item cartesian_to_cylindrical ($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, by importing the great_circle_distance() function: use Math::Trig 'great_circle_distance'; $distance = great_circle_distance($theta0, $phi0, $theta1, $phi1, [, $rho]); The I is the shortest distance between two points on a sphere. The distance is in C<$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 are longitudes: zero at the Greenwhich meridian, eastward positive, westward negative--and the I are latitudes: zero at the North Pole, northward positive, southward negative. B: this formula thinks in mathematics, not geographically: the I 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 (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 exports many mathematical routines in the caller environment and even overrides some (C, C). This is construed as a feature by the Authors, actually... ;-) The code is not optimized for speed, especially because we use C 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 to give an answer instead of giving a fatal runtime error. =head1 AUTHORS Jarkko Hietaniemi > and Raphael Manfredi >. =cut # eof