Math::Trig

Math::Trig(3)          Perl Programmers Reference Guide          Math::Trig(3)



NAME
       Math::Trig - trigonometric functions

SYNOPSIS
               use Math::Trig;

               $x = tan(0.9);
               $y = acos(3.7);
               $z = asin(2.4);

               $halfpi = pi/2;

               $rad = deg2rad(120);


DESCRIPTION
       Math::Trig defines many trigonometric functions not defined by the core
       Perl which defines only the sin() and cos().  The constant pi is also
       defined as are a few convenience functions for angle conversions.

TRIGONOMETRIC FUNCTIONS
       The tangent

       tan

       The cofunctions of the sine, cosine, and tangent (cosec/csc and
       cotan/cot are aliases)

       csc, cosec, sec, sec, cot, cotan

       The arcus (also known as the inverse) functions of the sine, cosine,
       and tangent

       asin, acos, atan

       The principal value of the arc tangent of y/x

       atan2(y, x)

       The arcus cofunctions of the sine, cosine, and tangent (acosec/acsc and
       acotan/acot are aliases)

       acsc, acosec, asec, acot, acotan

       The hyperbolic sine, cosine, and tangent

       sinh, cosh, tanh

       The cofunctions of the hyperbolic sine, cosine, and tangent
       (cosech/csch and cotanh/coth are aliases)

       csch, cosech, sech, coth, cotanh

       The arcus (also known as the inverse) functions of the hyperbolic sine,
       cosine, and tangent

       asinh, acosh, atanh

       The arcus cofunctions of the hyperbolic sine, cosine, and tangent
       (acsch/acosech and acoth/acotanh are aliases)

       acsch, acosech, asech, acoth, acotanh

       The trigonometric constant pi is also defined.

       $pi2 = 2 * pi;

       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 csc, cot, asec, acsc, acot, csch, coth, asech, acsch, the
       argument cannot be 0 (zero).  For the atanh, acoth, the argument cannot
       be 1 (one).  For the atanh, acoth, the argument cannot be -1 (minus
       one).  For the tan, sec, tanh, sech, the argument cannot be pi/2 + k *
       pi, where k is any integer.

       SIMPLE (REAL) ARGUMENTS, COMPLEX RESULTS

       Please note that some of the trigonometric functions can break out from
       the real axis into the complex plane. For example 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 the perldata manpage) as input for the
       trigonometric functions might produce as output results that no more
       are simple real numbers: instead they are complex numbers.

       The Math::Trig handles this by using the Math::Complex package which
       knows how to handle complex numbers, please see the Math::Complex
       manpage for more information. In practice you need not to worry about
       getting complex numbers as results because the 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 1.571 and
       the imaginary part of approximately -1.317.

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 pi radians or 360 degrees or 400 gradians.

RADIAL COORDINATE CONVERSIONS
       Radial coordinate systems are the spherical and the cylindrical
       systems, explained shortly in more detail.

       You can import radial coordinate conversion functions by using the
       :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);

       All angles are in radians.

       COORDINATE SYSTEMS

       Cartesian coordinates are the usual rectangular (x, y, z)-coordinates.

       Spherical coordinates, (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 rho, also known
       as the radial coordinate.  The angle in the xy-plane (around the
       z-axis) is theta, also known as the azimuthal coordinate.  The angle
       from the z-axis is phi, also known as the polar coordinate.  The `North
       Pole' is therefore 0, 0, rho, and the `Bay of Guinea' (think of the
       missing big chunk of Africa) 0, pi/2, rho.

       Beware: some texts define theta and phi the other way round, some texts
       define the phi to start from the horizontal plane, some texts use r in
       place of rho.

       Cylindrical coordinates, (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 rho, also
       known as the radial coordinate.  The angle in the xy-plane (around the
       z-axis) is theta, also known as the azimuthal coordinate.  The third
       coordinate is the z, pointing up from the theta-plane.

       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 pi angles being equal to -pi
       angles.

       cartesian_to_cylindrical

                   ($rho, $theta, $z) = cartesian_to_cylindrical($x, $y, $z);


       cartesian_to_spherical

                   ($rho, $theta, $phi) = cartesian_to_spherical($x, $y, $z);


       cylindrical_to_cartesian

                   ($x, $y, $z) = cylindrical_to_cartesian($rho, $theta, $z);


       cylindrical_to_spherical

                   ($rho_s, $theta, $phi) = cylindrical_to_spherical($rho_c, $theta, $z);

           Notice that when $z is not 0 $rho_s is not equal to $rho_c.

       spherical_to_cartesian

                   ($x, $y, $z) = spherical_to_cartesian($rho, $theta, $phi);


       spherical_to_cylindrical

                   ($rho_c, $theta, $z) = spherical_to_cylindrical($rho_s, $theta, $phi);

           Notice that when $z is not 0 $rho_c is not equal to $rho_s.

GREAT CIRCLE DISTANCES
       You can compute spherical distances, called great circle distances, by
       importing the great_circle_distance function:

               use Math::Trig 'great_circle_distance'

           $distance = great_circle_distance($theta0, $phi0, $theta1, $phi, [, $rho]);

       The great circle distance is the shortest distance between two points
       on a sphere.  The distance is in $rho units.  The $rho is optional, it
       defaults to 1 (the unit sphere), therefore the distance defaults to
       radians.

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 up to 0.3% because of the irregular (slightly
       aspherical) form of the Earth.

BUGS
       Saying use Math::Trig; exports many mathematical routines in the caller
       environment and even overrides some (sin, cos).  This is construed as a
       feature by the Authors, actually... ;-)

       The code is not optimized for speed, especially because we use
       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 asin(2) to give an answer
       instead of giving a fatal runtime error.

AUTHORS
       Jarkko Hietaniemi <jhi@iki.fi> and Raphael Manfredi
       <Raphael_Manfredi@grenoble.hp.com>.


























3rd Berkeley Distribution    perl 5.005, patch 02                Math::Trig(3)