MirOS Manual: Math::Trig(3p)


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

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);

             # Import constants pi2, pip2, pip4 (2*pi, pi/2, pi/4).
             use Math::Trig ':pi';

             # Import the conversions between cartesian/spherical/cylindrical.
             use Math::Trig ':radial';

             # Import the great circle formulas.
             use Math::Trig ':great_circle';

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 con-
     venience functions for angle conversions, and great circle
     formulas for spherical movement.

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)

perl v5.8.8                2005-02-05                           1

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

     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 situa-
     tions 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 ...

perl v5.8.8                2005-02-05                           2

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

     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.  atan2(0,
     0) is undefined.

     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 perldata) as
     input for the trigonometric functions might produce as out-
     put 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 Math::Complex 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 approxi-
     mately 1.571 and the imaginary part of approximately
     "-1.317".

PLANE ANGLE CONVERSIONS

     (Plane, 2-dimensional) angles may be converted with the fol-
     lowing functions.

             $radians  = deg2rad($degrees);
             $radians  = grad2rad($gradians);

perl v5.8.8                2005-02-05                           3

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

             $degrees  = rad2deg($radians);
             $degrees  = grad2deg($gradians);

             $gradians = deg2grad($degrees);
             $gradians = rad2grad($radians);

     The full circle is 2 pi radians or 360 degrees or 400 gradi-
     ans. 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().

RADIAL COORDINATE CONVERSIONS

     Radial coordinate systems are the spherical and the cylindr-
     ical 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 coor-
     dinate.  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 Gulf of
     Guinea (think of the missing big chunk of Africa) 0, pi/2,
     rho.  In geographical terms phi is latitude (northward posi-
     tive, southward negative) and theta is longitude (eastward
     positive, westward negative).

perl v5.8.8                2005-02-05                           4

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

     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 coordi-
     nates 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 AND DIRECTIONS

     You can compute spherical distances, called great circle
     distances, by importing the great_circle_distance() func-
     tion:

       use Math::Trig 'great_circle_distance';

       $distance = great_circle_distance($theta0, $phi0, $theta1, $phi1, [, $rho]);

perl v5.8.8                2005-02-05                           5

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

     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), there-
     fore the distance defaults to radians.

     If you think geographically the theta are longitudes: zero
     at the Greenwhich meridian, eastward positive, westward
     negative--and the phi are latitudes: zero at the North Pole,
     northward positive, southward negative.  NOTE: this formula
     thinks in mathematics, not geographically: the 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 geograph-
     ical coordinates from pi/2 (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 (also known
     as bearing) can be computed by the great_circle_direction()
     function:

       use Math::Trig 'great_circle_direction';

       $direction = great_circle_direction($theta0, $phi0, $theta1, $phi1);

     (Alias 'great_circle_bearing' is also available.) The result
     is in radians, zero indicating straight north, pi or -pi
     straight south, pi/2 straight west, and -pi/2 straight east.

     You can inversely compute the destination if you know the
     starting point, direction, and distance:

       use Math::Trig 'great_circle_destination';

       # thetad and phid are the destination coordinates,
       # dird is the final direction at the destination.

       ($thetad, $phid, $dird) =
         great_circle_destination($theta, $phi, $direction, $distance);

     or the midpoint if you know the end points:

       use Math::Trig 'great_circle_midpoint';

       ($thetam, $phim) =
         great_circle_midpoint($theta0, $phi0, $theta1, $phi1);

     The great_circle_midpoint() is just a special case of

       use Math::Trig 'great_circle_waypoint';

perl v5.8.8                2005-02-05                           6

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

       ($thetai, $phii) =
         great_circle_waypoint($theta0, $phi0, $theta1, $phi1, $way);

     Where the $way is a value from zero ($theta0, $phi0) to one
     ($theta1, $phi1).  Note that antipodal points (where their
     distance is pi radians) do not have waypoints between them
     (they would have an an "equator" between them), and there-
     fore "undef" is returned for antipodal points.  If the
     points are the same and the distance therefore zero and all
     waypoints therefore identical, the first point (either
     point) is returned.

     The thetas, phis, direction, and distance in the above are
     all in radians.

     You can import all the great circle formulas by

       use Math::Trig ':great_circle';

     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 pos-
     sible routes from Europe or North America to Asia do often
     cross the polar regions.

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.
             sub NESW { deg2rad($_[0]), deg2rad(90 - $_[1]) }
             my @L = NESW( -0.5, 51.3);
             my @T = NESW(139.8, 35.7);
             my $km = great_circle_distance(@L, @T, 6378); # About 9600 km.

     The direction you would have to go from London to Tokyo (in
     radians, straight north being zero, straight east being
     pi/2).

             use Math::Trig qw(great_circle_direction);

             my $rad = great_circle_direction(@L, @T); # About 0.547 or 0.174 pi.

     The midpoint between London and Tokyo being

             use Math::Trig qw(great_circle_midpoint);

             my @M = great_circle_midpoint(@L, @T);

perl v5.8.8                2005-02-05                           7

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

     or about 68.11N 24.74E, in the Finnish Lapland.

     CAVEAT FOR GREAT CIRCLE FORMULAS

     The answers may be off by few percentages because of the
     irregular (slightly aspherical) form of the Earth.  The
     errors are at worst about 0.55%, but generally below 0.3%.

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.

     Do not attempt navigation using these formulas.

AUTHORS

     Jarkko Hietaniemi <jhi@iki.fi> and Raphael Manfredi
     <Raphael_Manfredi@pobox.com>.

perl v5.8.8                2005-02-05                           8

Generated on 2014-07-04 21:17:45 by $MirOS: src/scripts/roff2htm,v 1.79 2014/02/10 00:36:11 tg Exp $

These manual pages and other documentation are copyrighted by their respective writers; their source is available at our CVSweb, AnonCVS, and other mirrors. The rest is Copyright © 2002‒2014 The MirOS Project, Germany.
This product includes material provided by Thorsten Glaser.

This manual page’s HTML representation is supposed to be valid XHTML/1.1; if not, please send a bug report – diffs preferred.