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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);

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 tan-
       gent (cosech/csch and cotanh/coth are aliases)

       csch, cosech, sech, coth, cotanh

       The arcus (also known as the inverse) functions of the



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


       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.






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


       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
       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 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
       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 gra-
       dians.  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().



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RADIAL COORDINATE CONVERSIONS
       Radial coordinate systems are the spherical and the cylin-
       drical 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-dimen-
       sional coordinates which define a point in three-dimen-
       sional 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.  In geographical terms phi is lati-
       tude (northward positive, southward negative) and theta is
       longitude (eastward positive, westward negative).

       BEWARE: some texts define theta and phi the other way
       round, some texts define the phi to start from the hori-
       zontal plane, some texts use r in place of rho.

       Cylindrical coordinates, (rho, theta, z), are three-dimen-
       sional coordinates which define a point in three-dimen-
       sional 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



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       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]);

       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.

       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 geographical 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 can be



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


       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.

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);

       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 loca-
       tions separated less than about five degrees) because it
       uses arc cosine which is rather ill-conditioned for values
       close to zero.

BUGS
       Saying "use Math::Trig;" exports many mathematical rou-
       tines in the caller environment and even overrides some



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       ("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 giv-
       ing a fatal runtime error.

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












































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