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



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 convenience 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).
       Note that atan2(0, 0) is not well-defined.

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

       asinh, acosh, atanh

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

       acsch, acosech, asech, acoth, acotanh

       The trigonometric constant pi and some of handy multiples of it are also defined.

       pi, pi2, pi4, pip2, pip4

   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.

       Note that atan2(0, 0) is not well-defined.

   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 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.

       deg2rad
               $radians  = deg2rad($degrees);

       grad2rad
               $radians  = grad2rad($gradians);

       rad2deg
               $degrees  = rad2deg($radians);

       grad2deg
               $degrees  = grad2deg($gradians);

       deg2grad
               $gradians = deg2grad($degrees);

       rad2grad
               $gradians = rad2grad($radians);

       The full circle is 2 pi radians or 36_ degrees or 4__ 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().

       rad2rad
               $radians_wrapped_by_2pi = rad2rad($radians);

       deg2deg
               $degrees_wrapped_by_360 = deg2deg($degrees);

       grad2grad
               $gradians_wrapped_by_400 = grad2grad($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 _, _, rho, and the Gulf of Guinea (think of the missing big chunk of
       Africa) _, pi/2, rho.  In geographical terms phi is latitude (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 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 AND DIRECTIONS
       A great circle is section of a circle that contains the circle diameter: the shortest distance
       between two (non-antipodal) points on the spherical surface goes along the great circle connecting
       those two points.

   great_circle_distance
       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, $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);

   great_circle_direction
       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);

   great_circle_bearing
       Alias 'great_circle_bearing' for 'great_circle_direction' is also available.

         use Math::Trig 'great_circle_bearing';

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

       The result of great_circle_direction is in radians, zero indicating straight north, pi or -pi
       straight south, pi/2 straight west, and -pi/2 straight east.

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

         use Math::Trig 'great_circle_destination';

         # $diro is the original direction,
         # for example from great_circle_bearing().
         # $distance is the angular distance in radians,
         # for example from great_circle_distance().
         # $thetad and $phid are the destination coordinates,
         # $dird is the final direction at the destination.

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

       or the midpoint if you know the end points:

   great_circle_midpoint
         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

   great_circle_waypoint
         use Math::Trig 'great_circle_waypoint';

         ($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 therefore "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 possible routes from Europe or North America to Asia do often cross
       the polar regions.  (The common Mercator projection does not show great circles as straight lines:
       straight lines in the Mercator projection are lines of constant bearing.)

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);

       or about 69 N 89 E, in the frozen wastes of Siberia.

       NOTE: you cannot get from A to B like this:

          Dist = great_circle_distance(A, B)
          Dir  = great_circle_direction(A, B)
          C    = great_circle_destination(A, Dist, Dir)

       and expect C to be B, because the bearing constantly changes when going from A to B (except in some
       special case like the meridians or the circles of latitudes) and in great_circle_destination() one
       gives a constant bearing to follow.

   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%.

   Real-valued asin and acos
       For small inputs asin() and acos() may return complex numbers even when real numbers would be enough
       and correct, this happens because of floating-point inaccuracies.  You can see these inaccuracies for
       example by trying theses:

         print cos(1e-6)**2+sin(1e-6)**2 - 1,"\n";
         printf "%.20f", cos(1e-6)**2+sin(1e-6)**2,"\n";

       which will print something like this

         -1.11022302462516e-16
         0.99999999999999988898

       even though the expected results are of course exactly zero and one.  The formulas used to compute
       asin() and acos() are quite sensitive to this, and therefore they might accidentally slip into the
       complex plane even when they should not.  To counter this there are two interfaces that are
       guaranteed to return a real-valued output.

       asin_real
               use Math::Trig qw(asin_real);

               $real_angle = asin_real($input_sin);

           Return a real-valued arcus sine if the input is between [-1, 1], inclusive the endpoints.  For
           inputs greater than one, pi/2 is returned.  For inputs less than minus one, -pi/2 is returned.

       acos_real
               use Math::Trig qw(acos_real);

               $real_angle = acos_real($input_cos);

           Return a real-valued arcus cosine if the input is between [-1, 1], inclusive the endpoints.  For
           inputs greater than one, zero is returned.  For inputs less than minus one, pi is returned.

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.

       Math::Complex

AUTHORS
       Jarkko Hietaniemi <jhi!at!iki.fi>, Raphael Manfredi <Raphael_Manfredi!at!pobox.com>, Zefram
       <zefram@fysh.org>

LICENSE
       This library is free software; you can redistribute it and/or modify it under the same terms as Perl
       itself.



perl v5.16.2                                     2012-10-11                                  Math::Trig(3pm)