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

       Math::Trig - trigonometric functions

	   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';

       "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

       The tangent


       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

       The following functions


       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 ...


	   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.

       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):


       That is, a complex number with the real part of approximately 1.571 and the imaginary part
       of approximately "-1.317".

       (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.	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().

	       $radians_wrapped_by_2pi = rad2rad($radians);

	       $degrees_wrapped_by_360 = deg2deg($degrees);

	       $gradians_wrapped_by_400 = grad2grad($gradians);

       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.

       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
       Gulf of Guinea (think of the missing big chunk of Africa) 0, 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.

       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.

	       ($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);

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

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

	       ($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.

       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.

       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);

       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' 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.

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

	 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:

	 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';

	 ($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.)

       To calculate the distance between London (51.3N 0.5W) and Tokyo (35.7N 139.8E) in

	   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.

       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


       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.

	       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.

	       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.

       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.


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

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

perl v5.16.3				    2013-03-04				  Math::Trig(3pm)
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