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

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 constantpiis also defined as are a few con- venience functions for angle conversions, andgreat circleformulasfor spherical movement.

The tangenttanThe cofunctions of the sine, cosine, and tangent (cosec/csc and cotan/cot are aliases)csc,cosec,sec,sec,cot,cotanThe arcus (also known as the inverse) functions of the sine, cosine, and tangentasin,acos,atanThe principal value of the arc tangent of y/xatan2(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,acotanThe hyperbolic sine, cosine, and tangentsinh,cosh,tanhThe cofunctions of the hyperbolic sine, cosine, and tangent (cosech/csch and cotanh/coth are aliases)csch,cosech,sech,coth,cotanhThe arcus (also known as the inverse) functions of the hyperbolic sine, cosine, and tangentasinh,acosh,atanhThe arcus cofunctions of the hyperbolic sine, cosine, and tangent (acsch/acosech and acoth/acotanh are aliases)acsch,acosech,asech,acoth,acotanhThe trigonometric constantpiis also defined. $pi2 = 2 *pi;ERRORS DUE TO DIVISION BY ZEROThe 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 bepi/2+k * pi, wherekis any integer. atan2(0, 0) is undefined.SIMPLE (REAL) ARGUMENTS, COMPLEX RESULTSPlease note that some of the trigonometric functions can break out from thereal axisinto thecomplex 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, 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 2piradians or360degrees or400gradi- 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 byrad2rad(),deg2deg(), andgrad2grad().

Radial coordinate systemsare thesphericaland thecylindr-icalsystems, 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, $phi);All angles are in radians.COORDINATE SYSTEMSCartesiancoordinates are the usual rectangular (x,y,z)-coordinates. Spherical coordinates, (rho,theta,pi), are three- dimensional coordinates which define a point in three- dimensional space. They are based on a sphere surface. The radius of the sphere isrho, also known as theradialcoor- dinate. The angle in thexy-plane (around thez-axis) istheta, also known as theazimuthalcoordinate. The angle from thez-axis isphi, also known as thepolarcoordinate. The North Pole is therefore0,0,rho, and the Gulf of Guinea (think of the missing big chunk of Africa)0,pi/2,rho. In geographical termsphiis latitude (northward posi- tive, southward negative) andthetais longitude (eastward positive, westward negative). perl v5.8.8 2005-02-05 4 Math::Trig(3p) Perl Programmers Reference Guide Math::Trig(3p)BEWARE: some texts definethetaandphithe other way round, some texts define thephito start from the horizontal plane, some texts userin place ofrho. Cylindrical coordinates, (rho,theta,z), are three- dimensional coordinates which define a point in three- dimensional space. They are based on a cylinder surface. The radius of the cylinder isrho, also known as theradialcoordinate. The angle in thexy-plane (around thez-axis) istheta, also known as theazimuthalcoordinate. The third coordinate is thez, pointing up from thetheta-plane.3-D ANGLE CONVERSIONSConversions to and from spherical and cylindrical coordi- nates are available. Please notice that the conversions are not necessarily reversible because of the equalities likepiangles being equal to-piangles. cartesian_to_cylindrical ($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.

You can compute spherical distances, calledgreat circledistances, by importing thegreat_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) Thegreat circle distanceis 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 thethetaare longitudes: zero at the Greenwhich meridian, eastward positive, westward negative--and thephiare latitudes: zero at the North Pole, northward positive, southward negative.NOTE: this formula thinks in mathematics, not geographically: thephizero 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 frompi/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 asbearing) can be computed by thegreat_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, $phi1); Thegreat_circle_midpoint() is just a special case of use Math::Trig 'great_circle_waypoint'; perl v5.8.8 2005-02-05 6 Math::Trig(3p) Perl Programmers Reference Guide Math::Trig(3p) ($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 ispiradians) 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.

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 FORMULASThe 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%.

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@iki.fi> and Raphael Manfredi <Raphael_Manfredi@pobox.com>. perl v5.8.8 2005-02-05 8

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