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Math::Trig(3) Perl Programmers Reference Guide Math::Trig(3)NAMEMath::Trig - trigonometric functionsSYNOPSISuse 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 constantpiis also defined as are a few convenience functions for angle conversions, andgreatcircleformulasfor spherical movement.TRIGONOMETRICFUNCTIONSThe 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)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;ERRORSDUETODIVISIONBYZEROThe 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 can- not 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,COMPLEXRESULTSPlease note that some of the trigonometric functions can break out from therealaxisinto thecomplexplane. 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".PLANEANGLECONVERSIONS(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 2piradians or360degrees or400gradians. 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().RADIALCOORDINATECONVERSIONSRadialcoordinatesystemsare thesphericaland thecylindricalsystems, 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, $phi);Allanglesareinradians.COORDINATESYSTEMSCartesiancoordinates are the usual rectangular(x,y,z)-coordinates. Spherical coordinates,(rho,theta,pi), are three-dimensional coordinates which define a point in three-dimen- sional space. They are based on a sphere surface. The radius of the sphere isrho, also known as theradialcoordinate. The angle in thexy-plane (around thez-axis) istheta, also known as theazimuthalcoordinate. The angle from thez-axis isphi, also known as thepolarcoordinate. The North Pole is therefore0,0,rho, and the Gulf of Guinea (think of the missing big chunk of Africa)0,pi/2,rho. In geographical termsphiis latitude (northward positive, southward negative) andthetais longitude (eastward positive, westward nega- tive).BEWARE: some texts definethetaandphithe other way round, some texts define thephito start from the hori- zontal plane, some texts userin place ofrho. Cylindrical coordinates,(rho,theta,z), are three-dimensional coordinates which define a point in three- dimensional space. They are based on a cylinder surface. The radius of the cylinder isrho, also known as theradialcoordinate. The angle in thexy-plane (around thez-axis) istheta, also known as theazimuthalcoordi- nate. The third coordinate is thez, pointing up from thetheta-plane.3-DANGLECONVERSIONSConversions to and from spherical and cylindrical coordinates are available. Please notice that the conver- sions are not necessarily reversible because of the equalities likepiangles being equal to-piangles. cartesian_to_cylindrical ($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.GREATCIRCLEDISTANCESANDDIRECTIONSYou can compute spherical distances, calledgreatcircledistances, by importing thegreat_circle_distance()function: use Math::Trig 'great_circle_distance'; $distance = great_circle_distance($theta0, $phi0, $theta1, $phi1, [, $rho]); Thegreatcircledistanceis 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 thethetaare longitudes: zero at the Greenwhich meridian, eastward positive, west- ward 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 geographical 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_cir-cle_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, $phi1); Thegreat_circle_midpoint()is just a special case of use Math::Trig 'great_circle_waypoint'; ($thetai, $phii) = great_circle_waypoint($theta0, $phi0, $theta1, $phi1, $way); Where the $way is a value from zero ($theta0, $phi0) to one ($theta1, $phi1). Note that antipodal points (where their distance ispiradians) do not have waypoints between them (they would have an an "equator" between them), and therefore "undef" is returned for antipodal points. If the points are the same and the dis- tance therefore zero and all waypoints therefore identical, the first point (either point) is returned. The thetas, phis, direction, and distance in the above are all in radians. You can import all the great circle formulas by use Math::Trig ':great_circle'; Notice that the resulting directions might be somewhat surprising if you are looking at a flat worldmap: in such map projections the great circles quite often do not look like the shortest routes-- but for example the shortest possible routes from Europe or North America to Asia do often cross the polar regions.EXAMPLESTo calculate the distance between London (51.3N 0.5W) and Tokyo (35.7N 139.8E) in kilometers: use Math::Trig qw(great_circle_distance deg2rad); # Notice the 90 - latitude: phi zero is at the North Pole. sub NESW { deg2rad($_[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 68.11N 24.74E, in the Finnish Lapland.CAVEATFORGREATCIRCLEFORMULASThe 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%.BUGSSaying "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.AUTHORSJarkko Hietaniemi <jhi> and Raphael Manfredi <ATiki.fiRaphael_Manfredi>. perl v5.8.8 2001-09-21 Math::Trig(3)ATpobox.com