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

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

NNAAMMEE #

 Math::Trig - trigonometric functions

SSYYNNOOPPSSIISS #

     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';

DDEESSCCRRIIPPTTIIOONN #

 "Math::Trig" defines many trigonometric functions not defined by the core
 Perl which defines only the "sin()" and "cos()".  The constant ppii is also
 defined as are a few convenience functions for angle conversions, and
 _g_r_e_a_t _c_i_r_c_l_e _f_o_r_m_u_l_a_s for spherical movement.

TTRRIIGGOONNOOMMEETTRRIICC FFUUNNCCTTIIOONNSS #

 The tangent

 ttaann

 The cofunctions of the sine, cosine, and tangent (cosec/csc and cotan/cot
 are aliases)

 ccsscc, ccoosseecc, sseecc, sseecc, ccoott, ccoottaann

 The arcus (also known as the inverse) functions of the sine, cosine, and
 tangent

 aassiinn, aaccooss, aattaann

 The principal value of the arc tangent of y/x

 aattaann22(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.

 aaccsscc, aaccoosseecc, aasseecc, aaccoott, aaccoottaann

 The hyperbolic sine, cosine, and tangent

 ssiinnhh, ccoosshh, ttaannhh

 The cofunctions of the hyperbolic sine, cosine, and tangent (cosech/csch
 and cotanh/coth are aliases)

 ccsscchh, ccoosseecchh, sseecchh, ccootthh, ccoottaannhh

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

 aassiinnhh, aaccoosshh, aattaannhh

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

 aaccsscchh, aaccoosseecchh, aasseecchh, aaccootthh, aaccoottaannhh

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

 ppii,, ppii22,, ppii44,, ppiipp22,, ppiipp44

EERRRROORRSS DDUUEE TTOO DDIIVVIISSIIOONN BBYY ZZEERROO #

 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 _p_i_/_2 _+ _k _* _p_i, where _k is any integer.

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

SSIIMMPPLLEE ((RREEAALL)) AARRGGUUMMEENNTTSS,, CCOOMMPPLLEEXX RREESSUULLTTSS #

 Please note that some of the trigonometric functions can break out from
 the rreeaall aaxxiiss into the ccoommpplleexx ppllaannee. 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".

PPLLAANNEE AANNGGLLEE CCOONNVVEERRSSIIOONNSS #

 (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 _p_i radians or _3_6_0 degrees or _4_0_0 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 rraadd22rraadd(()), ddeegg22ddeegg(()), and
 ggrraadd22ggrraadd(()).

 rad2rad
         $radians_wrapped_by_2pi = rad2rad($radians);

 deg2deg
         $degrees_wrapped_by_360 = deg2deg($degrees);

 grad2grad
         $gradians_wrapped_by_400 = grad2grad($gradians);

RRAADDIIAALL CCOOOORRDDIINNAATTEE CCOONNVVEERRSSIIOONNSS #

 RRaaddiiaall ccoooorrddiinnaattee ssyysstteemmss are the sspphheerriiccaall and the ccyylliinnddrriiccaall 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);

 AAllll aanngglleess aarree iinn rraaddiiaannss.

CCOOOORRDDIINNAATTEE SSYYSSTTEEMMSS #

 CCaarrtteessiiaann coordinates are the usual rectangular _(_x_, _y_, _z_)-coordinates.

 Spherical coordinates, _(_r_h_o_, _t_h_e_t_a_, _p_i_), 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 rrhhoo, also known
 as the _r_a_d_i_a_l coordinate.  The angle in the _x_y-plane (around the _z-axis)
 is tthheettaa, also known as the _a_z_i_m_u_t_h_a_l coordinate.  The angle from the
 _z-axis is pphhii, also known as the _p_o_l_a_r coordinate.  The North Pole is
 therefore _0_, _0_, _r_h_o, and the Gulf of Guinea (think of the missing big
 chunk of Africa) _0_, _p_i_/_2_, _r_h_o.  In geographical terms _p_h_i is latitude
 (northward positive, southward negative) and _t_h_e_t_a is longitude (eastward
 positive, westward negative).

 BBEEWWAARREE: some texts define _t_h_e_t_a and _p_h_i the other way round, some texts
 define the _p_h_i to start from the horizontal plane, some texts use _r in
 place of _r_h_o.

 Cylindrical coordinates, _(_r_h_o_, _t_h_e_t_a_, _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 rrhhoo, also
 known as the _r_a_d_i_a_l coordinate.  The angle in the _x_y-plane (around the
 _z-axis) is tthheettaa, also known as the _a_z_i_m_u_t_h_a_l coordinate.  The third
 coordinate is the _z, pointing up from the tthheettaa-plane.

33--DD AANNGGLLEE CCOONNVVEERRSSIIOONNSS #

 Conversions to and from spherical and cylindrical coordinates are
 available.  Please notice that the conversions are not necessarily
 reversible because of the equalities like _p_i angles being equal to _-_p_i
 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.

GGRREEAATT CCIIRRCCLLEE DDIISSTTAANNCCEESS AANNDD DDIIRREECCTTIIOONNSS #

 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.

ggrreeaatt__cciirrccllee__ddiissttaannccee You can compute spherical distances, called ggrreeaatt cciirrccllee ddiissttaanncceess, by importing the ggrreeaatt__cciirrccllee__ddiissttaannccee(()) function:

   use Math::Trig 'great_circle_distance';

   $distance = great_circle_distance($theta0, $phi0, $theta1, $phi1, [, $rho]);

 The _g_r_e_a_t _c_i_r_c_l_e _d_i_s_t_a_n_c_e 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 _t_h_e_t_a are longitudes: zero at the
 Greenwhich meridian, eastward positive, westward negative -- and the _p_h_i
 are latitudes: zero at the North Pole, northward positive, southward
 negative.  NNOOTTEE: this formula thinks in mathematics, not geographically:
 the _p_h_i 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 _p_i_/_2 (also known as 90 degrees).

   $distance = great_circle_distance($lon0, pi/2 - $lat0,
                                     $lon1, pi/2 - $lat1, $rho);

ggrreeaatt__cciirrccllee__ddiirreeccttiioonn The direction you must follow the great circle (also known as _b_e_a_r_i_n_g) can be computed by the ggrreeaatt__cciirrccllee__ddiirreeccttiioonn(()) function:

   use Math::Trig 'great_circle_direction';

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

ggrreeaatt__cciirrccllee__bbeeaarriinngg 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.

ggrreeaatt__cciirrccllee__ddeessttiinnaattiioonn 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:

ggrreeaatt__cciirrccllee__mmiiddppooiinntt use Math::Trig ‘great_circle_midpoint’;

   ($thetam, $phim) =
     great_circle_midpoint($theta0, $phi0, $theta1, $phi1);

 The ggrreeaatt__cciirrccllee__mmiiddppooiinntt(()) is just a special case of

ggrreeaatt__cciirrccllee__wwaayyppooiinntt 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 _p_i 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 nnoott show
 great circles as straight lines: straight lines in the Mercator
 projection are lines of constant bearing.)

EEXXAAMMPPLLEESS #

 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.

 NNOOTTEE: you ccaannnnoott 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 ggrreeaatt__cciirrccllee__ddeessttiinnaattiioonn(()) one gives a
 ccoonnssttaanntt bearing to follow.

CCAAVVEEAATT FFOORR GGRREEAATT CCIIRRCCLLEE FFOORRMMUULLAASS #

 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%.

RReeaall--vvaalluueedd aassiinn aanndd aaccooss For small inputs aassiinn(()) and aaccooss(()) 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 aassiinn(()) and aaccooss(()) 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],
     iinncclluussiivvee 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],
     iinncclluussiivvee the endpoints.  For inputs greater than one, zero is
     returned.  For inputs less than minus one, pi is returned.

BBUUGGSS #

 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

AAUUTTHHOORRSS #

 Jarkko Hietaniemi <_j_h_i_!_a_t_!_i_k_i_._f_i>, Raphael Manfredi
 <_R_a_p_h_a_e_l___M_a_n_f_r_e_d_i_!_a_t_!_p_o_b_o_x_._c_o_m>, Zefram <zefram@fysh.org>

LLIICCEENNSSEE #

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

perl v5.36.3 2013-03-25 Math::Trig(3p)