phpMan > man > Math::Trig

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.

ANGLES
    All angles are defined in radians, except where otherwise specified (for example in the deg/rad
    conversion functions).

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 *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().

    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 *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.

  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 (with $way = 0.5) 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.

SEE ALSO
    Math::Complex

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

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