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MAN page from Mandriva  2007 perl-5.8.8-7.1mdv2007.0.x86_64.rpm

Math::Trig

Section: Perl Programmers Reference Guide (3pm)
Updated: 2001-09-21
Index 

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 thecore Perl which defines only the "sin()" and "cos()". The constantpi is also defined as are a few convenience functions for angleconversions, and great circle formulas for spherical movement. 

TRIGONOMETRIC FUNCTIONS

The tangent
tan

The cofunctions of the sine, cosine, and tangent (cosec/csc and cotan/cotare 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/acscand acotan/acot are aliases)

acsc, acosec, asec, acot, acotan

The hyperbolic sine, cosine, and tangent

sinh, cosh, tanh

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

csch, cosech, sech, coth, cotanh

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

asinh, acosh, atanh

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

acsch, acosech, asech, acoth, acotanh

The trigonometric constant pi is also defined.

$pi2 = 2 * pi;

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 dividingby zero or taking logarithm of zero. These situations cause fatalruntime 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. atan2(0, 0) is undefined.

SIMPLE (REAL) ARGUMENTS, COMPLEX RESULTS

Please note that some of the trigonometric functions can break outfrom the real axis into the complex plane. For exampleasin(2) has no definition for plain real numbers but it hasdefinition for complex numbers.

In Perl terms this means that supplying the usual Perl numbers (alsoknown as scalars, please see perldata) as input for thetrigonometric functions might produce as output results that no moreare simple real numbers: instead they are complex numbers.

The "Math::Trig" handles this by using the "Math::Complex" packagewhich knows how to handle complex numbers, please see Math::Complexfor more information. In practice you need not to worry about gettingcomplex numbers as results because the "Math::Complex" takes care ofdetails 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 1.571and the imaginary part of approximately "-1.317". 

PLANE ANGLE CONVERSIONS

(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(), andgrad2grad(). 

RADIAL COORDINATE CONVERSIONS

Radial coordinate systems are the spherical and the cylindricalsystems, 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-dimensionalcoordinates which define a point in three-dimensional space. They arebased on a sphere surface. The radius of the sphere is rho, alsoknown as the radial coordinate. The angle in the xy-plane(around the z-axis) is theta, also known as the azimuthalcoordinate. The angle from the z-axis is phi, also known as thepolar coordinate. The North Pole is therefore 0, 0, rho, andthe Gulf of Guinea (think of the missing big chunk of Africa) 0,pi/2, rho. In geographical terms phi is latitude (northwardpositive, southward negative) and theta is longitude (eastwardpositive, westward negative).

BEWARE: some texts define theta and phi the other way round,some texts define the phi to start from the horizontal plane, sometexts use r in place of rho.

Cylindrical coordinates, (rho, theta, z), are three-dimensionalcoordinates which define a point in three-dimensional space. They arebased 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 azimuthalcoordinate. The third coordinate is the z, pointing up from thetheta-plane.

3-D ANGLE CONVERSIONS

Conversions to and from spherical and cylindrical coordinates areavailable. Please notice that the conversions are not necessarilyreversible 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

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 twopoints on a sphere. The distance is in $rho units. The $rho isoptional, it defaults to 1 (the unit sphere), therefore the distancedefaults to radians.

If you think geographically the theta are longitudes: zero at theGreenwhich meridian, eastward positive, westward negative---and thephi are latitudes: zero at the North Pole, northward positive,southward negative. NOTE: this formula thinks in mathematics, notgeographically: the phi zero is at the North Pole, not at theEquator on the west coast of Africa (Bay of Guinea). You need tosubtract your geographical coordinates from pi/2 (also known as 90degrees).

  $distance = great_circle_distance($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' is also available.)The result is in radians, zero indicating straight north, pi or -pistraight south, pi/2 straight west, and -pi/2 straight east.

You can inversely compute the destination if you know thestarting 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);

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 piradians) do not have waypoints between them (they would have an an``equator'' between them), and therefore "undef" is returned forantipodal points. If the points are the same and the distancetherefore 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 ifyou are looking at a flat worldmap: in such map projections the greatcircles quite often do not look like the shortest routes--- but forexample the shortest possible routes from Europe or North America toAsia do often cross the polar regions. 

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 68.11N 24.74E, in the Finnish Lapland.

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 worstabout 0.55%, but generally below 0.3%. 

BUGS

Saying "use Math::Trig;" exports many mathematical routines in thecaller environment and even overrides some ("sin", "cos"). This isconstrued 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 doingthe computations even when the arguments are not. This, however,cannot be completely avoided if we want things like asin(2) to givean answer instead of giving a fatal runtime error.

Do not attempt navigation using these formulas. 

AUTHORS

Jarkko Hietaniemi <jhiAATTiki.fi> and Raphael Manfredi <Raphael_ManfrediAATTpobox.com>.

 

Index

NAME
SYNOPSIS
DESCRIPTION
TRIGONOMETRIC FUNCTIONS
PLANE ANGLE CONVERSIONS
RADIAL COORDINATE CONVERSIONS
GREAT CIRCLE DISTANCES AND DIRECTIONS
EXAMPLES
BUGS
AUTHORS
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