MAN page from PLD perl-modules-5.6.1-71.i386.rpm

Math::Trig

Section: Perl Programmers Reference Guide (3pm)
Updated: perl v5.6.1
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);

 

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. 

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. 

SIMPLE (REAL) ARGUMENTS, COMPLEX RESULTS

Please note that some of the trigonometric functions can break outfrom the real axis into the complex plane. For example"asin(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 the perldata manpage) 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 the Math::Complex manpagefor 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 "1.571"and 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 `Bay 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

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);

 

EXAMPLES

To calculate the distance between London (51.3N 0.5W) and Tokyo (35.7N139.8E) in kilometers:

        use Math::Trig qw(great_circle_distance deg2rad);

        # Notice the 90 - latitude: phi zero is at the North Pole.        @L = (deg2rad(-0.5), deg2rad(90 - 51.3));        @T = (deg2rad(139.8),deg2rad(90 - 35.7));

        $km = great_circle_distance(@L, @T, 6378);

The answer may be off by few percentages because of the irregular(slightly aspherical) form of the Earth. The used formula

        lat0 = 90 degrees - phi0        lat1 = 90 degrees - phi1        d = R * arccos(cos(lat0) * cos(lat1) * cos(lon1 - lon01) +                       sin(lat0) * sin(lat1))

is also somewhat unreliable for small distances (for locationsseparated less than about five degrees) because it uses arc cosinewhich is rather ill-conditioned for values close to zero. 

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. 

AUTHORS

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

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Index

NAME

SYNOPSIS

DESCRIPTION

TRIGONOMETRIC FUNCTIONS

ERRORS DUE TO DIVISION BY ZERO

SIMPLE (REAL) ARGUMENTS, COMPLEX RESULTS

PLANE ANGLE CONVERSIONS

RADIAL COORDINATE CONVERSIONS

COORDINATE SYSTEMS

3-D ANGLE CONVERSIONS

GREAT CIRCLE DISTANCES

EXAMPLES

BUGS

AUTHORS

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