Math::Trig - trigonometric functions

        use Math::Trig;

        $x = tan(0.9);
        $y = acos(3.7);
        $z = asin(2.4);

        $halfpi = pi/2;

        $rad = deg2rad(1);

CWMath::Trig defines many trigonometric functions not defined by the core Perl which defines only the CWsin() and CWcos(). The constant pi is also defined as are a few convenience functions for angle conversions.

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)

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 arcus (also known as the inverse) functions of the hyperbolic sine, 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;

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 CWcsc, CWcot, CWasec, CWacsc, CWacot, CWcsch, CWcoth, CWasech, CWacsch, the argument cannot be CW0 (zero). For the CWatanh, CWacoth, the argument cannot be CW1 (one). For the CWatanh, CWacoth, the argument cannot be CW-1 (minus one). For the CWtan, CWsec, CWtanh, CWsech, the argument cannot be pi/2 + k * pi, where k is any integer.

Please note that some of the trigonometric functions can break out from the real axis into the complex plane. For example CWasin(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 CWMath::Trig handles this by using the CWMath::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 CWMath::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 CW1.571 and the imaginary part of approximately CW-1.317.

(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(), and grad2grad().

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 CW: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.

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 `Bay 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.

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 CW$z is not 0 CW$rho_s is not equal to CW$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 CW$z is not 0 CW$rho_c is not equal to CW$rho_s.

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 CW$rho units. The CW$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 negativeand 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);

The direction you must follow the great circle can be computed by the great_circle_direction() function:

  use Math::Trig 'great_circle_direction';

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

The result is in radians, zero indicating straight north, pi or -pi straight south, pi/2 straight west, and -pi/2 straight east.

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.

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.
        @L = (deg2rad(-0.5), deg2rad(90 - 51.3));
        @T = (deg2rad(139.8),deg2rad(90 - 35.7));

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

The direction you would have to go from London to Tokyo

        use Math::Trig qw(great_circle_direction);

        $rad = great_circle_direction(@L, @T);

The answers may be off by few percentages because of the irregular (slightly aspherical) form of the Earth. The formula used for grear circle distances

        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 locations separated less than about five degrees) because it uses arc cosine which is rather ill-conditioned for values close to zero.

Saying CWuse Math::Trig; exports many mathematical routines in the caller environment and even overrides some (CWsin, CWcos). This is construed as a feature by the Authors, actually... ;-)

The code is not optimized for speed, especially because we use CWMath::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 CWasin(2) to give an answer instead of giving a fatal runtime error.

Jarkko Hietaniemi <jhi@iki.fi> and Raphael Manfredi <Raphael_Manfredi@pobox.com>.