math::trig(3)

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 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.

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)

acsc, acosec, asec, acot, acotan

The hyperbolic sine, cosine, and tangent

sinh, cosh, tanh

The cofunctions of the hyperbolic sine, cosine, and tan
gent (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;

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.

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), "0;

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 approxi
mately 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 gra dians. 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_de

grees, 1);
$negative_degrees = rad2deg($negative_radians,

1);

You can also do the wrapping explicitly by rad2rad(), deg2deg(), and grad2grad().

RADIAL COORDINATE CONVERSIONS

Radial coordinate systems are the spherical and the cylin drical 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_carte

sian($rho, $theta, $z);
($rho_s, $theta, $phi) = cylindrical_to_spheri

cal($rho_c, $theta, $z);
($x, $y, $z) = spherical_to_cartesian($rho,

$theta, $phi);
($rho_c, $theta, $z) = spherical_to_cylindri

cal($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-dimen sional coordinates which define a point in three-dimen
sional 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 hori
zontal plane, some texts use r in place of rho.

Cylindrical coordinates, (rho, theta, z), are three-dimen sional coordinates which define a point in three-dimen
sional 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 coordi
nates 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_cylindri

cal($x, $y, $z);

cartesian_to_spherical

($rho, $theta, $phi) = cartesian_to_spheri

cal($x, $y, $z);

cylindrical_to_cartesian

($x, $y, $z) = cylindrical_to_cartesian($rho,

$theta, $z);

cylindrical_to_spherical

($rho_s, $theta, $phi) = cylindrical_to_spher

ical($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_cylindri

cal($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() func tion:

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);

The direction you must follow the great circle can be com
puted 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.

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.
@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);

CAVEAT FOR GREAT CIRCLE FORMULAS

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

BUGS

Saying "use Math::Trig;" exports many mathematical rou
tines 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 giv
ing a fatal runtime error.

AUTHORS

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

perl v5.8.0 2002-06-01 Math::Trig(3)