Math::Trig(3) Perl Programmers Reference Guide Math::Trig(3)NAMEMath::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);
DESCRIPTIONMath::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
tangent (cosech/csch and cotanh/coth are aliases)
csch, cosech, sech, coth, cotanh
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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.
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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 the perldata
manpage) 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 the Math::Complex manpage 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.
$radians = deg2rad($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.
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:
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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, $phi);
All angles are in radians.
COORDINATE SYSTEMS
Cartesian coordinates are the usual rectangular (x, y,
z)-coordinates.
Spherical coordinates, (rho, theta, pi), are three-
dimensional coordinates which define a point in three-
dimensional space. They are based on a sphere surface.
The radius of the sphere is rho, also known as the radial
coordinate. The angle in the xy-plane (around the z-axis)
is theta, also known as the azimuthal coordinate. The
angle from the z-axis is phi, also known as the polar
coordinate. The `North Pole' is therefore 0, 0, rho, and
the `Bay of Guinea' (think of the missing big chunk of
Africa) 0, pi/2, rho. In geographical terms phi is
latitude (northward positive, southward negative) and
theta is longitude (eastward positive, westward negative).
BEWARE: some texts define theta and phi the other way
round, some texts define the phi to start from the
horizontal plane, some texts use r in place of rho.
Cylindrical coordinates, (rho, theta, z), are three-
dimensional coordinates which define a point in three-
dimensional space. They are based on a cylinder surface.
The radius of the cylinder is rho, also known as the
radial coordinate. The angle in the xy-plane (around the
z-axis) is theta, also known as the azimuthal coordinate.
The third coordinate is the z, pointing up from the
theta-plane.
3-D ANGLE CONVERSIONS
Conversions to and from spherical and cylindrical
coordinates are available. Please notice that the
conversions are not necessarily reversible because of the
equalities like pi angles being equal to -pi angles.
cartesian_to_cylindrical
($rho, $theta, $z) = cartesian_to_cylindrical($x, $y, $z);
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cartesian_to_spherical
($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
two points on a sphere. The distance is in $rho units.
The $rho is optional, it defaults to 1 (the unit sphere),
therefore the distance defaults to radians.
If you think geographically the theta are longitudes: zero
at the Greenwhich meridian, eastward positive, westward
negative--and the phi are latitudes: zero at the North
Pole, northward positive, southward negative. NOTE: this
formula thinks in mathematics, not geographically: the phi
zero is at the North Pole, not at the Equator on the west
coast of Africa (Bay of Guinea). You need to subtract
your geographical coordinates from pi/2 (also known as 90
degrees).
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$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 (35.7N 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 answer may be off by few percentages because of the
irregular (slightly aspherical) form of the Earth.
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.
AUTHORS
Jarkko Hietaniemi <jhi@iki.fi> and Raphael Manfredi
<Raphael_Manfredi@grenoble.hp.com>.
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