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);
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 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 num
bers 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 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_degrees, 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, $z) = cylindrical_to_cartesian($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, $phi);
All angles are in radians.
COORDINATE SYSTEMS
Cartesian coordinates are the usual rectangular (x, y,
z)-coordinates.
Spherical coordinates, (rho, theta, pi), are three-dimen
sional coordinates which define a point in three-dimen
sional space. They are based on a sphere surface. The
radius of the sphere is rho, also known as the radial
coordinate. The angle in the xy-plane (around the z-axis)
is theta, also known as the azimuthal coordinate. The
angle from the z-axis is phi, also known as the polar
coordinate. The `North Pole' is therefore 0, 0, rho, and
the `Bay of Guinea' (think of the missing big chunk of
Africa) 0, pi/2, rho. In geographical terms phi is lati
tude (northward positive, southward negative) and theta is
longitude (eastward positive, westward negative).
BEWARE: some texts define theta and phi the other way
round, some texts define the phi to start from the hori
zontal plane, some texts use r in place of rho.
Cylindrical coordinates, (rho, theta, z), are three-dimen
sional coordinates which define a point in three-dimen
sional space. They are based on a cylinder surface. The
radius of the cylinder is rho, also known as the radial
coordinate. The angle in the xy-plane (around the z-axis)
is theta, also known as the azimuthal coordinate. The
third coordinate is the z, pointing up from the
theta-plane.
3-D ANGLE CONVERSIONS
Conversions to and from spherical and cylindrical coordi
nates are available. Please notice that the conversions
are not necessarily reversible because of the equalities
like pi angles being equal to -pi angles.
cartesian_to_cylindrical
($rho, $theta, $z) = cartesian_to_cylindrical($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" 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);
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. 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 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
giving a fatal runtime error.
AUTHORS
Jarkko Hietaniemi <jhi@iki.fi> and Raphael Manfredi
<Raphael_Manfredi@pobox.com>.
2001-02-22 perl v5.6.1 Math::Trig(3)
