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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 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;
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 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. 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 cylindrical 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, $phi);
All angles are in radians.
COORDINATE SYSTEMS
Cartesian coordinates are the usual rectangular (x, y, z)-coordinates.
Spherical coordinates, (rho, theta, pi), are three-dimensional coordinates
which define a point in three-dimensional space. They are based on a
sphere surface. The radius of the sphere is rho, also known as the radial
coordinate. The angle in the xy-plane (around the z-axis) is theta, also
known as the azimuthal coordinate. The angle from the z-axis is phi, also
known as the polar coordinate. The `North Pole' is therefore 0, 0, rho,
and the `Bay of Guinea' (think of the missing big chunk of Africa) 0, pi/2,
rho. In geographical terms phi is latitude (northward positive, southward
negative) and theta is longitude (eastward positive, westward negative).
BEWARE: some texts define theta and phi the other way round, some texts
define the phi to start from the horizontal plane, some texts use r in
place of rho.
Cylindrical coordinates, (rho, theta, z), are three-dimensional coordinates
which define a point in three-dimensional space. They are based on a
cylinder surface. The radius of the cylinder is rho, also known as the
radial coordinate. The angle in the xy-plane (around the z-axis) is theta,
also known as the azimuthal coordinate. The third coordinate is the z,
pointing up from the theta-plane.
3-D ANGLE CONVERSIONS
Conversions to and from spherical and cylindrical coordinates are
available. Please notice that the conversions are not necessarily
reversible because of the equalities like pi angles being equal to -pi
angles.
cartesian_to_cylindrical
($rho, $theta, $z) = cartesian_to_cylindrical($x, $y, $z);
cartesian_to_spherical
($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).
$distance = great_circle_distance($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 locations 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 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@pobox.com>.
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