Math::Trig(3p) Perl Programmers Reference Guide Math::Trig(3p)
Math::Trig - trigonometric functions
use Math::Trig;
$x = tan(0.9);
$y = acos(3.7);
$z = asin(2.4);
$halfpi = pi/2;
$rad = deg2rad(120);
# Import constants pi2, pip2, pip4 (2*pi, pi/2, pi/4).
use Math::Trig ':pi';
# Import the conversions between cartesian/spherical/cylindrical.
use Math::Trig ':radial';
# Import the great circle formulas.
use Math::Trig ':great_circle';
"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 con-
venience functions for angle conversions, and great circle
formulas for spherical movement.
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)
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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 situa-
tions 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 ...
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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. atan2(0,
0) is undefined.
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 out-
put 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), "\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, 2-dimensional) angles may be converted with the fol-
lowing functions.
$radians = deg2rad($degrees);
$radians = grad2rad($gradians);
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$degrees = rad2deg($radians);
$degrees = grad2deg($gradians);
$gradians = deg2grad($degrees);
$gradians = rad2grad($radians);
The full circle is 2 pi radians or 360 degrees or 400 gradi-
ans. 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 cylindr-
ical 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, $phi);
All angles are in radians.
COORDINATE SYSTEMS
Cartesian coordinates are the usual rectangular (x, y,
z)-coordinates.
Spherical coordinates, (rho, theta, pi), are three-
dimensional coordinates which define a point in three-
dimensional space. They are based on a sphere surface. The
radius of the sphere is rho, also known as the radial coor-
dinate. The angle in the xy-plane (around the z-axis) is
theta, also known as the azimuthal coordinate. The angle
from the z-axis is phi, also known as the polar coordinate.
The North Pole is therefore 0, 0, rho, and the Gulf of
Guinea (think of the missing big chunk of Africa) 0, pi/2,
rho. In geographical terms phi is latitude (northward posi-
tive, southward negative) and theta is longitude (eastward
positive, westward negative).
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BEWARE: some texts define theta and phi the other way round,
some texts define the phi to start from the horizontal
plane, some texts use r in place of rho.
Cylindrical coordinates, (rho, theta, z), are three-
dimensional coordinates which define a point in three-
dimensional space. They are based on a cylinder surface.
The radius of the cylinder is rho, also known as the radial
coordinate. The angle in the xy-plane (around the z-axis)
is theta, also known as the azimuthal coordinate. The third
coordinate is the z, pointing up from the theta-plane.
3-D ANGLE CONVERSIONS
Conversions to and from spherical and cylindrical coordi-
nates are available. Please notice that the conversions are
not necessarily reversible because of the equalities like pi
angles being equal to -pi angles.
cartesian_to_cylindrical
($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.
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]);
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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), there-
fore 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 geograph-
ical 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 (also known
as bearing) can be computed by the great_circle_direction()
function:
use Math::Trig 'great_circle_direction';
$direction = great_circle_direction($theta0, $phi0, $theta1, $phi1);
(Alias 'great_circle_bearing' is also available.) The result
is in radians, zero indicating straight north, pi or -pi
straight south, pi/2 straight west, and -pi/2 straight east.
You can inversely compute the destination if you know the
starting point, direction, and distance:
use Math::Trig 'great_circle_destination';
# thetad and phid are the destination coordinates,
# dird is the final direction at the destination.
($thetad, $phid, $dird) =
great_circle_destination($theta, $phi, $direction, $distance);
or the midpoint if you know the end points:
use Math::Trig 'great_circle_midpoint';
($thetam, $phim) =
great_circle_midpoint($theta0, $phi0, $theta1, $phi1);
The great_circle_midpoint() is just a special case of
use Math::Trig 'great_circle_waypoint';
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($thetai, $phii) =
great_circle_waypoint($theta0, $phi0, $theta1, $phi1, $way);
Where the $way is a value from zero ($theta0, $phi0) to one
($theta1, $phi1). Note that antipodal points (where their
distance is pi radians) do not have waypoints between them
(they would have an an "equator" between them), and there-
fore "undef" is returned for antipodal points. If the
points are the same and the distance therefore zero and all
waypoints therefore identical, the first point (either
point) is returned.
The thetas, phis, direction, and distance in the above are
all in radians.
You can import all the great circle formulas by
use Math::Trig ':great_circle';
Notice that the resulting directions might be somewhat
surprising if you are looking at a flat worldmap: in such
map projections the great circles quite often do not look
like the shortest routes-- but for example the shortest pos-
sible routes from Europe or North America to Asia do often
cross the polar regions.
To calculate the distance between London (51.3N 0.5W) and
Tokyo (35.7N 139.8E) in kilometers:
use Math::Trig qw(great_circle_distance deg2rad);
# Notice the 90 - latitude: phi zero is at the North Pole.
sub NESW { deg2rad($_[0]), deg2rad(90 - $_[1]) }
my @L = NESW( -0.5, 51.3);
my @T = NESW(139.8, 35.7);
my $km = great_circle_distance(@L, @T, 6378); # About 9600 km.
The direction you would have to go from London to Tokyo (in
radians, straight north being zero, straight east being
pi/2).
use Math::Trig qw(great_circle_direction);
my $rad = great_circle_direction(@L, @T); # About 0.547 or 0.174 pi.
The midpoint between London and Tokyo being
use Math::Trig qw(great_circle_midpoint);
my @M = great_circle_midpoint(@L, @T);
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or about 68.11N 24.74E, in the Finnish Lapland.
CAVEAT FOR GREAT CIRCLE FORMULAS
The answers may be off by few percentages because of the
irregular (slightly aspherical) form of the Earth. The
errors are at worst about 0.55%, but generally below 0.3%.
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.
Do not attempt navigation using these formulas.
Jarkko Hietaniemi <jhi@iki.fi> and Raphael Manfredi
<Raphael_Manfredi@pobox.com>.
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