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 convenience 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). Note that atan2(0, 0) is not welldefined.
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 area (also known as the inverse) functions of the hyperbolic sine, cosine,
and tangent
asinh,
acosh,
atanh
The area cofunctions of the hyperbolic sine, cosine, and tangent (acsch/acosech
and acoth/acotanh are aliases)
acsch,
acosech,
asech,
acoth,
acotanh
The trigonometric constant
pi and some of handy multiples of it are also
defined.
pi, pi2, pi4, pip2, pip4
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.
Note that atan2(0, 0) is not welldefined.
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), "\n";
should produce something like this (take or leave few last decimals):
1.57079632679491.31695789692482i
That is, a complex number with the real part of approximately 1.571 and the
imaginary part of approximately "1.317".
(Plane, 2dimensional) angles may be converted with the following functions.
 deg2rad

$radians = deg2rad($degrees);
 grad2rad

$radians = grad2rad($gradians);
 rad2deg

$degrees = rad2deg($radians);
 grad2deg

$degrees = grad2deg($gradians);
 deg2grad

$gradians = deg2grad($degrees);
 rad2grad

$gradians = rad2grad($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().
 rad2rad

$radians_wrapped_by_2pi = rad2rad($radians);
 deg2deg

$degrees_wrapped_by_360 = deg2deg($degrees);
 grad2grad

$gradians_wrapped_by_400 = grad2grad($gradians);
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.
Cartesian coordinates are the usual rectangular
(x, y,
z)coordinates.
Spherical coordinates,
(rho, theta, pi), are threedimensional
coordinates which define a point in threedimensional 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
xyplane (around the
zaxis) is
theta, also known as the
azimuthal coordinate.
The angle from the
zaxis 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 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 threedimensional
coordinates which define a point in threedimensional 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
xyplane (around the
zaxis) is
theta, also known as the
azimuthal coordinate.
The third coordinate is the
z, pointing up from the
thetaplane.
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.
A great circle is section of a circle that contains the circle diameter: the
shortest distance between two (nonantipodal) points on the spherical surface
goes along the great circle connecting those two points.
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);
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' for 'great_circle_direction' is also available.
use Math::Trig 'great_circle_bearing';
$direction = great_circle_bearing($theta0, $phi0, $theta1, $phi1);
The result of great_circle_direction 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';
# $diro is the original direction,
# for example from great_circle_bearing().
# $distance is the angular distance in radians,
# for example from great_circle_distance().
# $thetad and $phid are the destination coordinates,
# $dird is the final direction at the destination.
($thetad, $phid, $dird) =
great_circle_destination($theta, $phi, $diro, $distance);
or the midpoint if you know the end points:
use Math::Trig 'great_circle_midpoint';
($thetam, $phim) =
great_circle_midpoint($theta0, $phi0, $theta1, $phi1);
The
great_circle_midpoint() is just a special case of
use Math::Trig 'great_circle_waypoint';
($thetai, $phii) =
great_circle_waypoint($theta0, $phi0, $theta1, $phi1, $way);
Where the $way is a value from zero ($theta0, $phi0) to one ($theta1, $phi1).
Note that antipodal points (where their distance is
pi radians) do not
have waypoints between them (they would have an an "equator" between
them), and therefore "undef" is returned for antipodal points. If
the points are the same and the distance therefore zero and all waypoints
therefore identical, the first point (either point) is returned.
The thetas, phis, direction, and distance in the above are all in radians.
You can import all the great circle formulas by
use Math::Trig ':great_circle';
Notice that the resulting directions might be somewhat surprising if you are
looking at a flat worldmap: in such map projections the great circles quite
often do not look like the shortest routes  but for example the shortest
possible routes from Europe or North America to Asia do often cross the polar
regions. (The common Mercator projection does
not show great circles as
straight lines: straight lines in the Mercator projection are lines of
constant bearing.)
To calculate the distance between London (51.3N 0.5W) and Tokyo (35.7N 139.8E)
in kilometers:
use Math::Trig qw(great_circle_distance deg2rad);
# Notice the 90  latitude: phi zero is at the North Pole.
sub NESW { deg2rad($_[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);
or about 69 N 89 E, in the frozen wastes of Siberia.
NOTE: you
cannot get from A to B like this:
Dist = great_circle_distance(A, B)
Dir = great_circle_direction(A, B)
C = great_circle_destination(A, Dist, Dir)
and expect C to be B, because the bearing constantly changes when going from A
to B (except in some special case like the meridians or the circles of
latitudes) and in
great_circle_destination() one gives a
constant bearing to follow.
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%.
For small inputs
asin() and
acos() may return complex numbers even
when real numbers would be enough and correct, this happens because of
floatingpoint inaccuracies. You can see these inaccuracies for example by
trying theses:
print cos(1e6)**2+sin(1e6)**2  1,"\n";
printf "%.20f", cos(1e6)**2+sin(1e6)**2,"\n";
which will print something like this
1.11022302462516e16
0.99999999999999988898
even though the expected results are of course exactly zero and one. The
formulas used to compute
asin() and
acos() are quite sensitive
to this, and therefore they might accidentally slip into the complex plane
even when they should not. To counter this there are two interfaces that are
guaranteed to return a realvalued output.
 asin_real

use Math::Trig qw(asin_real);
$real_angle = asin_real($input_sin);
Return a realvalued arcus sine if the input is between [1, 1],
inclusive the endpoints. For inputs greater than one, pi/2 is
returned. For inputs less than minus one, pi/2 is returned.
 acos_real

use Math::Trig qw(acos_real);
$real_angle = acos_real($input_cos);
Return a realvalued arcus cosine if the input is between [1, 1],
inclusive the endpoints. For inputs greater than one, zero is
returned. For inputs less than minus one, pi is returned.
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.
Math::Complex
Jarkko Hietaniemi <
jhi!at!iki.fi>, Raphael Manfredi <
Raphael_Manfredi!at!pobox.com>, Zefram <zefram@fysh.org>
This library is free software; you can redistribute it and/or modify it under
the same terms as Perl itself.