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 well-defined.
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 well-defined.
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 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, 2-dimensional) 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, $phi);
All angles are in radians.
Cartesian coordinates are the usual rectangular (x, y, z)-coordinates.
Spherical coordinates, (rho, theta, pi), are three-dimensional
coordinates which define a point in three-dimensional space. They are
based on a sphere surface. The radius of the sphere is rho, also
known as the radial coordinate. The angle in the xy-plane
(around the z-axis) is theta, also known as the azimuthal
coordinate. The angle from the z-axis is phi, also known as the
polar coordinate. The North Pole is therefore 0, 0, rho, and
the Gulf of Guinea (think of the missing big chunk of Africa) 0,
pi/2, rho. In geographical terms phi is latitude (northward
positive, southward negative) and theta is longitude (eastward
positive, westward negative).
BEWARE: some texts define theta and phi the other way round,
some texts define the phi to start from the horizontal plane, some
texts use r in place of rho.
Cylindrical coordinates, (rho, theta, z), are three-dimensional
coordinates which define a point in three-dimensional space. They are
based on a cylinder surface. The radius of the cylinder is rho,
also known as the radial coordinate. The angle in the xy-plane
(around the z-axis) is theta, also known as the azimuthal
coordinate. The third coordinate is the z, pointing up from the
theta-plane.
Conversions to and from spherical and cylindrical coordinates are
available. Please notice that the conversions are not necessarily
reversible because of the equalities like pi angles being equal to
-pi angles.
- cartesian_to_cylindrical
-
($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 (non-antipodal) 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
floating-point inaccuracies. You can see these inaccuracies for
example by trying theses:
print cos(1e-6)**2+sin(1e-6)**2 - 1,"\n";
printf "%.20f", cos(1e-6)**2+sin(1e-6)**2,"\n";
which will print something like this
-1.11022302462516e-16
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 real-valued output.
- asin_real
-
use Math::Trig qw(asin_real);
$real_angle = asin_real($input_sin);
Return a real-valued 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 real-valued 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.
the Math::Complex manpage
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.
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