Math::Trig(3pm) Perl Programmers Reference Guide Math::Trig(3pm)
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);
# 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';
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, and great circle formulas for spherical movement.
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). 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
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.
Note that atan2(0, 0) is not well-defined.
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 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):
1.5707963267949-1.31695789692482i
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.
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 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, $phi);
All angles are in radians.
COORDINATE SYSTEMS
Cartesian coordinates are the usual rectangular (x, y, z)-coordinates.
Spherical coordinates, (rho, theta, pi), are three-dimensional
coordinates which define a point in three-dimensional space. They are
based on a sphere surface. The radius of the sphere is rho, also known
as the radial coordinate. The angle in the xy-plane (around the
z-axis) is theta, also known as the azimuthal coordinate. The angle
from the z-axis is phi, also known as the polar coordinate. The North
Pole is therefore 0, 0, rho, and the Gulf of Guinea (think of the
missing big chunk of Africa) 0, pi/2, rho. In geographical terms phi
is latitude (northward positive, southward negative) and theta is
longitude (eastward positive, westward negative).
BEWARE: some texts define theta and phi the other way round, some texts
define the phi to start from the horizontal plane, some texts use r in
place of rho.
Cylindrical coordinates, (rho, theta, z), are three-dimensional
coordinates which define a point in three-dimensional space. They are
based on a cylinder surface. The radius of the cylinder is rho, also
known as the radial coordinate. The angle in the xy-plane (around the
z-axis) is theta, also known as the azimuthal coordinate. The third
coordinate is the z, pointing up from the theta-plane.
3-D ANGLE CONVERSIONS
Conversions to and from spherical and cylindrical coordinates are
available. Please notice that the conversions are not necessarily
reversible because of the equalities like pi angles being equal to -pi
angles.
cartesian_to_cylindrical
($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 AND DIRECTIONS
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.
great_circle_distance
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);
great_circle_direction
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);
great_circle_bearing
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.
great_circle_destination
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, $distance);
or the midpoint if you know the end points:
great_circle_midpoint
use Math::Trig 'great_circle_midpoint';
($thetam, $phim) =
great_circle_midpoint($theta0, $phi0, $theta1, $phi1);
The great_circle_midpoint() is just a special case of
great_circle_waypoint
use Math::Trig 'great_circle_waypoint';
($thetai, $phii) =
great_circle_waypoint($theta0, $phi0, $theta1, $phi1, $way);
Where the $way is a value from zero ($theta0, $phi0) to one ($theta1,
$phi1). Note that antipodal points (where their distance is pi
radians) do not have waypoints between them (they would have an an
"equator" between them), and therefore "undef" is returned for
antipodal points. If the points are the same and the distance
therefore zero and all waypoints therefore identical, the first point
(either point) is returned.
The thetas, phis, direction, and distance in the above are all in
radians.
You can import all the great circle formulas by
use Math::Trig ':great_circle';
Notice that the resulting directions might be somewhat surprising if
you are looking at a flat worldmap: in such map projections the great
circles quite often do not look like the shortest routes -- but for
example the shortest possible routes from Europe or North America to
Asia do often cross the polar regions. (The common Mercator projection
does not show great circles as straight lines: straight lines in the
Mercator projection are lines of constant bearing.)
EXAMPLES
To calculate the distance between London (51.3N 0.5W) and Tokyo (35.7N
139.8E) in kilometers:
use Math::Trig qw(great_circle_distance deg2rad);
# Notice the 90 - latitude: phi zero is at the North Pole.
sub NESW { deg2rad($_[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.
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%.
Real-valued asin and acos
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.
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.
Do not attempt navigation using these formulas.
Math::Complex
AUTHORS
Jarkko Hietaniemi <jhi!at!iki.fi>, Raphael Manfredi
<Raphael_Manfredi!at!pobox.com>, Zefram <zefram AT fysh.org>
LICENSE
This library is free software; you can redistribute it and/or modify it
under the same terms as Perl itself.
perl v5.16.3 2013-03-04 Math::Trig(3pm)
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