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)
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. 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 com-plex complex
plex 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 get-ting getting
ting 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 approx-imately approximately
imately "-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, $phi);
All angles are in radians.
COORDINATE SYSTEMS
Cartesian coordinates are the usual rectangular (x, y, z)-coordinates.
Spherical coordinates, (rho, theta, pi), are three-dimensional coordinates which define a point in
three-dimensional space. They are based on a sphere surface. The radius of the sphere is rho, also
known as the radial coordinate. The angle in the xy-plane (around the z-axis) is theta, also known
as the azimuthal coordinate. The angle from the z-axis is phi, also known as the polar coordinate.
The North Pole is therefore 0, 0, rho, and the Gulf of Guinea (think of the missing big chunk of
Africa) 0, pi/2, rho. In geographical terms phi is latitude (northward positive, southward negative)
and theta is longitude (eastward positive, westward negative).
BEWARE: some texts define theta and phi the other way round, some texts define the phi to start from
the horizontal plane, some texts use r in place of rho.
Cylindrical coordinates, (rho, theta, z), are three-dimensional coordinates which define a point in
three-dimensional space. They are based on a cylinder surface. The radius of the cylinder is rho,
also known as the radial coordinate. The angle in the xy-plane (around the z-axis) is theta, also
known as the azimuthal coordinate. The third coordinate is the z, pointing up from the theta-plane.
3-D ANGLE CONVERSIONS
Conversions to and from spherical and cylindrical coordinates are available. Please notice that the
conversions are not necessarily reversible because of the equalities like pi angles being equal to
-pi angles.
cartesian_to_cylindrical
($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
You can compute spherical distances, called great circle distances, by importing the great_cir-cle_distance() great_circle_distance()
cle_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 posi-tive, positive,
tive, 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, $phi1);
(Alias 'great_circle_bearing' is also available.) The result is in radians, zero indicating straight
north, pi or -pi straight south, pi/2 straight west, and -pi/2 straight east.
You can inversely compute the destination if you know the starting point, direction, and distance:
use Math::Trig 'great_circle_destination';
# thetad and phid are the destination coordinates,
# dird is the final direction at the destination.
($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, $phi1);
The great_circle_midpoint() is just a special case of
use Math::Trig 'great_circle_waypoint';
($thetai, $phii) =
great_circle_waypoint($theta0, $phi0, $theta1, $phi1, $way);
Where the $way is a value from zero ($theta0, $phi0) to one ($theta1, $phi1). Note that antipodal
points (where their distance is pi radians) do not have waypoints between them (they would have an an
"equator" between them), and therefore "undef" is returned for antipodal points. If the points are
the same and the distance therefore zero and all waypoints therefore identical, the first point
(either point) is returned.
The thetas, phis, direction, and distance in the above are all in radians.
You can import all the great circle formulas by
use Math::Trig ':great_circle';
Notice that the resulting directions might be somewhat surprising if you are looking at a flat
worldmap: in such map projections the great circles quite often do not look like the shortest
routes-- but for example the shortest possible routes from Europe or North America to Asia do often
cross the polar regions.
EXAMPLES
To calculate the distance between London (51.3N 0.5W) and Tokyo (35.7N 139.8E) in kilometers:
use Math::Trig qw(great_circle_distance deg2rad);
# Notice the 90 - latitude: phi zero is at the North Pole.
sub NESW { deg2rad($_[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 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%.
BUGS
Saying "use Math::Trig;" exports many mathematical routines in the caller environment and even over-rides overrides
rides 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 run-time runtime
time error.
Do not attempt navigation using these formulas.
AUTHORS
Jarkko Hietaniemi <jhi@iki.fi> and Raphael Manfredi <Raphael_Manfredi@pobox.com>.
perl v5.8.8 2001-09-21 Math::Trig(3pm)
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