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.

COORDINATE SYSTEMS

Cartesian coordinates are the usual rectangular (x, y, z)-coordinates.

Spherical coordinates, (rho, theta, phi), 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 rho, 0, 0, and the Gulf of Guinea (think of the missing big chunk of Africa) rho, 0, pi/2. 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_distance

Returns the great circle distance between two points on a sphere.

$distance = great_circle_distance($theta0, $phi0, $theta1, $phi1, [, $rho]);

Where ($theta0, $phi0) and ($theta1, $phi1) are the spherical coordinates of the two points, respectively. The distance is in $rho units. The $rho is optional. It defaults to 1 (the unit sphere).

If you are using geographic coordinates, latitude and longitude, you need to adjust for the fact that latitude is zero at the equator increasing towards the north and decreasing towards the south. Assuming ($lat0, $lon0) and ($lat1, $lon1) are the geographic coordinates in radians of the two points, the distance can be computed with

$distance = great_circle_distance($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 $way indicates the position of the waypoint along the great circle arc through the starting point ($theta0, $phi0) and the end point ($theta1, $phi1) relative to the distance from the starting point to the end point. So $way = 0 gives the starting point, $way = 1 gives the end point, $way < 0 gives a point "behind" the starting point, and $way > 1 gives a point beyond the end point. $way defaults to 0.5 if not given.

Note that antipodal points (where their distance is pi radians) do not have unique waypoints between them, and therefore "undef" is returned in such cases. If the points are the same, so the distance between them is zero, all waypoints are identical to the starting/end point.

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.)

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.