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# Math::Trig

trigonometric functions

## SYNOPSIS

```perl
use Math::Trig;

$x = tan(0.9);
$y = acos(3.7);
$z = asin(2.4);

$halfpi = pi/2;

$rad = deg2rad(120);

## Import constants pi2, pi4, pip2, pip4 (2*pi, 4*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

```text
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

```text
cot(0): Division by zero.
(Because in the definition of cot(0), the divisor sin(0) is 0)
Died at ...
```

or

```text
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](../core/perldata.md)) 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](Complex.md)
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:

```perl
print asin(2), "\n";
```

should produce something like this (take or leave few last decimals):

```text
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

    ```perl
    $radians  = deg2rad($degrees);
    ```

- grad2rad

    ```perl
    $radians  = grad2rad($gradians);
    ```

- rad2deg

    ```perl
    $degrees  = rad2deg($radians);
    ```

- grad2deg

    ```perl
    $degrees  = grad2deg($gradians);
    ```

- deg2grad

    ```perl
    $gradians = deg2grad($degrees);
    ```

- rad2grad

    ```perl
    $gradians = rad2grad($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:

```perl
$zillions_of_radians  = deg2rad($zillions_of_degrees, 1);
$negative_degrees     = rad2deg($negative_radians, 1);
```

You can also do the wrapping explicitly by rad2rad(), deg2deg(), and
grad2grad().

- rad2rad

    ```perl
    $radians_wrapped_by_2pi = rad2rad($radians);
    ```

- deg2deg

    ```perl
    $degrees_wrapped_by_360 = deg2deg($degrees);
    ```

- grad2grad

    ```perl
    $gradians_wrapped_by_400 = grad2grad($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:

```perl
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, 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

    ```perl
    ($rho, $theta, $z) = cartesian_to_cylindrical($x, $y, $z);
    ```

- cartesian\_to\_spherical

    ```perl
    ($rho, $theta, $phi) = cartesian_to_spherical($x, $y, $z);
    ```

- cylindrical\_to\_cartesian

    ```perl
    ($x, $y, $z) = cylindrical_to_cartesian($rho, $theta, $z);
    ```

- cylindrical\_to\_spherical

    ```perl
    ($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

    ```perl
    ($x, $y, $z) = spherical_to_cartesian($rho, $theta, $phi);
    ```

- spherical\_to\_cylindrical

    ```perl
    ($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

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

```perl
$distance = great_circle_distance($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

```perl
$distance = great_circle_distance($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:

```perl
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.

```perl
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:

```perl
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

```perl
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

```perl
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

```perl
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:

```perl
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).

```perl
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

```perl
use Math::Trig qw(great_circle_midpoint rad2deg);

my @M = great_circle_midpoint(@L, @T);
sub SWNE { rad2deg( $_[0] ), 90 - rad2deg( $_[1] ) }
my @lonlat = SWNE(@M);
```

or about 69 N 89 E, on the Putorana Plateau of Siberia.

**NOTE**: you **cannot** get from A to B like this:

```text
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:

```perl
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

```text
-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

    ```perl
    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

    ```perl
    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.

## SEE ALSO

[Math::Complex](Complex.md)

## AUTHORS

Jarkko Hietaniemi <`jhi!at!iki.fi`>,
Raphael Manfredi <`Raphael_Manfredi!at!pobox.com`>,
Zefram <zefram@fysh.org>

## LICENSE

This library is free software; you can redistribute it and/or modify
it under the same terms as Perl itself.