# Math::Trig(3p) Perl Programmers Reference Guide Math::Trig(3p) #

Math::Trig(3p) Perl Programmers Reference Guide Math::Trig(3p)

## NNAAMMEE #

```
Math::Trig - trigonometric functions
```

## SSYYNNOOPPSSIISS #

```
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';
```

## DDEESSCCRRIIPPTTIIOONN #

```
"Math::Trig" defines many trigonometric functions not defined by the core
Perl which defines only the "sin()" and "cos()". The constant ppii is also
defined as are a few convenience functions for angle conversions, and
_g_r_e_a_t _c_i_r_c_l_e _f_o_r_m_u_l_a_s for spherical movement.
```

## TTRRIIGGOONNOOMMEETTRRIICC FFUUNNCCTTIIOONNSS #

```
The tangent
ttaann
The cofunctions of the sine, cosine, and tangent (cosec/csc and cotan/cot
are aliases)
ccsscc, ccoosseecc, sseecc, sseecc, ccoott, ccoottaann
The arcus (also known as the inverse) functions of the sine, cosine, and
tangent
aassiinn, aaccooss, aattaann
The principal value of the arc tangent of y/x
aattaann22(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.
aaccsscc, aaccoosseecc, aasseecc, aaccoott, aaccoottaann
The hyperbolic sine, cosine, and tangent
ssiinnhh, ccoosshh, ttaannhh
The cofunctions of the hyperbolic sine, cosine, and tangent (cosech/csch
and cotanh/coth are aliases)
ccsscchh, ccoosseecchh, sseecchh, ccootthh, ccoottaannhh
The area (also known as the inverse) functions of the hyperbolic sine,
cosine, and tangent
aassiinnhh, aaccoosshh, aattaannhh
The area cofunctions of the hyperbolic sine, cosine, and tangent
(acsch/acosech and acoth/acotanh are aliases)
aaccsscchh, aaccoosseecchh, aasseecchh, aaccootthh, aaccoottaannhh
The trigonometric constant ppii and some of handy multiples of it are also
defined.
ppii,, ppii22,, ppii44,, ppiipp22,, ppiipp44
```

## EERRRROORRSS DDUUEE TTOO DDIIVVIISSIIOONN BBYY ZZEERROO #

```
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 _p_i_/_2 _+ _k _* _p_i, where _k is any integer.
Note that atan2(0, 0) is not well-defined.
```

## SSIIMMPPLLEE ((RREEAALL)) AARRGGUUMMEENNTTSS,, CCOOMMPPLLEEXX RREESSUULLTTSS #

```
Please note that some of the trigonometric functions can break out from
the rreeaall aaxxiiss into the ccoommpplleexx ppllaannee. 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".
```

## PPLLAANNEE AANNGGLLEE CCOONNVVEERRSSIIOONNSS #

```
(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 _p_i radians or _3_6_0 degrees or _4_0_0 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 rraadd22rraadd(()), ddeegg22ddeegg(()), and
ggrraadd22ggrraadd(()).
rad2rad
$radians_wrapped_by_2pi = rad2rad($radians);
deg2deg
$degrees_wrapped_by_360 = deg2deg($degrees);
grad2grad
$gradians_wrapped_by_400 = grad2grad($gradians);
```

## RRAADDIIAALL CCOOOORRDDIINNAATTEE CCOONNVVEERRSSIIOONNSS #

```
RRaaddiiaall ccoooorrddiinnaattee ssyysstteemmss are the sspphheerriiccaall and the ccyylliinnddrriiccaall 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);
AAllll aanngglleess aarree iinn rraaddiiaannss.
```

## CCOOOORRDDIINNAATTEE SSYYSSTTEEMMSS #

```
CCaarrtteessiiaann coordinates are the usual rectangular _(_x_, _y_, _z_)-coordinates.
Spherical coordinates, _(_r_h_o_, _t_h_e_t_a_, _p_i_), 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 rrhhoo, also known
as the _r_a_d_i_a_l coordinate. The angle in the _x_y-plane (around the _z-axis)
is tthheettaa, also known as the _a_z_i_m_u_t_h_a_l coordinate. The angle from the
_z-axis is pphhii, also known as the _p_o_l_a_r coordinate. The North Pole is
therefore _0_, _0_, _r_h_o, and the Gulf of Guinea (think of the missing big
chunk of Africa) _0_, _p_i_/_2_, _r_h_o. In geographical terms _p_h_i is latitude
(northward positive, southward negative) and _t_h_e_t_a is longitude (eastward
positive, westward negative).
BBEEWWAARREE: some texts define _t_h_e_t_a and _p_h_i the other way round, some texts
define the _p_h_i to start from the horizontal plane, some texts use _r in
place of _r_h_o.
Cylindrical coordinates, _(_r_h_o_, _t_h_e_t_a_, _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 rrhhoo, also
known as the _r_a_d_i_a_l coordinate. The angle in the _x_y-plane (around the
_z-axis) is tthheettaa, also known as the _a_z_i_m_u_t_h_a_l coordinate. The third
coordinate is the _z, pointing up from the tthheettaa-plane.
```

## 33--DD AANNGGLLEE CCOONNVVEERRSSIIOONNSS #

```
Conversions to and from spherical and cylindrical coordinates are
available. Please notice that the conversions are not necessarily
reversible because of the equalities like _p_i angles being equal to _-_p_i
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.
```

## GGRREEAATT CCIIRRCCLLEE DDIISSTTAANNCCEESS AANNDD DDIIRREECCTTIIOONNSS #

```
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.
```

ggrreeaatt__cciirrccllee__ddiissttaannccee You can compute spherical distances, called ggrreeaatt cciirrccllee ddiissttaanncceess, by importing the ggrreeaatt__cciirrccllee__ddiissttaannccee(()) function:

```
use Math::Trig 'great_circle_distance';
$distance = great_circle_distance($theta0, $phi0, $theta1, $phi1, [, $rho]);
The _g_r_e_a_t _c_i_r_c_l_e _d_i_s_t_a_n_c_e 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 _t_h_e_t_a are longitudes: zero at the
Greenwhich meridian, eastward positive, westward negative -- and the _p_h_i
are latitudes: zero at the North Pole, northward positive, southward
negative. NNOOTTEE: this formula thinks in mathematics, not geographically:
the _p_h_i 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 _p_i_/_2 (also known as 90 degrees).
$distance = great_circle_distance($lon0, pi/2 - $lat0,
$lon1, pi/2 - $lat1, $rho);
```

ggrreeaatt__cciirrccllee__ddiirreeccttiioonn The direction you must follow the great circle (also known as _b_e_a_r_i_n_g) can be computed by the ggrreeaatt__cciirrccllee__ddiirreeccttiioonn(()) function:

```
use Math::Trig 'great_circle_direction';
$direction = great_circle_direction($theta0, $phi0, $theta1, $phi1);
```

ggrreeaatt__cciirrccllee__bbeeaarriinngg 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.
```

ggrreeaatt__cciirrccllee__ddeessttiinnaattiioonn 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:
```

ggrreeaatt__cciirrccllee__mmiiddppooiinntt use Math::Trig ‘great_circle_midpoint’;

```
($thetam, $phim) =
great_circle_midpoint($theta0, $phi0, $theta1, $phi1);
The ggrreeaatt__cciirrccllee__mmiiddppooiinntt(()) is just a special case of
```

ggrreeaatt__cciirrccllee__wwaayyppooiinntt 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 _p_i 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 nnoott show
great circles as straight lines: straight lines in the Mercator
projection are lines of constant bearing.)
```

## EEXXAAMMPPLLEESS #

```
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.
NNOOTTEE: you ccaannnnoott 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 ggrreeaatt__cciirrccllee__ddeessttiinnaattiioonn(()) one gives a
ccoonnssttaanntt bearing to follow.
```

## CCAAVVEEAATT FFOORR GGRREEAATT CCIIRRCCLLEE FFOORRMMUULLAASS #

```
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%.
```

RReeaall--vvaalluueedd aassiinn aanndd aaccooss For small inputs aassiinn(()) and aaccooss(()) 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 aassiinn(()) and aaccooss(()) 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],
iinncclluussiivvee 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],
iinncclluussiivvee the endpoints. For inputs greater than one, zero is
returned. For inputs less than minus one, pi is returned.
```

## BBUUGGSS #

```
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
```

## AAUUTTHHOORRSS #

```
Jarkko Hietaniemi <_j_h_i_!_a_t_!_i_k_i_._f_i>, Raphael Manfredi
<_R_a_p_h_a_e_l___M_a_n_f_r_e_d_i_!_a_t_!_p_o_b_o_x_._c_o_m>, Zefram <zefram@fysh.org>
```

## LLIICCEENNSSEE #

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

perl v5.36.3 2013-03-25 Math::Trig(3p)