Commit | Line | Data |
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5aabfad6 | 1 | # |
2 | # Trigonometric functions, mostly inherited from Math::Complex. | |
d54bf66f | 3 | # -- Jarkko Hietaniemi, since April 1997 |
5cd24f17 | 4 | # -- Raphael Manfredi, September 1996 (indirectly: because of Math::Complex) |
5aabfad6 | 5 | # |
6 | ||
7 | require Exporter; | |
8 | package Math::Trig; | |
9 | ||
affad850 | 10 | use 5.005; |
5aabfad6 | 11 | use strict; |
12 | ||
b57c8994 | 13 | use Math::Complex 1.48; |
affad850 | 14 | use Math::Complex qw(:trig :pi); |
5aabfad6 | 15 | |
affad850 | 16 | use vars qw($VERSION $PACKAGE @ISA @EXPORT @EXPORT_OK %EXPORT_TAGS); |
5aabfad6 | 17 | |
18 | @ISA = qw(Exporter); | |
19 | ||
b57c8994 | 20 | $VERSION = 1.13; |
5aabfad6 | 21 | |
ace5de91 | 22 | my @angcnv = qw(rad2deg rad2grad |
d139edd6 JH |
23 | deg2rad deg2grad |
24 | grad2rad grad2deg); | |
5aabfad6 | 25 | |
f3828575 SP |
26 | my @areal = qw(asin_real acos_real); |
27 | ||
5aabfad6 | 28 | @EXPORT = (@{$Math::Complex::EXPORT_TAGS{'trig'}}, |
f3828575 | 29 | @angcnv, @areal); |
5aabfad6 | 30 | |
d54bf66f JH |
31 | my @rdlcnv = qw(cartesian_to_cylindrical |
32 | cartesian_to_spherical | |
33 | cylindrical_to_cartesian | |
34 | cylindrical_to_spherical | |
35 | spherical_to_cartesian | |
36 | spherical_to_cylindrical); | |
37 | ||
bf5f1b4c JH |
38 | my @greatcircle = qw( |
39 | great_circle_distance | |
40 | great_circle_direction | |
41 | great_circle_bearing | |
42 | great_circle_waypoint | |
43 | great_circle_midpoint | |
44 | great_circle_destination | |
45 | ); | |
d54bf66f | 46 | |
affad850 | 47 | my @pi = qw(pi pi2 pi4 pip2 pip4); |
bf5f1b4c | 48 | |
1515bec6 | 49 | @EXPORT_OK = (@rdlcnv, @greatcircle, @pi, 'Inf'); |
bf5f1b4c JH |
50 | |
51 | # See e.g. the following pages: | |
52 | # http://www.movable-type.co.uk/scripts/LatLong.html | |
53 | # http://williams.best.vwh.net/avform.htm | |
54 | ||
55 | %EXPORT_TAGS = ('radial' => [ @rdlcnv ], | |
56 | 'great_circle' => [ @greatcircle ], | |
57 | 'pi' => [ @pi ]); | |
d54bf66f | 58 | |
affad850 SP |
59 | sub _DR () { pi2/360 } |
60 | sub _RD () { 360/pi2 } | |
61 | sub _DG () { 400/360 } | |
62 | sub _GD () { 360/400 } | |
63 | sub _RG () { 400/pi2 } | |
64 | sub _GR () { pi2/400 } | |
5aabfad6 | 65 | |
66 | # | |
67 | # Truncating remainder. | |
68 | # | |
69 | ||
affad850 | 70 | sub _remt ($$) { |
5aabfad6 | 71 | # Oh yes, POSIX::fmod() would be faster. Possibly. If it is available. |
72 | $_[0] - $_[1] * int($_[0] / $_[1]); | |
73 | } | |
74 | ||
75 | # | |
76 | # Angle conversions. | |
77 | # | |
78 | ||
affad850 | 79 | sub rad2rad($) { _remt($_[0], pi2) } |
9db5a202 | 80 | |
affad850 | 81 | sub deg2deg($) { _remt($_[0], 360) } |
9db5a202 | 82 | |
affad850 | 83 | sub grad2grad($) { _remt($_[0], 400) } |
5aabfad6 | 84 | |
affad850 | 85 | sub rad2deg ($;$) { my $d = _RD * $_[0]; $_[1] ? $d : deg2deg($d) } |
5aabfad6 | 86 | |
affad850 | 87 | sub deg2rad ($;$) { my $d = _DR * $_[0]; $_[1] ? $d : rad2rad($d) } |
5aabfad6 | 88 | |
affad850 | 89 | sub grad2deg ($;$) { my $d = _GD * $_[0]; $_[1] ? $d : deg2deg($d) } |
5aabfad6 | 90 | |
affad850 | 91 | sub deg2grad ($;$) { my $d = _DG * $_[0]; $_[1] ? $d : grad2grad($d) } |
5aabfad6 | 92 | |
affad850 | 93 | sub rad2grad ($;$) { my $d = _RG * $_[0]; $_[1] ? $d : grad2grad($d) } |
9db5a202 | 94 | |
affad850 | 95 | sub grad2rad ($;$) { my $d = _GR * $_[0]; $_[1] ? $d : rad2rad($d) } |
5aabfad6 | 96 | |
f3828575 SP |
97 | # |
98 | # acos and asin functions which always return a real number | |
99 | # | |
100 | ||
101 | sub acos_real { | |
102 | return 0 if $_[0] >= 1; | |
103 | return pi if $_[0] <= -1; | |
104 | return acos($_[0]); | |
105 | } | |
106 | ||
107 | sub asin_real { | |
108 | return &pip2 if $_[0] >= 1; | |
109 | return -&pip2 if $_[0] <= -1; | |
110 | return asin($_[0]); | |
111 | } | |
112 | ||
d54bf66f JH |
113 | sub cartesian_to_spherical { |
114 | my ( $x, $y, $z ) = @_; | |
115 | ||
116 | my $rho = sqrt( $x * $x + $y * $y + $z * $z ); | |
117 | ||
118 | return ( $rho, | |
119 | atan2( $y, $x ), | |
f3828575 | 120 | $rho ? acos_real( $z / $rho ) : 0 ); |
d54bf66f JH |
121 | } |
122 | ||
123 | sub spherical_to_cartesian { | |
124 | my ( $rho, $theta, $phi ) = @_; | |
125 | ||
126 | return ( $rho * cos( $theta ) * sin( $phi ), | |
127 | $rho * sin( $theta ) * sin( $phi ), | |
128 | $rho * cos( $phi ) ); | |
129 | } | |
130 | ||
131 | sub spherical_to_cylindrical { | |
132 | my ( $x, $y, $z ) = spherical_to_cartesian( @_ ); | |
133 | ||
134 | return ( sqrt( $x * $x + $y * $y ), $_[1], $z ); | |
135 | } | |
136 | ||
137 | sub cartesian_to_cylindrical { | |
138 | my ( $x, $y, $z ) = @_; | |
139 | ||
140 | return ( sqrt( $x * $x + $y * $y ), atan2( $y, $x ), $z ); | |
141 | } | |
142 | ||
143 | sub cylindrical_to_cartesian { | |
144 | my ( $rho, $theta, $z ) = @_; | |
145 | ||
146 | return ( $rho * cos( $theta ), $rho * sin( $theta ), $z ); | |
147 | } | |
148 | ||
149 | sub cylindrical_to_spherical { | |
150 | return ( cartesian_to_spherical( cylindrical_to_cartesian( @_ ) ) ); | |
151 | } | |
152 | ||
153 | sub great_circle_distance { | |
154 | my ( $theta0, $phi0, $theta1, $phi1, $rho ) = @_; | |
155 | ||
156 | $rho = 1 unless defined $rho; # Default to the unit sphere. | |
157 | ||
158 | my $lat0 = pip2 - $phi0; | |
159 | my $lat1 = pip2 - $phi1; | |
160 | ||
161 | return $rho * | |
f3828575 SP |
162 | acos_real( cos( $lat0 ) * cos( $lat1 ) * cos( $theta0 - $theta1 ) + |
163 | sin( $lat0 ) * sin( $lat1 ) ); | |
d54bf66f JH |
164 | } |
165 | ||
7e5f197a JH |
166 | sub great_circle_direction { |
167 | my ( $theta0, $phi0, $theta1, $phi1 ) = @_; | |
168 | ||
d139edd6 JH |
169 | my $distance = &great_circle_distance; |
170 | ||
7e5f197a JH |
171 | my $lat0 = pip2 - $phi0; |
172 | my $lat1 = pip2 - $phi1; | |
173 | ||
174 | my $direction = | |
f3828575 SP |
175 | acos_real((sin($lat1) - sin($lat0) * cos($distance)) / |
176 | (cos($lat0) * sin($distance))); | |
177 | ||
d139edd6 JH |
178 | $direction = pi2 - $direction |
179 | if sin($theta1 - $theta0) < 0; | |
7e5f197a JH |
180 | |
181 | return rad2rad($direction); | |
182 | } | |
183 | ||
bf5f1b4c JH |
184 | *great_circle_bearing = \&great_circle_direction; |
185 | ||
186 | sub great_circle_waypoint { | |
187 | my ( $theta0, $phi0, $theta1, $phi1, $point ) = @_; | |
188 | ||
189 | $point = 0.5 unless defined $point; | |
190 | ||
191 | my $d = great_circle_distance( $theta0, $phi0, $theta1, $phi1 ); | |
192 | ||
193 | return undef if $d == pi; | |
194 | ||
195 | my $sd = sin($d); | |
196 | ||
197 | return ($theta0, $phi0) if $sd == 0; | |
198 | ||
199 | my $A = sin((1 - $point) * $d) / $sd; | |
200 | my $B = sin( $point * $d) / $sd; | |
201 | ||
202 | my $lat0 = pip2 - $phi0; | |
203 | my $lat1 = pip2 - $phi1; | |
204 | ||
205 | my $x = $A * cos($lat0) * cos($theta0) + $B * cos($lat1) * cos($theta1); | |
206 | my $y = $A * cos($lat0) * sin($theta0) + $B * cos($lat1) * sin($theta1); | |
207 | my $z = $A * sin($lat0) + $B * sin($lat1); | |
208 | ||
209 | my $theta = atan2($y, $x); | |
f3828575 | 210 | my $phi = acos_real($z); |
bf5f1b4c JH |
211 | |
212 | return ($theta, $phi); | |
213 | } | |
214 | ||
215 | sub great_circle_midpoint { | |
216 | great_circle_waypoint(@_[0..3], 0.5); | |
217 | } | |
218 | ||
219 | sub great_circle_destination { | |
220 | my ( $theta0, $phi0, $dir0, $dst ) = @_; | |
221 | ||
222 | my $lat0 = pip2 - $phi0; | |
223 | ||
f3828575 SP |
224 | my $phi1 = asin_real(sin($lat0)*cos($dst) + |
225 | cos($lat0)*sin($dst)*cos($dir0)); | |
bf5f1b4c JH |
226 | my $theta1 = $theta0 + atan2(sin($dir0)*sin($dst)*cos($lat0), |
227 | cos($dst)-sin($lat0)*sin($phi1)); | |
228 | ||
229 | my $dir1 = great_circle_bearing($theta1, $phi1, $theta0, $phi0) + pi; | |
230 | ||
231 | $dir1 -= pi2 if $dir1 > pi2; | |
232 | ||
233 | return ($theta1, $phi1, $dir1); | |
234 | } | |
235 | ||
ea0630ea HS |
236 | 1; |
237 | ||
238 | __END__ | |
d54bf66f JH |
239 | =pod |
240 | ||
5aabfad6 | 241 | =head1 NAME |
242 | ||
243 | Math::Trig - trigonometric functions | |
244 | ||
245 | =head1 SYNOPSIS | |
246 | ||
affad850 | 247 | use Math::Trig; |
3cb6de81 | 248 | |
affad850 SP |
249 | $x = tan(0.9); |
250 | $y = acos(3.7); | |
251 | $z = asin(2.4); | |
3cb6de81 | 252 | |
affad850 | 253 | $halfpi = pi/2; |
5aabfad6 | 254 | |
affad850 | 255 | $rad = deg2rad(120); |
5aabfad6 | 256 | |
affad850 SP |
257 | # Import constants pi2, pip2, pip4 (2*pi, pi/2, pi/4). |
258 | use Math::Trig ':pi'; | |
bf5f1b4c | 259 | |
affad850 SP |
260 | # Import the conversions between cartesian/spherical/cylindrical. |
261 | use Math::Trig ':radial'; | |
bf5f1b4c JH |
262 | |
263 | # Import the great circle formulas. | |
affad850 | 264 | use Math::Trig ':great_circle'; |
bf5f1b4c | 265 | |
5aabfad6 | 266 | =head1 DESCRIPTION |
267 | ||
268 | C<Math::Trig> defines many trigonometric functions not defined by the | |
4ae80833 | 269 | core Perl which defines only the C<sin()> and C<cos()>. The constant |
5aabfad6 | 270 | B<pi> is also defined as are a few convenience functions for angle |
bf5f1b4c | 271 | conversions, and I<great circle formulas> for spherical movement. |
5aabfad6 | 272 | |
273 | =head1 TRIGONOMETRIC FUNCTIONS | |
274 | ||
275 | The tangent | |
276 | ||
d54bf66f JH |
277 | =over 4 |
278 | ||
279 | =item B<tan> | |
280 | ||
281 | =back | |
5aabfad6 | 282 | |
283 | The cofunctions of the sine, cosine, and tangent (cosec/csc and cotan/cot | |
284 | are aliases) | |
285 | ||
d54bf66f | 286 | B<csc>, B<cosec>, B<sec>, B<sec>, B<cot>, B<cotan> |
5aabfad6 | 287 | |
288 | The arcus (also known as the inverse) functions of the sine, cosine, | |
289 | and tangent | |
290 | ||
d54bf66f | 291 | B<asin>, B<acos>, B<atan> |
5aabfad6 | 292 | |
293 | The principal value of the arc tangent of y/x | |
294 | ||
d54bf66f | 295 | B<atan2>(y, x) |
5aabfad6 | 296 | |
297 | The arcus cofunctions of the sine, cosine, and tangent (acosec/acsc | |
affad850 | 298 | and acotan/acot are aliases). Note that atan2(0, 0) is not well-defined. |
5aabfad6 | 299 | |
d54bf66f | 300 | B<acsc>, B<acosec>, B<asec>, B<acot>, B<acotan> |
5aabfad6 | 301 | |
302 | The hyperbolic sine, cosine, and tangent | |
303 | ||
d54bf66f | 304 | B<sinh>, B<cosh>, B<tanh> |
5aabfad6 | 305 | |
306 | The cofunctions of the hyperbolic sine, cosine, and tangent (cosech/csch | |
307 | and cotanh/coth are aliases) | |
308 | ||
d54bf66f | 309 | B<csch>, B<cosech>, B<sech>, B<coth>, B<cotanh> |
5aabfad6 | 310 | |
311 | The arcus (also known as the inverse) functions of the hyperbolic | |
312 | sine, cosine, and tangent | |
313 | ||
d54bf66f | 314 | B<asinh>, B<acosh>, B<atanh> |
5aabfad6 | 315 | |
316 | The arcus cofunctions of the hyperbolic sine, cosine, and tangent | |
317 | (acsch/acosech and acoth/acotanh are aliases) | |
318 | ||
d54bf66f | 319 | B<acsch>, B<acosech>, B<asech>, B<acoth>, B<acotanh> |
5aabfad6 | 320 | |
affad850 SP |
321 | The trigonometric constant B<pi> and some of handy multiples |
322 | of it are also defined. | |
5aabfad6 | 323 | |
affad850 | 324 | B<pi, pi2, pi4, pip2, pip4> |
5aabfad6 | 325 | |
5cd24f17 | 326 | =head2 ERRORS DUE TO DIVISION BY ZERO |
327 | ||
328 | The following functions | |
329 | ||
affad850 SP |
330 | acoth |
331 | acsc | |
332 | acsch | |
333 | asec | |
334 | asech | |
335 | atanh | |
336 | cot | |
337 | coth | |
338 | csc | |
339 | csch | |
340 | sec | |
341 | sech | |
342 | tan | |
343 | tanh | |
5cd24f17 | 344 | |
345 | cannot be computed for all arguments because that would mean dividing | |
8c03c583 JH |
346 | by zero or taking logarithm of zero. These situations cause fatal |
347 | runtime errors looking like this | |
5cd24f17 | 348 | |
affad850 SP |
349 | cot(0): Division by zero. |
350 | (Because in the definition of cot(0), the divisor sin(0) is 0) | |
351 | Died at ... | |
5cd24f17 | 352 | |
8c03c583 JH |
353 | or |
354 | ||
affad850 SP |
355 | atanh(-1): Logarithm of zero. |
356 | Died at... | |
8c03c583 JH |
357 | |
358 | For the C<csc>, C<cot>, C<asec>, C<acsc>, C<acot>, C<csch>, C<coth>, | |
359 | C<asech>, C<acsch>, the argument cannot be C<0> (zero). For the | |
360 | C<atanh>, C<acoth>, the argument cannot be C<1> (one). For the | |
361 | C<atanh>, C<acoth>, the argument cannot be C<-1> (minus one). For the | |
362 | C<tan>, C<sec>, C<tanh>, C<sech>, the argument cannot be I<pi/2 + k * | |
affad850 SP |
363 | pi>, where I<k> is any integer. |
364 | ||
365 | Note that atan2(0, 0) is not well-defined. | |
5cd24f17 | 366 | |
367 | =head2 SIMPLE (REAL) ARGUMENTS, COMPLEX RESULTS | |
5aabfad6 | 368 | |
369 | Please note that some of the trigonometric functions can break out | |
370 | from the B<real axis> into the B<complex plane>. For example | |
371 | C<asin(2)> has no definition for plain real numbers but it has | |
372 | definition for complex numbers. | |
373 | ||
374 | In Perl terms this means that supplying the usual Perl numbers (also | |
375 | known as scalars, please see L<perldata>) as input for the | |
376 | trigonometric functions might produce as output results that no more | |
377 | are simple real numbers: instead they are complex numbers. | |
378 | ||
379 | The C<Math::Trig> handles this by using the C<Math::Complex> package | |
380 | which knows how to handle complex numbers, please see L<Math::Complex> | |
381 | for more information. In practice you need not to worry about getting | |
382 | complex numbers as results because the C<Math::Complex> takes care of | |
383 | details like for example how to display complex numbers. For example: | |
384 | ||
affad850 | 385 | print asin(2), "\n"; |
3cb6de81 | 386 | |
5aabfad6 | 387 | should produce something like this (take or leave few last decimals): |
388 | ||
affad850 | 389 | 1.5707963267949-1.31695789692482i |
5aabfad6 | 390 | |
5cd24f17 | 391 | That is, a complex number with the real part of approximately C<1.571> |
392 | and the imaginary part of approximately C<-1.317>. | |
5aabfad6 | 393 | |
d54bf66f | 394 | =head1 PLANE ANGLE CONVERSIONS |
5aabfad6 | 395 | |
396 | (Plane, 2-dimensional) angles may be converted with the following functions. | |
397 | ||
affad850 SP |
398 | =over |
399 | ||
400 | =item deg2rad | |
401 | ||
402 | $radians = deg2rad($degrees); | |
403 | ||
404 | =item grad2rad | |
405 | ||
406 | $radians = grad2rad($gradians); | |
407 | ||
408 | =item rad2deg | |
409 | ||
410 | $degrees = rad2deg($radians); | |
3cb6de81 | 411 | |
affad850 | 412 | =item grad2deg |
3cb6de81 | 413 | |
affad850 SP |
414 | $degrees = grad2deg($gradians); |
415 | ||
416 | =item deg2grad | |
417 | ||
418 | $gradians = deg2grad($degrees); | |
419 | ||
420 | =item rad2grad | |
421 | ||
422 | $gradians = rad2grad($radians); | |
423 | ||
424 | =back | |
5aabfad6 | 425 | |
5cd24f17 | 426 | The full circle is 2 I<pi> radians or I<360> degrees or I<400> gradians. |
9db5a202 JH |
427 | The result is by default wrapped to be inside the [0, {2pi,360,400}[ circle. |
428 | If you don't want this, supply a true second argument: | |
429 | ||
affad850 SP |
430 | $zillions_of_radians = deg2rad($zillions_of_degrees, 1); |
431 | $negative_degrees = rad2deg($negative_radians, 1); | |
9db5a202 JH |
432 | |
433 | You can also do the wrapping explicitly by rad2rad(), deg2deg(), and | |
434 | grad2grad(). | |
5aabfad6 | 435 | |
affad850 SP |
436 | =over 4 |
437 | ||
438 | =item rad2rad | |
439 | ||
440 | $radians_wrapped_by_2pi = rad2rad($radians); | |
441 | ||
442 | =item deg2deg | |
443 | ||
444 | $degrees_wrapped_by_360 = deg2deg($degrees); | |
445 | ||
446 | =item grad2grad | |
447 | ||
448 | $gradians_wrapped_by_400 = grad2grad($gradians); | |
449 | ||
450 | =back | |
451 | ||
d54bf66f JH |
452 | =head1 RADIAL COORDINATE CONVERSIONS |
453 | ||
454 | B<Radial coordinate systems> are the B<spherical> and the B<cylindrical> | |
455 | systems, explained shortly in more detail. | |
456 | ||
457 | You can import radial coordinate conversion functions by using the | |
458 | C<:radial> tag: | |
459 | ||
460 | use Math::Trig ':radial'; | |
461 | ||
462 | ($rho, $theta, $z) = cartesian_to_cylindrical($x, $y, $z); | |
463 | ($rho, $theta, $phi) = cartesian_to_spherical($x, $y, $z); | |
464 | ($x, $y, $z) = cylindrical_to_cartesian($rho, $theta, $z); | |
465 | ($rho_s, $theta, $phi) = cylindrical_to_spherical($rho_c, $theta, $z); | |
466 | ($x, $y, $z) = spherical_to_cartesian($rho, $theta, $phi); | |
467 | ($rho_c, $theta, $z) = spherical_to_cylindrical($rho_s, $theta, $phi); | |
468 | ||
469 | B<All angles are in radians>. | |
470 | ||
471 | =head2 COORDINATE SYSTEMS | |
472 | ||
bf5f1b4c | 473 | B<Cartesian> coordinates are the usual rectangular I<(x, y, z)>-coordinates. |
d54bf66f JH |
474 | |
475 | Spherical coordinates, I<(rho, theta, pi)>, are three-dimensional | |
476 | coordinates which define a point in three-dimensional space. They are | |
477 | based on a sphere surface. The radius of the sphere is B<rho>, also | |
478 | known as the I<radial> coordinate. The angle in the I<xy>-plane | |
479 | (around the I<z>-axis) is B<theta>, also known as the I<azimuthal> | |
480 | coordinate. The angle from the I<z>-axis is B<phi>, also known as the | |
2d6f5264 JH |
481 | I<polar> coordinate. The North Pole is therefore I<0, 0, rho>, and |
482 | the Gulf of Guinea (think of the missing big chunk of Africa) I<0, | |
4b0d1da8 JH |
483 | pi/2, rho>. In geographical terms I<phi> is latitude (northward |
484 | positive, southward negative) and I<theta> is longitude (eastward | |
485 | positive, westward negative). | |
d54bf66f | 486 | |
4b0d1da8 | 487 | B<BEWARE>: some texts define I<theta> and I<phi> the other way round, |
d54bf66f JH |
488 | some texts define the I<phi> to start from the horizontal plane, some |
489 | texts use I<r> in place of I<rho>. | |
490 | ||
491 | Cylindrical coordinates, I<(rho, theta, z)>, are three-dimensional | |
492 | coordinates which define a point in three-dimensional space. They are | |
493 | based on a cylinder surface. The radius of the cylinder is B<rho>, | |
494 | also known as the I<radial> coordinate. The angle in the I<xy>-plane | |
495 | (around the I<z>-axis) is B<theta>, also known as the I<azimuthal> | |
496 | coordinate. The third coordinate is the I<z>, pointing up from the | |
497 | B<theta>-plane. | |
498 | ||
499 | =head2 3-D ANGLE CONVERSIONS | |
500 | ||
501 | Conversions to and from spherical and cylindrical coordinates are | |
502 | available. Please notice that the conversions are not necessarily | |
503 | reversible because of the equalities like I<pi> angles being equal to | |
504 | I<-pi> angles. | |
505 | ||
506 | =over 4 | |
507 | ||
508 | =item cartesian_to_cylindrical | |
509 | ||
affad850 | 510 | ($rho, $theta, $z) = cartesian_to_cylindrical($x, $y, $z); |
d54bf66f JH |
511 | |
512 | =item cartesian_to_spherical | |
513 | ||
affad850 | 514 | ($rho, $theta, $phi) = cartesian_to_spherical($x, $y, $z); |
d54bf66f JH |
515 | |
516 | =item cylindrical_to_cartesian | |
517 | ||
affad850 | 518 | ($x, $y, $z) = cylindrical_to_cartesian($rho, $theta, $z); |
d54bf66f JH |
519 | |
520 | =item cylindrical_to_spherical | |
521 | ||
affad850 | 522 | ($rho_s, $theta, $phi) = cylindrical_to_spherical($rho_c, $theta, $z); |
d54bf66f JH |
523 | |
524 | Notice that when C<$z> is not 0 C<$rho_s> is not equal to C<$rho_c>. | |
525 | ||
526 | =item spherical_to_cartesian | |
527 | ||
affad850 | 528 | ($x, $y, $z) = spherical_to_cartesian($rho, $theta, $phi); |
d54bf66f JH |
529 | |
530 | =item spherical_to_cylindrical | |
531 | ||
affad850 | 532 | ($rho_c, $theta, $z) = spherical_to_cylindrical($rho_s, $theta, $phi); |
d54bf66f JH |
533 | |
534 | Notice that when C<$z> is not 0 C<$rho_c> is not equal to C<$rho_s>. | |
535 | ||
536 | =back | |
537 | ||
7e5f197a | 538 | =head1 GREAT CIRCLE DISTANCES AND DIRECTIONS |
d54bf66f | 539 | |
affad850 SP |
540 | A great circle is section of a circle that contains the circle |
541 | diameter: the shortest distance between two (non-antipodal) points on | |
542 | the spherical surface goes along the great circle connecting those two | |
543 | points. | |
544 | ||
545 | =head2 great_circle_distance | |
546 | ||
d54bf66f | 547 | You can compute spherical distances, called B<great circle distances>, |
7e5f197a | 548 | by importing the great_circle_distance() function: |
d54bf66f | 549 | |
7e5f197a | 550 | use Math::Trig 'great_circle_distance'; |
d54bf66f | 551 | |
4b0d1da8 | 552 | $distance = great_circle_distance($theta0, $phi0, $theta1, $phi1, [, $rho]); |
d54bf66f JH |
553 | |
554 | The I<great circle distance> is the shortest distance between two | |
555 | points on a sphere. The distance is in C<$rho> units. The C<$rho> is | |
556 | optional, it defaults to 1 (the unit sphere), therefore the distance | |
557 | defaults to radians. | |
558 | ||
4b0d1da8 | 559 | If you think geographically the I<theta> are longitudes: zero at the |
f3828575 | 560 | Greenwhich meridian, eastward positive, westward negative -- and the |
2d06e7d7 | 561 | I<phi> are latitudes: zero at the North Pole, northward positive, |
4b0d1da8 | 562 | southward negative. B<NOTE>: this formula thinks in mathematics, not |
2d06e7d7 JH |
563 | geographically: the I<phi> zero is at the North Pole, not at the |
564 | Equator on the west coast of Africa (Bay of Guinea). You need to | |
565 | subtract your geographical coordinates from I<pi/2> (also known as 90 | |
566 | degrees). | |
4b0d1da8 JH |
567 | |
568 | $distance = great_circle_distance($lon0, pi/2 - $lat0, | |
569 | $lon1, pi/2 - $lat1, $rho); | |
570 | ||
affad850 SP |
571 | =head2 great_circle_direction |
572 | ||
bf5f1b4c JH |
573 | The direction you must follow the great circle (also known as I<bearing>) |
574 | can be computed by the great_circle_direction() function: | |
7e5f197a JH |
575 | |
576 | use Math::Trig 'great_circle_direction'; | |
577 | ||
578 | $direction = great_circle_direction($theta0, $phi0, $theta1, $phi1); | |
579 | ||
affad850 SP |
580 | =head2 great_circle_bearing |
581 | ||
582 | Alias 'great_circle_bearing' for 'great_circle_direction' is also available. | |
583 | ||
584 | use Math::Trig 'great_circle_bearing'; | |
585 | ||
586 | $direction = great_circle_bearing($theta0, $phi0, $theta1, $phi1); | |
587 | ||
588 | The result of great_circle_direction is in radians, zero indicating | |
589 | straight north, pi or -pi straight south, pi/2 straight west, and | |
590 | -pi/2 straight east. | |
7e5f197a | 591 | |
bf5f1b4c JH |
592 | You can inversely compute the destination if you know the |
593 | starting point, direction, and distance: | |
594 | ||
affad850 SP |
595 | =head2 great_circle_destination |
596 | ||
bf5f1b4c JH |
597 | use Math::Trig 'great_circle_destination'; |
598 | ||
599 | # thetad and phid are the destination coordinates, | |
600 | # dird is the final direction at the destination. | |
601 | ||
602 | ($thetad, $phid, $dird) = | |
603 | great_circle_destination($theta, $phi, $direction, $distance); | |
604 | ||
605 | or the midpoint if you know the end points: | |
606 | ||
affad850 SP |
607 | =head2 great_circle_midpoint |
608 | ||
bf5f1b4c JH |
609 | use Math::Trig 'great_circle_midpoint'; |
610 | ||
611 | ($thetam, $phim) = | |
612 | great_circle_midpoint($theta0, $phi0, $theta1, $phi1); | |
613 | ||
614 | The great_circle_midpoint() is just a special case of | |
615 | ||
affad850 SP |
616 | =head2 great_circle_waypoint |
617 | ||
bf5f1b4c JH |
618 | use Math::Trig 'great_circle_waypoint'; |
619 | ||
620 | ($thetai, $phii) = | |
621 | great_circle_waypoint($theta0, $phi0, $theta1, $phi1, $way); | |
622 | ||
623 | Where the $way is a value from zero ($theta0, $phi0) to one ($theta1, | |
624 | $phi1). Note that antipodal points (where their distance is I<pi> | |
625 | radians) do not have waypoints between them (they would have an an | |
626 | "equator" between them), and therefore C<undef> is returned for | |
627 | antipodal points. If the points are the same and the distance | |
628 | therefore zero and all waypoints therefore identical, the first point | |
629 | (either point) is returned. | |
630 | ||
631 | The thetas, phis, direction, and distance in the above are all in radians. | |
632 | ||
633 | You can import all the great circle formulas by | |
634 | ||
635 | use Math::Trig ':great_circle'; | |
636 | ||
7e5f197a JH |
637 | Notice that the resulting directions might be somewhat surprising if |
638 | you are looking at a flat worldmap: in such map projections the great | |
f3828575 | 639 | circles quite often do not look like the shortest routes -- but for |
7e5f197a | 640 | example the shortest possible routes from Europe or North America to |
f3828575 SP |
641 | Asia do often cross the polar regions. (The common Mercator projection |
642 | does B<not> show great circles as straight lines: straight lines in the | |
643 | Mercator projection are lines of constant bearing.) | |
7e5f197a | 644 | |
51301382 | 645 | =head1 EXAMPLES |
d54bf66f | 646 | |
7e5f197a JH |
647 | To calculate the distance between London (51.3N 0.5W) and Tokyo |
648 | (35.7N 139.8E) in kilometers: | |
d54bf66f | 649 | |
affad850 | 650 | use Math::Trig qw(great_circle_distance deg2rad); |
d54bf66f | 651 | |
affad850 SP |
652 | # Notice the 90 - latitude: phi zero is at the North Pole. |
653 | sub NESW { deg2rad($_[0]), deg2rad(90 - $_[1]) } | |
654 | my @L = NESW( -0.5, 51.3); | |
655 | my @T = NESW(139.8, 35.7); | |
656 | my $km = great_circle_distance(@L, @T, 6378); # About 9600 km. | |
d54bf66f | 657 | |
bf5f1b4c JH |
658 | The direction you would have to go from London to Tokyo (in radians, |
659 | straight north being zero, straight east being pi/2). | |
7e5f197a | 660 | |
affad850 | 661 | use Math::Trig qw(great_circle_direction); |
7e5f197a | 662 | |
affad850 | 663 | my $rad = great_circle_direction(@L, @T); # About 0.547 or 0.174 pi. |
7e5f197a | 664 | |
bf5f1b4c | 665 | The midpoint between London and Tokyo being |
7e5f197a | 666 | |
affad850 | 667 | use Math::Trig qw(great_circle_midpoint); |
bf5f1b4c | 668 | |
affad850 | 669 | my @M = great_circle_midpoint(@L, @T); |
bf5f1b4c | 670 | |
d020892c | 671 | or about 68.93N 89.16E, in the frozen wastes of Siberia. |
41bd693c | 672 | |
bf5f1b4c | 673 | =head2 CAVEAT FOR GREAT CIRCLE FORMULAS |
41bd693c | 674 | |
bf5f1b4c JH |
675 | The answers may be off by few percentages because of the irregular |
676 | (slightly aspherical) form of the Earth. The errors are at worst | |
677 | about 0.55%, but generally below 0.3%. | |
d54bf66f | 678 | |
f3828575 SP |
679 | =head2 Real-valued asin and acos |
680 | ||
681 | For small inputs asin() and acos() may return complex numbers even | |
682 | when real numbers would be enough and correct, this happens because of | |
683 | floating-point inaccuracies. You can see these inaccuracies for | |
684 | example by trying theses: | |
685 | ||
686 | print cos(1e-6)**2+sin(1e-6)**2 - 1,"\n"; | |
687 | printf "%.20f", cos(1e-6)**2+sin(1e-6)**2,"\n"; | |
688 | ||
689 | which will print something like this | |
690 | ||
691 | -1.11022302462516e-16 | |
692 | 0.99999999999999988898 | |
693 | ||
694 | even though the expected results are of course exactly zero and one. | |
695 | The formulas used to compute asin() and acos() are quite sensitive to | |
696 | this, and therefore they might accidentally slip into the complex | |
697 | plane even when they should not. To counter this there are two | |
698 | interfaces that are guaranteed to return a real-valued output. | |
699 | ||
700 | =over 4 | |
701 | ||
702 | =item asin_real | |
703 | ||
704 | use Math::Trig qw(asin_real); | |
705 | ||
706 | $real_angle = asin_real($input_sin); | |
707 | ||
708 | Return a real-valued arcus sine if the input is between [-1, 1], | |
709 | B<inclusive> the endpoints. For inputs greater than one, pi/2 | |
710 | is returned. For inputs less than minus one, -pi/2 is returned. | |
711 | ||
712 | =item acos_real | |
713 | ||
714 | use Math::Trig qw(acos_real); | |
715 | ||
716 | $real_angle = acos_real($input_cos); | |
717 | ||
718 | Return a real-valued arcus cosine if the input is between [-1, 1], | |
719 | B<inclusive> the endpoints. For inputs greater than one, zero | |
720 | is returned. For inputs less than minus one, pi is returned. | |
721 | ||
722 | =back | |
723 | ||
5cd24f17 | 724 | =head1 BUGS |
5aabfad6 | 725 | |
5cd24f17 | 726 | Saying C<use Math::Trig;> exports many mathematical routines in the |
727 | caller environment and even overrides some (C<sin>, C<cos>). This is | |
728 | construed as a feature by the Authors, actually... ;-) | |
5aabfad6 | 729 | |
5cd24f17 | 730 | The code is not optimized for speed, especially because we use |
731 | C<Math::Complex> and thus go quite near complex numbers while doing | |
732 | the computations even when the arguments are not. This, however, | |
733 | cannot be completely avoided if we want things like C<asin(2)> to give | |
734 | an answer instead of giving a fatal runtime error. | |
5aabfad6 | 735 | |
bf5f1b4c JH |
736 | Do not attempt navigation using these formulas. |
737 | ||
7637cd07 SP |
738 | L<Math::Complex> |
739 | ||
5cd24f17 | 740 | =head1 AUTHORS |
5aabfad6 | 741 | |
affad850 SP |
742 | Jarkko Hietaniemi <F<jhi!at!iki.fi>> and |
743 | Raphael Manfredi <F<Raphael_Manfredi!at!pobox.com>>. | |
5aabfad6 | 744 | |
1515bec6 SP |
745 | =head1 LICENSE |
746 | ||
747 | This library is free software; you can redistribute it and/or modify | |
748 | it under the same terms as Perl itself. | |
749 | ||
5aabfad6 | 750 | =cut |
751 | ||
752 | # eof |