Commit | Line | Data |
---|---|---|
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 | ||
affad850 SP |
13 | use Math::Complex 1.36; |
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 | ||
affad850 | 20 | $VERSION = 1.04; |
5aabfad6 | 21 | |
ace5de91 | 22 | my @angcnv = qw(rad2deg rad2grad |
d139edd6 JH |
23 | deg2rad deg2grad |
24 | grad2rad grad2deg); | |
5aabfad6 | 25 | |
26 | @EXPORT = (@{$Math::Complex::EXPORT_TAGS{'trig'}}, | |
27 | @angcnv); | |
28 | ||
d54bf66f JH |
29 | my @rdlcnv = qw(cartesian_to_cylindrical |
30 | cartesian_to_spherical | |
31 | cylindrical_to_cartesian | |
32 | cylindrical_to_spherical | |
33 | spherical_to_cartesian | |
34 | spherical_to_cylindrical); | |
35 | ||
bf5f1b4c JH |
36 | my @greatcircle = qw( |
37 | great_circle_distance | |
38 | great_circle_direction | |
39 | great_circle_bearing | |
40 | great_circle_waypoint | |
41 | great_circle_midpoint | |
42 | great_circle_destination | |
43 | ); | |
d54bf66f | 44 | |
affad850 | 45 | my @pi = qw(pi pi2 pi4 pip2 pip4); |
bf5f1b4c JH |
46 | |
47 | @EXPORT_OK = (@rdlcnv, @greatcircle, @pi); | |
48 | ||
49 | # See e.g. the following pages: | |
50 | # http://www.movable-type.co.uk/scripts/LatLong.html | |
51 | # http://williams.best.vwh.net/avform.htm | |
52 | ||
53 | %EXPORT_TAGS = ('radial' => [ @rdlcnv ], | |
54 | 'great_circle' => [ @greatcircle ], | |
55 | 'pi' => [ @pi ]); | |
d54bf66f | 56 | |
affad850 SP |
57 | sub _DR () { pi2/360 } |
58 | sub _RD () { 360/pi2 } | |
59 | sub _DG () { 400/360 } | |
60 | sub _GD () { 360/400 } | |
61 | sub _RG () { 400/pi2 } | |
62 | sub _GR () { pi2/400 } | |
5aabfad6 | 63 | |
64 | # | |
65 | # Truncating remainder. | |
66 | # | |
67 | ||
affad850 | 68 | sub _remt ($$) { |
5aabfad6 | 69 | # Oh yes, POSIX::fmod() would be faster. Possibly. If it is available. |
70 | $_[0] - $_[1] * int($_[0] / $_[1]); | |
71 | } | |
72 | ||
73 | # | |
74 | # Angle conversions. | |
75 | # | |
76 | ||
affad850 | 77 | sub rad2rad($) { _remt($_[0], pi2) } |
9db5a202 | 78 | |
affad850 | 79 | sub deg2deg($) { _remt($_[0], 360) } |
9db5a202 | 80 | |
affad850 | 81 | sub grad2grad($) { _remt($_[0], 400) } |
5aabfad6 | 82 | |
affad850 | 83 | sub rad2deg ($;$) { my $d = _RD * $_[0]; $_[1] ? $d : deg2deg($d) } |
5aabfad6 | 84 | |
affad850 | 85 | sub deg2rad ($;$) { my $d = _DR * $_[0]; $_[1] ? $d : rad2rad($d) } |
5aabfad6 | 86 | |
affad850 | 87 | sub grad2deg ($;$) { my $d = _GD * $_[0]; $_[1] ? $d : deg2deg($d) } |
5aabfad6 | 88 | |
affad850 | 89 | sub deg2grad ($;$) { my $d = _DG * $_[0]; $_[1] ? $d : grad2grad($d) } |
5aabfad6 | 90 | |
affad850 | 91 | sub rad2grad ($;$) { my $d = _RG * $_[0]; $_[1] ? $d : grad2grad($d) } |
9db5a202 | 92 | |
affad850 | 93 | sub grad2rad ($;$) { my $d = _GR * $_[0]; $_[1] ? $d : rad2rad($d) } |
5aabfad6 | 94 | |
d54bf66f JH |
95 | sub cartesian_to_spherical { |
96 | my ( $x, $y, $z ) = @_; | |
97 | ||
98 | my $rho = sqrt( $x * $x + $y * $y + $z * $z ); | |
99 | ||
100 | return ( $rho, | |
101 | atan2( $y, $x ), | |
102 | $rho ? acos( $z / $rho ) : 0 ); | |
103 | } | |
104 | ||
105 | sub spherical_to_cartesian { | |
106 | my ( $rho, $theta, $phi ) = @_; | |
107 | ||
108 | return ( $rho * cos( $theta ) * sin( $phi ), | |
109 | $rho * sin( $theta ) * sin( $phi ), | |
110 | $rho * cos( $phi ) ); | |
111 | } | |
112 | ||
113 | sub spherical_to_cylindrical { | |
114 | my ( $x, $y, $z ) = spherical_to_cartesian( @_ ); | |
115 | ||
116 | return ( sqrt( $x * $x + $y * $y ), $_[1], $z ); | |
117 | } | |
118 | ||
119 | sub cartesian_to_cylindrical { | |
120 | my ( $x, $y, $z ) = @_; | |
121 | ||
122 | return ( sqrt( $x * $x + $y * $y ), atan2( $y, $x ), $z ); | |
123 | } | |
124 | ||
125 | sub cylindrical_to_cartesian { | |
126 | my ( $rho, $theta, $z ) = @_; | |
127 | ||
128 | return ( $rho * cos( $theta ), $rho * sin( $theta ), $z ); | |
129 | } | |
130 | ||
131 | sub cylindrical_to_spherical { | |
132 | return ( cartesian_to_spherical( cylindrical_to_cartesian( @_ ) ) ); | |
133 | } | |
134 | ||
135 | sub great_circle_distance { | |
136 | my ( $theta0, $phi0, $theta1, $phi1, $rho ) = @_; | |
137 | ||
138 | $rho = 1 unless defined $rho; # Default to the unit sphere. | |
139 | ||
140 | my $lat0 = pip2 - $phi0; | |
141 | my $lat1 = pip2 - $phi1; | |
142 | ||
143 | return $rho * | |
144 | acos(cos( $lat0 ) * cos( $lat1 ) * cos( $theta0 - $theta1 ) + | |
145 | sin( $lat0 ) * sin( $lat1 ) ); | |
146 | } | |
147 | ||
7e5f197a JH |
148 | sub great_circle_direction { |
149 | my ( $theta0, $phi0, $theta1, $phi1 ) = @_; | |
150 | ||
d139edd6 JH |
151 | my $distance = &great_circle_distance; |
152 | ||
7e5f197a JH |
153 | my $lat0 = pip2 - $phi0; |
154 | my $lat1 = pip2 - $phi1; | |
155 | ||
156 | my $direction = | |
d139edd6 JH |
157 | acos((sin($lat1) - sin($lat0) * cos($distance)) / |
158 | (cos($lat0) * sin($distance))); | |
159 | ||
160 | $direction = pi2 - $direction | |
161 | if sin($theta1 - $theta0) < 0; | |
7e5f197a JH |
162 | |
163 | return rad2rad($direction); | |
164 | } | |
165 | ||
bf5f1b4c JH |
166 | *great_circle_bearing = \&great_circle_direction; |
167 | ||
168 | sub great_circle_waypoint { | |
169 | my ( $theta0, $phi0, $theta1, $phi1, $point ) = @_; | |
170 | ||
171 | $point = 0.5 unless defined $point; | |
172 | ||
173 | my $d = great_circle_distance( $theta0, $phi0, $theta1, $phi1 ); | |
174 | ||
175 | return undef if $d == pi; | |
176 | ||
177 | my $sd = sin($d); | |
178 | ||
179 | return ($theta0, $phi0) if $sd == 0; | |
180 | ||
181 | my $A = sin((1 - $point) * $d) / $sd; | |
182 | my $B = sin( $point * $d) / $sd; | |
183 | ||
184 | my $lat0 = pip2 - $phi0; | |
185 | my $lat1 = pip2 - $phi1; | |
186 | ||
187 | my $x = $A * cos($lat0) * cos($theta0) + $B * cos($lat1) * cos($theta1); | |
188 | my $y = $A * cos($lat0) * sin($theta0) + $B * cos($lat1) * sin($theta1); | |
189 | my $z = $A * sin($lat0) + $B * sin($lat1); | |
190 | ||
191 | my $theta = atan2($y, $x); | |
618e05e9 | 192 | my $phi = acos($z); |
bf5f1b4c JH |
193 | |
194 | return ($theta, $phi); | |
195 | } | |
196 | ||
197 | sub great_circle_midpoint { | |
198 | great_circle_waypoint(@_[0..3], 0.5); | |
199 | } | |
200 | ||
201 | sub great_circle_destination { | |
202 | my ( $theta0, $phi0, $dir0, $dst ) = @_; | |
203 | ||
204 | my $lat0 = pip2 - $phi0; | |
205 | ||
206 | my $phi1 = asin(sin($lat0)*cos($dst)+cos($lat0)*sin($dst)*cos($dir0)); | |
207 | my $theta1 = $theta0 + atan2(sin($dir0)*sin($dst)*cos($lat0), | |
208 | cos($dst)-sin($lat0)*sin($phi1)); | |
209 | ||
210 | my $dir1 = great_circle_bearing($theta1, $phi1, $theta0, $phi0) + pi; | |
211 | ||
212 | $dir1 -= pi2 if $dir1 > pi2; | |
213 | ||
214 | return ($theta1, $phi1, $dir1); | |
215 | } | |
216 | ||
ea0630ea HS |
217 | 1; |
218 | ||
219 | __END__ | |
d54bf66f JH |
220 | =pod |
221 | ||
5aabfad6 | 222 | =head1 NAME |
223 | ||
224 | Math::Trig - trigonometric functions | |
225 | ||
226 | =head1 SYNOPSIS | |
227 | ||
affad850 | 228 | use Math::Trig; |
3cb6de81 | 229 | |
affad850 SP |
230 | $x = tan(0.9); |
231 | $y = acos(3.7); | |
232 | $z = asin(2.4); | |
3cb6de81 | 233 | |
affad850 | 234 | $halfpi = pi/2; |
5aabfad6 | 235 | |
affad850 | 236 | $rad = deg2rad(120); |
5aabfad6 | 237 | |
affad850 SP |
238 | # Import constants pi2, pip2, pip4 (2*pi, pi/2, pi/4). |
239 | use Math::Trig ':pi'; | |
bf5f1b4c | 240 | |
affad850 SP |
241 | # Import the conversions between cartesian/spherical/cylindrical. |
242 | use Math::Trig ':radial'; | |
bf5f1b4c JH |
243 | |
244 | # Import the great circle formulas. | |
affad850 | 245 | use Math::Trig ':great_circle'; |
bf5f1b4c | 246 | |
5aabfad6 | 247 | =head1 DESCRIPTION |
248 | ||
249 | C<Math::Trig> defines many trigonometric functions not defined by the | |
4ae80833 | 250 | core Perl which defines only the C<sin()> and C<cos()>. The constant |
5aabfad6 | 251 | B<pi> is also defined as are a few convenience functions for angle |
bf5f1b4c | 252 | conversions, and I<great circle formulas> for spherical movement. |
5aabfad6 | 253 | |
254 | =head1 TRIGONOMETRIC FUNCTIONS | |
255 | ||
256 | The tangent | |
257 | ||
d54bf66f JH |
258 | =over 4 |
259 | ||
260 | =item B<tan> | |
261 | ||
262 | =back | |
5aabfad6 | 263 | |
264 | The cofunctions of the sine, cosine, and tangent (cosec/csc and cotan/cot | |
265 | are aliases) | |
266 | ||
d54bf66f | 267 | B<csc>, B<cosec>, B<sec>, B<sec>, B<cot>, B<cotan> |
5aabfad6 | 268 | |
269 | The arcus (also known as the inverse) functions of the sine, cosine, | |
270 | and tangent | |
271 | ||
d54bf66f | 272 | B<asin>, B<acos>, B<atan> |
5aabfad6 | 273 | |
274 | The principal value of the arc tangent of y/x | |
275 | ||
d54bf66f | 276 | B<atan2>(y, x) |
5aabfad6 | 277 | |
278 | The arcus cofunctions of the sine, cosine, and tangent (acosec/acsc | |
affad850 | 279 | and acotan/acot are aliases). Note that atan2(0, 0) is not well-defined. |
5aabfad6 | 280 | |
d54bf66f | 281 | B<acsc>, B<acosec>, B<asec>, B<acot>, B<acotan> |
5aabfad6 | 282 | |
283 | The hyperbolic sine, cosine, and tangent | |
284 | ||
d54bf66f | 285 | B<sinh>, B<cosh>, B<tanh> |
5aabfad6 | 286 | |
287 | The cofunctions of the hyperbolic sine, cosine, and tangent (cosech/csch | |
288 | and cotanh/coth are aliases) | |
289 | ||
d54bf66f | 290 | B<csch>, B<cosech>, B<sech>, B<coth>, B<cotanh> |
5aabfad6 | 291 | |
292 | The arcus (also known as the inverse) functions of the hyperbolic | |
293 | sine, cosine, and tangent | |
294 | ||
d54bf66f | 295 | B<asinh>, B<acosh>, B<atanh> |
5aabfad6 | 296 | |
297 | The arcus cofunctions of the hyperbolic sine, cosine, and tangent | |
298 | (acsch/acosech and acoth/acotanh are aliases) | |
299 | ||
d54bf66f | 300 | B<acsch>, B<acosech>, B<asech>, B<acoth>, B<acotanh> |
5aabfad6 | 301 | |
affad850 SP |
302 | The trigonometric constant B<pi> and some of handy multiples |
303 | of it are also defined. | |
5aabfad6 | 304 | |
affad850 | 305 | B<pi, pi2, pi4, pip2, pip4> |
5aabfad6 | 306 | |
5cd24f17 | 307 | =head2 ERRORS DUE TO DIVISION BY ZERO |
308 | ||
309 | The following functions | |
310 | ||
affad850 SP |
311 | acoth |
312 | acsc | |
313 | acsch | |
314 | asec | |
315 | asech | |
316 | atanh | |
317 | cot | |
318 | coth | |
319 | csc | |
320 | csch | |
321 | sec | |
322 | sech | |
323 | tan | |
324 | tanh | |
5cd24f17 | 325 | |
326 | cannot be computed for all arguments because that would mean dividing | |
8c03c583 JH |
327 | by zero or taking logarithm of zero. These situations cause fatal |
328 | runtime errors looking like this | |
5cd24f17 | 329 | |
affad850 SP |
330 | cot(0): Division by zero. |
331 | (Because in the definition of cot(0), the divisor sin(0) is 0) | |
332 | Died at ... | |
5cd24f17 | 333 | |
8c03c583 JH |
334 | or |
335 | ||
affad850 SP |
336 | atanh(-1): Logarithm of zero. |
337 | Died at... | |
8c03c583 JH |
338 | |
339 | For the C<csc>, C<cot>, C<asec>, C<acsc>, C<acot>, C<csch>, C<coth>, | |
340 | C<asech>, C<acsch>, the argument cannot be C<0> (zero). For the | |
341 | C<atanh>, C<acoth>, the argument cannot be C<1> (one). For the | |
342 | C<atanh>, C<acoth>, the argument cannot be C<-1> (minus one). For the | |
343 | C<tan>, C<sec>, C<tanh>, C<sech>, the argument cannot be I<pi/2 + k * | |
affad850 SP |
344 | pi>, where I<k> is any integer. |
345 | ||
346 | Note that atan2(0, 0) is not well-defined. | |
5cd24f17 | 347 | |
348 | =head2 SIMPLE (REAL) ARGUMENTS, COMPLEX RESULTS | |
5aabfad6 | 349 | |
350 | Please note that some of the trigonometric functions can break out | |
351 | from the B<real axis> into the B<complex plane>. For example | |
352 | C<asin(2)> has no definition for plain real numbers but it has | |
353 | definition for complex numbers. | |
354 | ||
355 | In Perl terms this means that supplying the usual Perl numbers (also | |
356 | known as scalars, please see L<perldata>) as input for the | |
357 | trigonometric functions might produce as output results that no more | |
358 | are simple real numbers: instead they are complex numbers. | |
359 | ||
360 | The C<Math::Trig> handles this by using the C<Math::Complex> package | |
361 | which knows how to handle complex numbers, please see L<Math::Complex> | |
362 | for more information. In practice you need not to worry about getting | |
363 | complex numbers as results because the C<Math::Complex> takes care of | |
364 | details like for example how to display complex numbers. For example: | |
365 | ||
affad850 | 366 | print asin(2), "\n"; |
3cb6de81 | 367 | |
5aabfad6 | 368 | should produce something like this (take or leave few last decimals): |
369 | ||
affad850 | 370 | 1.5707963267949-1.31695789692482i |
5aabfad6 | 371 | |
5cd24f17 | 372 | That is, a complex number with the real part of approximately C<1.571> |
373 | and the imaginary part of approximately C<-1.317>. | |
5aabfad6 | 374 | |
d54bf66f | 375 | =head1 PLANE ANGLE CONVERSIONS |
5aabfad6 | 376 | |
377 | (Plane, 2-dimensional) angles may be converted with the following functions. | |
378 | ||
affad850 SP |
379 | =over |
380 | ||
381 | =item deg2rad | |
382 | ||
383 | $radians = deg2rad($degrees); | |
384 | ||
385 | =item grad2rad | |
386 | ||
387 | $radians = grad2rad($gradians); | |
388 | ||
389 | =item rad2deg | |
390 | ||
391 | $degrees = rad2deg($radians); | |
3cb6de81 | 392 | |
affad850 | 393 | =item grad2deg |
3cb6de81 | 394 | |
affad850 SP |
395 | $degrees = grad2deg($gradians); |
396 | ||
397 | =item deg2grad | |
398 | ||
399 | $gradians = deg2grad($degrees); | |
400 | ||
401 | =item rad2grad | |
402 | ||
403 | $gradians = rad2grad($radians); | |
404 | ||
405 | =back | |
5aabfad6 | 406 | |
5cd24f17 | 407 | The full circle is 2 I<pi> radians or I<360> degrees or I<400> gradians. |
9db5a202 JH |
408 | The result is by default wrapped to be inside the [0, {2pi,360,400}[ circle. |
409 | If you don't want this, supply a true second argument: | |
410 | ||
affad850 SP |
411 | $zillions_of_radians = deg2rad($zillions_of_degrees, 1); |
412 | $negative_degrees = rad2deg($negative_radians, 1); | |
9db5a202 JH |
413 | |
414 | You can also do the wrapping explicitly by rad2rad(), deg2deg(), and | |
415 | grad2grad(). | |
5aabfad6 | 416 | |
affad850 SP |
417 | =over 4 |
418 | ||
419 | =item rad2rad | |
420 | ||
421 | $radians_wrapped_by_2pi = rad2rad($radians); | |
422 | ||
423 | =item deg2deg | |
424 | ||
425 | $degrees_wrapped_by_360 = deg2deg($degrees); | |
426 | ||
427 | =item grad2grad | |
428 | ||
429 | $gradians_wrapped_by_400 = grad2grad($gradians); | |
430 | ||
431 | =back | |
432 | ||
d54bf66f JH |
433 | =head1 RADIAL COORDINATE CONVERSIONS |
434 | ||
435 | B<Radial coordinate systems> are the B<spherical> and the B<cylindrical> | |
436 | systems, explained shortly in more detail. | |
437 | ||
438 | You can import radial coordinate conversion functions by using the | |
439 | C<:radial> tag: | |
440 | ||
441 | use Math::Trig ':radial'; | |
442 | ||
443 | ($rho, $theta, $z) = cartesian_to_cylindrical($x, $y, $z); | |
444 | ($rho, $theta, $phi) = cartesian_to_spherical($x, $y, $z); | |
445 | ($x, $y, $z) = cylindrical_to_cartesian($rho, $theta, $z); | |
446 | ($rho_s, $theta, $phi) = cylindrical_to_spherical($rho_c, $theta, $z); | |
447 | ($x, $y, $z) = spherical_to_cartesian($rho, $theta, $phi); | |
448 | ($rho_c, $theta, $z) = spherical_to_cylindrical($rho_s, $theta, $phi); | |
449 | ||
450 | B<All angles are in radians>. | |
451 | ||
452 | =head2 COORDINATE SYSTEMS | |
453 | ||
bf5f1b4c | 454 | B<Cartesian> coordinates are the usual rectangular I<(x, y, z)>-coordinates. |
d54bf66f JH |
455 | |
456 | Spherical coordinates, I<(rho, theta, pi)>, are three-dimensional | |
457 | coordinates which define a point in three-dimensional space. They are | |
458 | based on a sphere surface. The radius of the sphere is B<rho>, also | |
459 | known as the I<radial> coordinate. The angle in the I<xy>-plane | |
460 | (around the I<z>-axis) is B<theta>, also known as the I<azimuthal> | |
461 | coordinate. The angle from the I<z>-axis is B<phi>, also known as the | |
2d6f5264 JH |
462 | I<polar> coordinate. The North Pole is therefore I<0, 0, rho>, and |
463 | the Gulf of Guinea (think of the missing big chunk of Africa) I<0, | |
4b0d1da8 JH |
464 | pi/2, rho>. In geographical terms I<phi> is latitude (northward |
465 | positive, southward negative) and I<theta> is longitude (eastward | |
466 | positive, westward negative). | |
d54bf66f | 467 | |
4b0d1da8 | 468 | B<BEWARE>: some texts define I<theta> and I<phi> the other way round, |
d54bf66f JH |
469 | some texts define the I<phi> to start from the horizontal plane, some |
470 | texts use I<r> in place of I<rho>. | |
471 | ||
472 | Cylindrical coordinates, I<(rho, theta, z)>, are three-dimensional | |
473 | coordinates which define a point in three-dimensional space. They are | |
474 | based on a cylinder surface. The radius of the cylinder is B<rho>, | |
475 | also known as the I<radial> coordinate. The angle in the I<xy>-plane | |
476 | (around the I<z>-axis) is B<theta>, also known as the I<azimuthal> | |
477 | coordinate. The third coordinate is the I<z>, pointing up from the | |
478 | B<theta>-plane. | |
479 | ||
480 | =head2 3-D ANGLE CONVERSIONS | |
481 | ||
482 | Conversions to and from spherical and cylindrical coordinates are | |
483 | available. Please notice that the conversions are not necessarily | |
484 | reversible because of the equalities like I<pi> angles being equal to | |
485 | I<-pi> angles. | |
486 | ||
487 | =over 4 | |
488 | ||
489 | =item cartesian_to_cylindrical | |
490 | ||
affad850 | 491 | ($rho, $theta, $z) = cartesian_to_cylindrical($x, $y, $z); |
d54bf66f JH |
492 | |
493 | =item cartesian_to_spherical | |
494 | ||
affad850 | 495 | ($rho, $theta, $phi) = cartesian_to_spherical($x, $y, $z); |
d54bf66f JH |
496 | |
497 | =item cylindrical_to_cartesian | |
498 | ||
affad850 | 499 | ($x, $y, $z) = cylindrical_to_cartesian($rho, $theta, $z); |
d54bf66f JH |
500 | |
501 | =item cylindrical_to_spherical | |
502 | ||
affad850 | 503 | ($rho_s, $theta, $phi) = cylindrical_to_spherical($rho_c, $theta, $z); |
d54bf66f JH |
504 | |
505 | Notice that when C<$z> is not 0 C<$rho_s> is not equal to C<$rho_c>. | |
506 | ||
507 | =item spherical_to_cartesian | |
508 | ||
affad850 | 509 | ($x, $y, $z) = spherical_to_cartesian($rho, $theta, $phi); |
d54bf66f JH |
510 | |
511 | =item spherical_to_cylindrical | |
512 | ||
affad850 | 513 | ($rho_c, $theta, $z) = spherical_to_cylindrical($rho_s, $theta, $phi); |
d54bf66f JH |
514 | |
515 | Notice that when C<$z> is not 0 C<$rho_c> is not equal to C<$rho_s>. | |
516 | ||
517 | =back | |
518 | ||
7e5f197a | 519 | =head1 GREAT CIRCLE DISTANCES AND DIRECTIONS |
d54bf66f | 520 | |
affad850 SP |
521 | A great circle is section of a circle that contains the circle |
522 | diameter: the shortest distance between two (non-antipodal) points on | |
523 | the spherical surface goes along the great circle connecting those two | |
524 | points. | |
525 | ||
526 | =head2 great_circle_distance | |
527 | ||
d54bf66f | 528 | You can compute spherical distances, called B<great circle distances>, |
7e5f197a | 529 | by importing the great_circle_distance() function: |
d54bf66f | 530 | |
7e5f197a | 531 | use Math::Trig 'great_circle_distance'; |
d54bf66f | 532 | |
4b0d1da8 | 533 | $distance = great_circle_distance($theta0, $phi0, $theta1, $phi1, [, $rho]); |
d54bf66f JH |
534 | |
535 | The I<great circle distance> is the shortest distance between two | |
536 | points on a sphere. The distance is in C<$rho> units. The C<$rho> is | |
537 | optional, it defaults to 1 (the unit sphere), therefore the distance | |
538 | defaults to radians. | |
539 | ||
4b0d1da8 JH |
540 | If you think geographically the I<theta> are longitudes: zero at the |
541 | Greenwhich meridian, eastward positive, westward negative--and the | |
2d06e7d7 | 542 | I<phi> are latitudes: zero at the North Pole, northward positive, |
4b0d1da8 | 543 | southward negative. B<NOTE>: this formula thinks in mathematics, not |
2d06e7d7 JH |
544 | geographically: the I<phi> zero is at the North Pole, not at the |
545 | Equator on the west coast of Africa (Bay of Guinea). You need to | |
546 | subtract your geographical coordinates from I<pi/2> (also known as 90 | |
547 | degrees). | |
4b0d1da8 JH |
548 | |
549 | $distance = great_circle_distance($lon0, pi/2 - $lat0, | |
550 | $lon1, pi/2 - $lat1, $rho); | |
551 | ||
affad850 SP |
552 | =head2 great_circle_direction |
553 | ||
bf5f1b4c JH |
554 | The direction you must follow the great circle (also known as I<bearing>) |
555 | can be computed by the great_circle_direction() function: | |
7e5f197a JH |
556 | |
557 | use Math::Trig 'great_circle_direction'; | |
558 | ||
559 | $direction = great_circle_direction($theta0, $phi0, $theta1, $phi1); | |
560 | ||
affad850 SP |
561 | =head2 great_circle_bearing |
562 | ||
563 | Alias 'great_circle_bearing' for 'great_circle_direction' is also available. | |
564 | ||
565 | use Math::Trig 'great_circle_bearing'; | |
566 | ||
567 | $direction = great_circle_bearing($theta0, $phi0, $theta1, $phi1); | |
568 | ||
569 | The result of great_circle_direction is in radians, zero indicating | |
570 | straight north, pi or -pi straight south, pi/2 straight west, and | |
571 | -pi/2 straight east. | |
7e5f197a | 572 | |
bf5f1b4c JH |
573 | You can inversely compute the destination if you know the |
574 | starting point, direction, and distance: | |
575 | ||
affad850 SP |
576 | =head2 great_circle_destination |
577 | ||
bf5f1b4c JH |
578 | use Math::Trig 'great_circle_destination'; |
579 | ||
580 | # thetad and phid are the destination coordinates, | |
581 | # dird is the final direction at the destination. | |
582 | ||
583 | ($thetad, $phid, $dird) = | |
584 | great_circle_destination($theta, $phi, $direction, $distance); | |
585 | ||
586 | or the midpoint if you know the end points: | |
587 | ||
affad850 SP |
588 | =head2 great_circle_midpoint |
589 | ||
bf5f1b4c JH |
590 | use Math::Trig 'great_circle_midpoint'; |
591 | ||
592 | ($thetam, $phim) = | |
593 | great_circle_midpoint($theta0, $phi0, $theta1, $phi1); | |
594 | ||
595 | The great_circle_midpoint() is just a special case of | |
596 | ||
affad850 SP |
597 | =head2 great_circle_waypoint |
598 | ||
bf5f1b4c JH |
599 | use Math::Trig 'great_circle_waypoint'; |
600 | ||
601 | ($thetai, $phii) = | |
602 | great_circle_waypoint($theta0, $phi0, $theta1, $phi1, $way); | |
603 | ||
604 | Where the $way is a value from zero ($theta0, $phi0) to one ($theta1, | |
605 | $phi1). Note that antipodal points (where their distance is I<pi> | |
606 | radians) do not have waypoints between them (they would have an an | |
607 | "equator" between them), and therefore C<undef> is returned for | |
608 | antipodal points. If the points are the same and the distance | |
609 | therefore zero and all waypoints therefore identical, the first point | |
610 | (either point) is returned. | |
611 | ||
612 | The thetas, phis, direction, and distance in the above are all in radians. | |
613 | ||
614 | You can import all the great circle formulas by | |
615 | ||
616 | use Math::Trig ':great_circle'; | |
617 | ||
7e5f197a JH |
618 | Notice that the resulting directions might be somewhat surprising if |
619 | you are looking at a flat worldmap: in such map projections the great | |
620 | circles quite often do not look like the shortest routes-- but for | |
621 | example the shortest possible routes from Europe or North America to | |
622 | Asia do often cross the polar regions. | |
623 | ||
51301382 | 624 | =head1 EXAMPLES |
d54bf66f | 625 | |
7e5f197a JH |
626 | To calculate the distance between London (51.3N 0.5W) and Tokyo |
627 | (35.7N 139.8E) in kilometers: | |
d54bf66f | 628 | |
affad850 | 629 | use Math::Trig qw(great_circle_distance deg2rad); |
d54bf66f | 630 | |
affad850 SP |
631 | # Notice the 90 - latitude: phi zero is at the North Pole. |
632 | sub NESW { deg2rad($_[0]), deg2rad(90 - $_[1]) } | |
633 | my @L = NESW( -0.5, 51.3); | |
634 | my @T = NESW(139.8, 35.7); | |
635 | my $km = great_circle_distance(@L, @T, 6378); # About 9600 km. | |
d54bf66f | 636 | |
bf5f1b4c JH |
637 | The direction you would have to go from London to Tokyo (in radians, |
638 | straight north being zero, straight east being pi/2). | |
7e5f197a | 639 | |
affad850 | 640 | use Math::Trig qw(great_circle_direction); |
7e5f197a | 641 | |
affad850 | 642 | my $rad = great_circle_direction(@L, @T); # About 0.547 or 0.174 pi. |
7e5f197a | 643 | |
bf5f1b4c | 644 | The midpoint between London and Tokyo being |
7e5f197a | 645 | |
affad850 | 646 | use Math::Trig qw(great_circle_midpoint); |
bf5f1b4c | 647 | |
affad850 | 648 | my @M = great_circle_midpoint(@L, @T); |
bf5f1b4c | 649 | |
618e05e9 | 650 | or about 89.16N 68.93E, practically at the North Pole. |
41bd693c | 651 | |
bf5f1b4c | 652 | =head2 CAVEAT FOR GREAT CIRCLE FORMULAS |
41bd693c | 653 | |
bf5f1b4c JH |
654 | The answers may be off by few percentages because of the irregular |
655 | (slightly aspherical) form of the Earth. The errors are at worst | |
656 | about 0.55%, but generally below 0.3%. | |
d54bf66f | 657 | |
5cd24f17 | 658 | =head1 BUGS |
5aabfad6 | 659 | |
5cd24f17 | 660 | Saying C<use Math::Trig;> exports many mathematical routines in the |
661 | caller environment and even overrides some (C<sin>, C<cos>). This is | |
662 | construed as a feature by the Authors, actually... ;-) | |
5aabfad6 | 663 | |
5cd24f17 | 664 | The code is not optimized for speed, especially because we use |
665 | C<Math::Complex> and thus go quite near complex numbers while doing | |
666 | the computations even when the arguments are not. This, however, | |
667 | cannot be completely avoided if we want things like C<asin(2)> to give | |
668 | an answer instead of giving a fatal runtime error. | |
5aabfad6 | 669 | |
bf5f1b4c JH |
670 | Do not attempt navigation using these formulas. |
671 | ||
5cd24f17 | 672 | =head1 AUTHORS |
5aabfad6 | 673 | |
affad850 SP |
674 | Jarkko Hietaniemi <F<jhi!at!iki.fi>> and |
675 | Raphael Manfredi <F<Raphael_Manfredi!at!pobox.com>>. | |
5aabfad6 | 676 | |
677 | =cut | |
678 | ||
679 | # eof |