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1 | # |
2 | # Trigonometric functions, mostly inherited from Math::Complex. |
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3 | # -- Jarkko Hietaniemi, since April 1997 |
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4 | # -- Raphael Manfredi, September 1996 (indirectly: because of Math::Complex) |
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5 | # |
6 | |
7 | require Exporter; |
8 | package Math::Trig; |
9 | |
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10 | use 5.006; |
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11 | use strict; |
12 | |
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13 | use Math::Complex 1.35; |
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14 | use Math::Complex qw(:trig); |
15 | |
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16 | our($VERSION, $PACKAGE, @ISA, @EXPORT, @EXPORT_OK, %EXPORT_TAGS); |
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17 | |
18 | @ISA = qw(Exporter); |
19 | |
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20 | $VERSION = 1.03; |
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21 | |
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22 | my @angcnv = qw(rad2deg rad2grad |
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23 | deg2rad deg2grad |
24 | grad2rad grad2deg); |
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25 | |
26 | @EXPORT = (@{$Math::Complex::EXPORT_TAGS{'trig'}}, |
27 | @angcnv); |
28 | |
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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 | |
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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 | ); |
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44 | |
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45 | my @pi = qw(pi2 pip2 pip4); |
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 ]); |
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56 | |
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57 | sub pi2 () { 2 * pi } |
58 | sub pip2 () { pi / 2 } |
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59 | sub pip4 () { pi / 4 } |
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60 | |
61 | sub DR () { pi2/360 } |
62 | sub RD () { 360/pi2 } |
63 | sub DG () { 400/360 } |
64 | sub GD () { 360/400 } |
65 | sub RG () { 400/pi2 } |
66 | sub GR () { pi2/400 } |
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67 | |
68 | # |
69 | # Truncating remainder. |
70 | # |
71 | |
72 | sub remt ($$) { |
73 | # Oh yes, POSIX::fmod() would be faster. Possibly. If it is available. |
74 | $_[0] - $_[1] * int($_[0] / $_[1]); |
75 | } |
76 | |
77 | # |
78 | # Angle conversions. |
79 | # |
80 | |
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81 | sub rad2rad($) { remt($_[0], pi2) } |
82 | |
83 | sub deg2deg($) { remt($_[0], 360) } |
84 | |
85 | sub grad2grad($) { remt($_[0], 400) } |
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86 | |
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87 | sub rad2deg ($;$) { my $d = RD * $_[0]; $_[1] ? $d : deg2deg($d) } |
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88 | |
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89 | sub deg2rad ($;$) { my $d = DR * $_[0]; $_[1] ? $d : rad2rad($d) } |
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90 | |
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91 | sub grad2deg ($;$) { my $d = GD * $_[0]; $_[1] ? $d : deg2deg($d) } |
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92 | |
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93 | sub deg2grad ($;$) { my $d = DG * $_[0]; $_[1] ? $d : grad2grad($d) } |
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94 | |
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95 | sub rad2grad ($;$) { my $d = RG * $_[0]; $_[1] ? $d : grad2grad($d) } |
96 | |
97 | sub grad2rad ($;$) { my $d = GR * $_[0]; $_[1] ? $d : rad2rad($d) } |
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98 | |
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99 | sub cartesian_to_spherical { |
100 | my ( $x, $y, $z ) = @_; |
101 | |
102 | my $rho = sqrt( $x * $x + $y * $y + $z * $z ); |
103 | |
104 | return ( $rho, |
105 | atan2( $y, $x ), |
106 | $rho ? acos( $z / $rho ) : 0 ); |
107 | } |
108 | |
109 | sub spherical_to_cartesian { |
110 | my ( $rho, $theta, $phi ) = @_; |
111 | |
112 | return ( $rho * cos( $theta ) * sin( $phi ), |
113 | $rho * sin( $theta ) * sin( $phi ), |
114 | $rho * cos( $phi ) ); |
115 | } |
116 | |
117 | sub spherical_to_cylindrical { |
118 | my ( $x, $y, $z ) = spherical_to_cartesian( @_ ); |
119 | |
120 | return ( sqrt( $x * $x + $y * $y ), $_[1], $z ); |
121 | } |
122 | |
123 | sub cartesian_to_cylindrical { |
124 | my ( $x, $y, $z ) = @_; |
125 | |
126 | return ( sqrt( $x * $x + $y * $y ), atan2( $y, $x ), $z ); |
127 | } |
128 | |
129 | sub cylindrical_to_cartesian { |
130 | my ( $rho, $theta, $z ) = @_; |
131 | |
132 | return ( $rho * cos( $theta ), $rho * sin( $theta ), $z ); |
133 | } |
134 | |
135 | sub cylindrical_to_spherical { |
136 | return ( cartesian_to_spherical( cylindrical_to_cartesian( @_ ) ) ); |
137 | } |
138 | |
139 | sub great_circle_distance { |
140 | my ( $theta0, $phi0, $theta1, $phi1, $rho ) = @_; |
141 | |
142 | $rho = 1 unless defined $rho; # Default to the unit sphere. |
143 | |
144 | my $lat0 = pip2 - $phi0; |
145 | my $lat1 = pip2 - $phi1; |
146 | |
147 | return $rho * |
148 | acos(cos( $lat0 ) * cos( $lat1 ) * cos( $theta0 - $theta1 ) + |
149 | sin( $lat0 ) * sin( $lat1 ) ); |
150 | } |
151 | |
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152 | sub great_circle_direction { |
153 | my ( $theta0, $phi0, $theta1, $phi1 ) = @_; |
154 | |
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155 | my $distance = &great_circle_distance; |
156 | |
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157 | my $lat0 = pip2 - $phi0; |
158 | my $lat1 = pip2 - $phi1; |
159 | |
160 | my $direction = |
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161 | acos((sin($lat1) - sin($lat0) * cos($distance)) / |
162 | (cos($lat0) * sin($distance))); |
163 | |
164 | $direction = pi2 - $direction |
165 | if sin($theta1 - $theta0) < 0; |
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166 | |
167 | return rad2rad($direction); |
168 | } |
169 | |
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170 | *great_circle_bearing = \&great_circle_direction; |
171 | |
172 | sub great_circle_waypoint { |
173 | my ( $theta0, $phi0, $theta1, $phi1, $point ) = @_; |
174 | |
175 | $point = 0.5 unless defined $point; |
176 | |
177 | my $d = great_circle_distance( $theta0, $phi0, $theta1, $phi1 ); |
178 | |
179 | return undef if $d == pi; |
180 | |
181 | my $sd = sin($d); |
182 | |
183 | return ($theta0, $phi0) if $sd == 0; |
184 | |
185 | my $A = sin((1 - $point) * $d) / $sd; |
186 | my $B = sin( $point * $d) / $sd; |
187 | |
188 | my $lat0 = pip2 - $phi0; |
189 | my $lat1 = pip2 - $phi1; |
190 | |
191 | my $x = $A * cos($lat0) * cos($theta0) + $B * cos($lat1) * cos($theta1); |
192 | my $y = $A * cos($lat0) * sin($theta0) + $B * cos($lat1) * sin($theta1); |
193 | my $z = $A * sin($lat0) + $B * sin($lat1); |
194 | |
195 | my $theta = atan2($y, $x); |
196 | my $phi = atan2($z, sqrt($x*$x + $y*$y)); |
197 | |
198 | return ($theta, $phi); |
199 | } |
200 | |
201 | sub great_circle_midpoint { |
202 | great_circle_waypoint(@_[0..3], 0.5); |
203 | } |
204 | |
205 | sub great_circle_destination { |
206 | my ( $theta0, $phi0, $dir0, $dst ) = @_; |
207 | |
208 | my $lat0 = pip2 - $phi0; |
209 | |
210 | my $phi1 = asin(sin($lat0)*cos($dst)+cos($lat0)*sin($dst)*cos($dir0)); |
211 | my $theta1 = $theta0 + atan2(sin($dir0)*sin($dst)*cos($lat0), |
212 | cos($dst)-sin($lat0)*sin($phi1)); |
213 | |
214 | my $dir1 = great_circle_bearing($theta1, $phi1, $theta0, $phi0) + pi; |
215 | |
216 | $dir1 -= pi2 if $dir1 > pi2; |
217 | |
218 | return ($theta1, $phi1, $dir1); |
219 | } |
220 | |
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221 | 1; |
222 | |
223 | __END__ |
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224 | =pod |
225 | |
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226 | =head1 NAME |
227 | |
228 | Math::Trig - trigonometric functions |
229 | |
230 | =head1 SYNOPSIS |
231 | |
232 | use Math::Trig; |
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233 | |
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234 | $x = tan(0.9); |
235 | $y = acos(3.7); |
236 | $z = asin(2.4); |
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237 | |
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238 | $halfpi = pi/2; |
239 | |
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240 | $rad = deg2rad(120); |
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241 | |
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242 | # Import constants pi2, pip2, pip4 (2*pi, pi/2, pi/4). |
243 | use Math::Trig ':pi'; |
244 | |
245 | # Import the conversions between cartesian/spherical/cylindrical. |
246 | use Math::Trig ':radial'; |
247 | |
248 | # Import the great circle formulas. |
249 | use Math::Trig ':great_circle'; |
250 | |
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251 | =head1 DESCRIPTION |
252 | |
253 | C<Math::Trig> defines many trigonometric functions not defined by the |
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254 | core Perl which defines only the C<sin()> and C<cos()>. The constant |
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255 | B<pi> is also defined as are a few convenience functions for angle |
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256 | conversions, and I<great circle formulas> for spherical movement. |
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257 | |
258 | =head1 TRIGONOMETRIC FUNCTIONS |
259 | |
260 | The tangent |
261 | |
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262 | =over 4 |
263 | |
264 | =item B<tan> |
265 | |
266 | =back |
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267 | |
268 | The cofunctions of the sine, cosine, and tangent (cosec/csc and cotan/cot |
269 | are aliases) |
270 | |
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271 | B<csc>, B<cosec>, B<sec>, B<sec>, B<cot>, B<cotan> |
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272 | |
273 | The arcus (also known as the inverse) functions of the sine, cosine, |
274 | and tangent |
275 | |
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276 | B<asin>, B<acos>, B<atan> |
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277 | |
278 | The principal value of the arc tangent of y/x |
279 | |
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280 | B<atan2>(y, x) |
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281 | |
282 | The arcus cofunctions of the sine, cosine, and tangent (acosec/acsc |
283 | and acotan/acot are aliases) |
284 | |
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285 | B<acsc>, B<acosec>, B<asec>, B<acot>, B<acotan> |
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286 | |
287 | The hyperbolic sine, cosine, and tangent |
288 | |
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289 | B<sinh>, B<cosh>, B<tanh> |
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290 | |
291 | The cofunctions of the hyperbolic sine, cosine, and tangent (cosech/csch |
292 | and cotanh/coth are aliases) |
293 | |
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294 | B<csch>, B<cosech>, B<sech>, B<coth>, B<cotanh> |
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295 | |
296 | The arcus (also known as the inverse) functions of the hyperbolic |
297 | sine, cosine, and tangent |
298 | |
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299 | B<asinh>, B<acosh>, B<atanh> |
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300 | |
301 | The arcus cofunctions of the hyperbolic sine, cosine, and tangent |
302 | (acsch/acosech and acoth/acotanh are aliases) |
303 | |
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304 | B<acsch>, B<acosech>, B<asech>, B<acoth>, B<acotanh> |
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305 | |
306 | The trigonometric constant B<pi> is also defined. |
307 | |
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308 | $pi2 = 2 * B<pi>; |
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309 | |
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310 | =head2 ERRORS DUE TO DIVISION BY ZERO |
311 | |
312 | The following functions |
313 | |
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314 | acoth |
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315 | acsc |
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316 | acsch |
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317 | asec |
318 | asech |
319 | atanh |
320 | cot |
321 | coth |
322 | csc |
323 | csch |
324 | sec |
325 | sech |
326 | tan |
327 | tanh |
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328 | |
329 | cannot be computed for all arguments because that would mean dividing |
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330 | by zero or taking logarithm of zero. These situations cause fatal |
331 | runtime errors looking like this |
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332 | |
333 | cot(0): Division by zero. |
334 | (Because in the definition of cot(0), the divisor sin(0) is 0) |
335 | Died at ... |
336 | |
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337 | or |
338 | |
339 | atanh(-1): Logarithm of zero. |
340 | Died at... |
341 | |
342 | For the C<csc>, C<cot>, C<asec>, C<acsc>, C<acot>, C<csch>, C<coth>, |
343 | C<asech>, C<acsch>, the argument cannot be C<0> (zero). For the |
344 | C<atanh>, C<acoth>, the argument cannot be C<1> (one). For the |
345 | C<atanh>, C<acoth>, the argument cannot be C<-1> (minus one). For the |
346 | C<tan>, C<sec>, C<tanh>, C<sech>, the argument cannot be I<pi/2 + k * |
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347 | pi>, where I<k> is any integer. atan2(0, 0) is undefined. |
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348 | |
349 | =head2 SIMPLE (REAL) ARGUMENTS, COMPLEX RESULTS |
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350 | |
351 | Please note that some of the trigonometric functions can break out |
352 | from the B<real axis> into the B<complex plane>. For example |
353 | C<asin(2)> has no definition for plain real numbers but it has |
354 | definition for complex numbers. |
355 | |
356 | In Perl terms this means that supplying the usual Perl numbers (also |
357 | known as scalars, please see L<perldata>) as input for the |
358 | trigonometric functions might produce as output results that no more |
359 | are simple real numbers: instead they are complex numbers. |
360 | |
361 | The C<Math::Trig> handles this by using the C<Math::Complex> package |
362 | which knows how to handle complex numbers, please see L<Math::Complex> |
363 | for more information. In practice you need not to worry about getting |
364 | complex numbers as results because the C<Math::Complex> takes care of |
365 | details like for example how to display complex numbers. For example: |
366 | |
367 | print asin(2), "\n"; |
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368 | |
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369 | should produce something like this (take or leave few last decimals): |
370 | |
371 | 1.5707963267949-1.31695789692482i |
372 | |
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373 | That is, a complex number with the real part of approximately C<1.571> |
374 | and the imaginary part of approximately C<-1.317>. |
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375 | |
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376 | =head1 PLANE ANGLE CONVERSIONS |
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377 | |
378 | (Plane, 2-dimensional) angles may be converted with the following functions. |
379 | |
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380 | $radians = deg2rad($degrees); |
381 | $radians = grad2rad($gradians); |
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382 | |
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383 | $degrees = rad2deg($radians); |
384 | $degrees = grad2deg($gradians); |
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385 | |
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386 | $gradians = deg2grad($degrees); |
387 | $gradians = rad2grad($radians); |
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388 | |
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389 | The full circle is 2 I<pi> radians or I<360> degrees or I<400> gradians. |
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390 | The result is by default wrapped to be inside the [0, {2pi,360,400}[ circle. |
391 | If you don't want this, supply a true second argument: |
392 | |
393 | $zillions_of_radians = deg2rad($zillions_of_degrees, 1); |
394 | $negative_degrees = rad2deg($negative_radians, 1); |
395 | |
396 | You can also do the wrapping explicitly by rad2rad(), deg2deg(), and |
397 | grad2grad(). |
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398 | |
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399 | =head1 RADIAL COORDINATE CONVERSIONS |
400 | |
401 | B<Radial coordinate systems> are the B<spherical> and the B<cylindrical> |
402 | systems, explained shortly in more detail. |
403 | |
404 | You can import radial coordinate conversion functions by using the |
405 | C<:radial> tag: |
406 | |
407 | use Math::Trig ':radial'; |
408 | |
409 | ($rho, $theta, $z) = cartesian_to_cylindrical($x, $y, $z); |
410 | ($rho, $theta, $phi) = cartesian_to_spherical($x, $y, $z); |
411 | ($x, $y, $z) = cylindrical_to_cartesian($rho, $theta, $z); |
412 | ($rho_s, $theta, $phi) = cylindrical_to_spherical($rho_c, $theta, $z); |
413 | ($x, $y, $z) = spherical_to_cartesian($rho, $theta, $phi); |
414 | ($rho_c, $theta, $z) = spherical_to_cylindrical($rho_s, $theta, $phi); |
415 | |
416 | B<All angles are in radians>. |
417 | |
418 | =head2 COORDINATE SYSTEMS |
419 | |
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420 | B<Cartesian> coordinates are the usual rectangular I<(x, y, z)>-coordinates. |
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421 | |
422 | Spherical coordinates, I<(rho, theta, pi)>, are three-dimensional |
423 | coordinates which define a point in three-dimensional space. They are |
424 | based on a sphere surface. The radius of the sphere is B<rho>, also |
425 | known as the I<radial> coordinate. The angle in the I<xy>-plane |
426 | (around the I<z>-axis) is B<theta>, also known as the I<azimuthal> |
427 | coordinate. The angle from the I<z>-axis is B<phi>, also known as the |
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428 | I<polar> coordinate. The North Pole is therefore I<0, 0, rho>, and |
429 | the Gulf of Guinea (think of the missing big chunk of Africa) I<0, |
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430 | pi/2, rho>. In geographical terms I<phi> is latitude (northward |
431 | positive, southward negative) and I<theta> is longitude (eastward |
432 | positive, westward negative). |
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433 | |
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434 | B<BEWARE>: some texts define I<theta> and I<phi> the other way round, |
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435 | some texts define the I<phi> to start from the horizontal plane, some |
436 | texts use I<r> in place of I<rho>. |
437 | |
438 | Cylindrical coordinates, I<(rho, theta, z)>, are three-dimensional |
439 | coordinates which define a point in three-dimensional space. They are |
440 | based on a cylinder surface. The radius of the cylinder is B<rho>, |
441 | also known as the I<radial> coordinate. The angle in the I<xy>-plane |
442 | (around the I<z>-axis) is B<theta>, also known as the I<azimuthal> |
443 | coordinate. The third coordinate is the I<z>, pointing up from the |
444 | B<theta>-plane. |
445 | |
446 | =head2 3-D ANGLE CONVERSIONS |
447 | |
448 | Conversions to and from spherical and cylindrical coordinates are |
449 | available. Please notice that the conversions are not necessarily |
450 | reversible because of the equalities like I<pi> angles being equal to |
451 | I<-pi> angles. |
452 | |
453 | =over 4 |
454 | |
455 | =item cartesian_to_cylindrical |
456 | |
457 | ($rho, $theta, $z) = cartesian_to_cylindrical($x, $y, $z); |
458 | |
459 | =item cartesian_to_spherical |
460 | |
461 | ($rho, $theta, $phi) = cartesian_to_spherical($x, $y, $z); |
462 | |
463 | =item cylindrical_to_cartesian |
464 | |
465 | ($x, $y, $z) = cylindrical_to_cartesian($rho, $theta, $z); |
466 | |
467 | =item cylindrical_to_spherical |
468 | |
469 | ($rho_s, $theta, $phi) = cylindrical_to_spherical($rho_c, $theta, $z); |
470 | |
471 | Notice that when C<$z> is not 0 C<$rho_s> is not equal to C<$rho_c>. |
472 | |
473 | =item spherical_to_cartesian |
474 | |
475 | ($x, $y, $z) = spherical_to_cartesian($rho, $theta, $phi); |
476 | |
477 | =item spherical_to_cylindrical |
478 | |
479 | ($rho_c, $theta, $z) = spherical_to_cylindrical($rho_s, $theta, $phi); |
480 | |
481 | Notice that when C<$z> is not 0 C<$rho_c> is not equal to C<$rho_s>. |
482 | |
483 | =back |
484 | |
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485 | =head1 GREAT CIRCLE DISTANCES AND DIRECTIONS |
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486 | |
487 | You can compute spherical distances, called B<great circle distances>, |
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488 | by importing the great_circle_distance() function: |
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489 | |
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490 | use Math::Trig 'great_circle_distance'; |
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491 | |
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492 | $distance = great_circle_distance($theta0, $phi0, $theta1, $phi1, [, $rho]); |
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493 | |
494 | The I<great circle distance> is the shortest distance between two |
495 | points on a sphere. The distance is in C<$rho> units. The C<$rho> is |
496 | optional, it defaults to 1 (the unit sphere), therefore the distance |
497 | defaults to radians. |
498 | |
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499 | If you think geographically the I<theta> are longitudes: zero at the |
500 | Greenwhich meridian, eastward positive, westward negative--and the |
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501 | I<phi> are latitudes: zero at the North Pole, northward positive, |
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502 | southward negative. B<NOTE>: this formula thinks in mathematics, not |
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503 | geographically: the I<phi> zero is at the North Pole, not at the |
504 | Equator on the west coast of Africa (Bay of Guinea). You need to |
505 | subtract your geographical coordinates from I<pi/2> (also known as 90 |
506 | degrees). |
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507 | |
508 | $distance = great_circle_distance($lon0, pi/2 - $lat0, |
509 | $lon1, pi/2 - $lat1, $rho); |
510 | |
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511 | The direction you must follow the great circle (also known as I<bearing>) |
512 | can be computed by the great_circle_direction() function: |
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513 | |
514 | use Math::Trig 'great_circle_direction'; |
515 | |
516 | $direction = great_circle_direction($theta0, $phi0, $theta1, $phi1); |
517 | |
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518 | (Alias 'great_circle_bearing' is also available.) |
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519 | The result is in radians, zero indicating straight north, pi or -pi |
520 | straight south, pi/2 straight west, and -pi/2 straight east. |
521 | |
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522 | You can inversely compute the destination if you know the |
523 | starting point, direction, and distance: |
524 | |
525 | use Math::Trig 'great_circle_destination'; |
526 | |
527 | # thetad and phid are the destination coordinates, |
528 | # dird is the final direction at the destination. |
529 | |
530 | ($thetad, $phid, $dird) = |
531 | great_circle_destination($theta, $phi, $direction, $distance); |
532 | |
533 | or the midpoint if you know the end points: |
534 | |
535 | use Math::Trig 'great_circle_midpoint'; |
536 | |
537 | ($thetam, $phim) = |
538 | great_circle_midpoint($theta0, $phi0, $theta1, $phi1); |
539 | |
540 | The great_circle_midpoint() is just a special case of |
541 | |
542 | use Math::Trig 'great_circle_waypoint'; |
543 | |
544 | ($thetai, $phii) = |
545 | great_circle_waypoint($theta0, $phi0, $theta1, $phi1, $way); |
546 | |
547 | Where the $way is a value from zero ($theta0, $phi0) to one ($theta1, |
548 | $phi1). Note that antipodal points (where their distance is I<pi> |
549 | radians) do not have waypoints between them (they would have an an |
550 | "equator" between them), and therefore C<undef> is returned for |
551 | antipodal points. If the points are the same and the distance |
552 | therefore zero and all waypoints therefore identical, the first point |
553 | (either point) is returned. |
554 | |
555 | The thetas, phis, direction, and distance in the above are all in radians. |
556 | |
557 | You can import all the great circle formulas by |
558 | |
559 | use Math::Trig ':great_circle'; |
560 | |
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561 | Notice that the resulting directions might be somewhat surprising if |
562 | you are looking at a flat worldmap: in such map projections the great |
563 | circles quite often do not look like the shortest routes-- but for |
564 | example the shortest possible routes from Europe or North America to |
565 | Asia do often cross the polar regions. |
566 | |
51301382 |
567 | =head1 EXAMPLES |
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568 | |
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569 | To calculate the distance between London (51.3N 0.5W) and Tokyo |
570 | (35.7N 139.8E) in kilometers: |
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571 | |
572 | use Math::Trig qw(great_circle_distance deg2rad); |
573 | |
574 | # Notice the 90 - latitude: phi zero is at the North Pole. |
bf5f1b4c |
575 | sub NESW { deg2rad($_[0]), deg2rad(90 - $_[1]) } |
576 | my @L = NESW( -0.5, 51.3); |
577 | my @T = NESW(139.8, 35.7); |
578 | my $km = great_circle_distance(@L, @T, 6378); # About 9600 km. |
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579 | |
bf5f1b4c |
580 | The direction you would have to go from London to Tokyo (in radians, |
581 | straight north being zero, straight east being pi/2). |
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582 | |
583 | use Math::Trig qw(great_circle_direction); |
584 | |
bf5f1b4c |
585 | my $rad = great_circle_direction(@L, @T); # About 0.547 or 0.174 pi. |
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586 | |
bf5f1b4c |
587 | The midpoint between London and Tokyo being |
7e5f197a |
588 | |
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589 | use Math::Trig qw(great_circle_midpoint); |
590 | |
591 | my @M = great_circle_midpoint(@L, @T); |
592 | |
593 | or about 68.11N 24.74E, in the Finnish Lapland. |
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594 | |
bf5f1b4c |
595 | =head2 CAVEAT FOR GREAT CIRCLE FORMULAS |
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596 | |
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597 | The answers may be off by few percentages because of the irregular |
598 | (slightly aspherical) form of the Earth. The errors are at worst |
599 | about 0.55%, but generally below 0.3%. |
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600 | |
5cd24f17 |
601 | =head1 BUGS |
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602 | |
5cd24f17 |
603 | Saying C<use Math::Trig;> exports many mathematical routines in the |
604 | caller environment and even overrides some (C<sin>, C<cos>). This is |
605 | construed as a feature by the Authors, actually... ;-) |
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606 | |
5cd24f17 |
607 | The code is not optimized for speed, especially because we use |
608 | C<Math::Complex> and thus go quite near complex numbers while doing |
609 | the computations even when the arguments are not. This, however, |
610 | cannot be completely avoided if we want things like C<asin(2)> to give |
611 | an answer instead of giving a fatal runtime error. |
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612 | |
bf5f1b4c |
613 | Do not attempt navigation using these formulas. |
614 | |
5cd24f17 |
615 | =head1 AUTHORS |
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616 | |
ace5de91 |
617 | Jarkko Hietaniemi <F<jhi@iki.fi>> and |
6e238990 |
618 | Raphael Manfredi <F<Raphael_Manfredi@pobox.com>>. |
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619 | |
620 | =cut |
621 | |
622 | # eof |