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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 | |
10 | use strict; |
11 | |
12 | use Math::Complex qw(:trig); |
13 | |
14 | use vars qw($VERSION $PACKAGE |
15 | @ISA |
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16 | @EXPORT @EXPORT_OK %EXPORT_TAGS); |
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17 | |
18 | @ISA = qw(Exporter); |
19 | |
20 | $VERSION = 1.00; |
21 | |
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22 | my @angcnv = qw(rad2deg rad2grad |
23 | deg2rad deg2grad |
24 | grad2rad grad2deg); |
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25 | |
26 | @EXPORT = (@{$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 | |
36 | @EXPORT_OK = (@rdlcnv, 'great_circle_distance'); |
37 | |
38 | %EXPORT_TAGS = ('radial' => [ @rdlcnv ]); |
39 | |
40 | use constant pi2 => 2 * pi; |
41 | use constant pip2 => pi / 2; |
42 | use constant DR => pi2/360; |
43 | use constant RD => 360/pi2; |
44 | use constant DG => 400/360; |
45 | use constant GD => 360/400; |
46 | use constant RG => 400/pi2; |
47 | use constant GR => pi2/400; |
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48 | |
49 | # |
50 | # Truncating remainder. |
51 | # |
52 | |
53 | sub remt ($$) { |
54 | # Oh yes, POSIX::fmod() would be faster. Possibly. If it is available. |
55 | $_[0] - $_[1] * int($_[0] / $_[1]); |
56 | } |
57 | |
58 | # |
59 | # Angle conversions. |
60 | # |
61 | |
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62 | sub rad2deg ($) { remt(RD * $_[0], 360) } |
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63 | |
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64 | sub deg2rad ($) { remt(DR * $_[0], pi2) } |
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65 | |
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66 | sub grad2deg ($) { remt(GD * $_[0], 360) } |
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67 | |
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68 | sub deg2grad ($) { remt(DG * $_[0], 400) } |
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69 | |
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70 | sub rad2grad ($) { remt(RG * $_[0], 400) } |
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71 | |
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72 | sub grad2rad ($) { remt(GR * $_[0], pi2) } |
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73 | |
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74 | sub cartesian_to_spherical { |
75 | my ( $x, $y, $z ) = @_; |
76 | |
77 | my $rho = sqrt( $x * $x + $y * $y + $z * $z ); |
78 | |
79 | return ( $rho, |
80 | atan2( $y, $x ), |
81 | $rho ? acos( $z / $rho ) : 0 ); |
82 | } |
83 | |
84 | sub spherical_to_cartesian { |
85 | my ( $rho, $theta, $phi ) = @_; |
86 | |
87 | return ( $rho * cos( $theta ) * sin( $phi ), |
88 | $rho * sin( $theta ) * sin( $phi ), |
89 | $rho * cos( $phi ) ); |
90 | } |
91 | |
92 | sub spherical_to_cylindrical { |
93 | my ( $x, $y, $z ) = spherical_to_cartesian( @_ ); |
94 | |
95 | return ( sqrt( $x * $x + $y * $y ), $_[1], $z ); |
96 | } |
97 | |
98 | sub cartesian_to_cylindrical { |
99 | my ( $x, $y, $z ) = @_; |
100 | |
101 | return ( sqrt( $x * $x + $y * $y ), atan2( $y, $x ), $z ); |
102 | } |
103 | |
104 | sub cylindrical_to_cartesian { |
105 | my ( $rho, $theta, $z ) = @_; |
106 | |
107 | return ( $rho * cos( $theta ), $rho * sin( $theta ), $z ); |
108 | } |
109 | |
110 | sub cylindrical_to_spherical { |
111 | return ( cartesian_to_spherical( cylindrical_to_cartesian( @_ ) ) ); |
112 | } |
113 | |
114 | sub great_circle_distance { |
115 | my ( $theta0, $phi0, $theta1, $phi1, $rho ) = @_; |
116 | |
117 | $rho = 1 unless defined $rho; # Default to the unit sphere. |
118 | |
119 | my $lat0 = pip2 - $phi0; |
120 | my $lat1 = pip2 - $phi1; |
121 | |
122 | return $rho * |
123 | acos(cos( $lat0 ) * cos( $lat1 ) * cos( $theta0 - $theta1 ) + |
124 | sin( $lat0 ) * sin( $lat1 ) ); |
125 | } |
126 | |
127 | =pod |
128 | |
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129 | =head1 NAME |
130 | |
131 | Math::Trig - trigonometric functions |
132 | |
133 | =head1 SYNOPSIS |
134 | |
135 | use Math::Trig; |
136 | |
137 | $x = tan(0.9); |
138 | $y = acos(3.7); |
139 | $z = asin(2.4); |
140 | |
141 | $halfpi = pi/2; |
142 | |
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143 | $rad = deg2rad(120); |
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144 | |
145 | =head1 DESCRIPTION |
146 | |
147 | C<Math::Trig> defines many trigonometric functions not defined by the |
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148 | core Perl which defines only the C<sin()> and C<cos()>. The constant |
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149 | B<pi> is also defined as are a few convenience functions for angle |
150 | conversions. |
151 | |
152 | =head1 TRIGONOMETRIC FUNCTIONS |
153 | |
154 | The tangent |
155 | |
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156 | =over 4 |
157 | |
158 | =item B<tan> |
159 | |
160 | =back |
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161 | |
162 | The cofunctions of the sine, cosine, and tangent (cosec/csc and cotan/cot |
163 | are aliases) |
164 | |
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165 | B<csc>, B<cosec>, B<sec>, B<sec>, B<cot>, B<cotan> |
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166 | |
167 | The arcus (also known as the inverse) functions of the sine, cosine, |
168 | and tangent |
169 | |
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170 | B<asin>, B<acos>, B<atan> |
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171 | |
172 | The principal value of the arc tangent of y/x |
173 | |
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174 | B<atan2>(y, x) |
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175 | |
176 | The arcus cofunctions of the sine, cosine, and tangent (acosec/acsc |
177 | and acotan/acot are aliases) |
178 | |
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179 | B<acsc>, B<acosec>, B<asec>, B<acot>, B<acotan> |
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180 | |
181 | The hyperbolic sine, cosine, and tangent |
182 | |
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183 | B<sinh>, B<cosh>, B<tanh> |
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184 | |
185 | The cofunctions of the hyperbolic sine, cosine, and tangent (cosech/csch |
186 | and cotanh/coth are aliases) |
187 | |
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188 | B<csch>, B<cosech>, B<sech>, B<coth>, B<cotanh> |
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189 | |
190 | The arcus (also known as the inverse) functions of the hyperbolic |
191 | sine, cosine, and tangent |
192 | |
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193 | B<asinh>, B<acosh>, B<atanh> |
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194 | |
195 | The arcus cofunctions of the hyperbolic sine, cosine, and tangent |
196 | (acsch/acosech and acoth/acotanh are aliases) |
197 | |
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198 | B<acsch>, B<acosech>, B<asech>, B<acoth>, B<acotanh> |
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199 | |
200 | The trigonometric constant B<pi> is also defined. |
201 | |
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202 | $pi2 = 2 * B<pi>; |
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203 | |
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204 | =head2 ERRORS DUE TO DIVISION BY ZERO |
205 | |
206 | The following functions |
207 | |
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208 | acoth |
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209 | acsc |
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210 | acsch |
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211 | asec |
212 | asech |
213 | atanh |
214 | cot |
215 | coth |
216 | csc |
217 | csch |
218 | sec |
219 | sech |
220 | tan |
221 | tanh |
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222 | |
223 | cannot be computed for all arguments because that would mean dividing |
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224 | by zero or taking logarithm of zero. These situations cause fatal |
225 | runtime errors looking like this |
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226 | |
227 | cot(0): Division by zero. |
228 | (Because in the definition of cot(0), the divisor sin(0) is 0) |
229 | Died at ... |
230 | |
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231 | or |
232 | |
233 | atanh(-1): Logarithm of zero. |
234 | Died at... |
235 | |
236 | For the C<csc>, C<cot>, C<asec>, C<acsc>, C<acot>, C<csch>, C<coth>, |
237 | C<asech>, C<acsch>, the argument cannot be C<0> (zero). For the |
238 | C<atanh>, C<acoth>, the argument cannot be C<1> (one). For the |
239 | C<atanh>, C<acoth>, the argument cannot be C<-1> (minus one). For the |
240 | C<tan>, C<sec>, C<tanh>, C<sech>, the argument cannot be I<pi/2 + k * |
241 | pi>, where I<k> is any integer. |
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242 | |
243 | =head2 SIMPLE (REAL) ARGUMENTS, COMPLEX RESULTS |
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244 | |
245 | Please note that some of the trigonometric functions can break out |
246 | from the B<real axis> into the B<complex plane>. For example |
247 | C<asin(2)> has no definition for plain real numbers but it has |
248 | definition for complex numbers. |
249 | |
250 | In Perl terms this means that supplying the usual Perl numbers (also |
251 | known as scalars, please see L<perldata>) as input for the |
252 | trigonometric functions might produce as output results that no more |
253 | are simple real numbers: instead they are complex numbers. |
254 | |
255 | The C<Math::Trig> handles this by using the C<Math::Complex> package |
256 | which knows how to handle complex numbers, please see L<Math::Complex> |
257 | for more information. In practice you need not to worry about getting |
258 | complex numbers as results because the C<Math::Complex> takes care of |
259 | details like for example how to display complex numbers. For example: |
260 | |
261 | print asin(2), "\n"; |
262 | |
263 | should produce something like this (take or leave few last decimals): |
264 | |
265 | 1.5707963267949-1.31695789692482i |
266 | |
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267 | That is, a complex number with the real part of approximately C<1.571> |
268 | and the imaginary part of approximately C<-1.317>. |
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269 | |
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270 | =head1 PLANE ANGLE CONVERSIONS |
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271 | |
272 | (Plane, 2-dimensional) angles may be converted with the following functions. |
273 | |
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274 | $radians = deg2rad($degrees); |
275 | $radians = grad2rad($gradians); |
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276 | |
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277 | $degrees = rad2deg($radians); |
278 | $degrees = grad2deg($gradians); |
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279 | |
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280 | $gradians = deg2grad($degrees); |
281 | $gradians = rad2grad($radians); |
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282 | |
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283 | The full circle is 2 I<pi> radians or I<360> degrees or I<400> gradians. |
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284 | |
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285 | =head1 RADIAL COORDINATE CONVERSIONS |
286 | |
287 | B<Radial coordinate systems> are the B<spherical> and the B<cylindrical> |
288 | systems, explained shortly in more detail. |
289 | |
290 | You can import radial coordinate conversion functions by using the |
291 | C<:radial> tag: |
292 | |
293 | use Math::Trig ':radial'; |
294 | |
295 | ($rho, $theta, $z) = cartesian_to_cylindrical($x, $y, $z); |
296 | ($rho, $theta, $phi) = cartesian_to_spherical($x, $y, $z); |
297 | ($x, $y, $z) = cylindrical_to_cartesian($rho, $theta, $z); |
298 | ($rho_s, $theta, $phi) = cylindrical_to_spherical($rho_c, $theta, $z); |
299 | ($x, $y, $z) = spherical_to_cartesian($rho, $theta, $phi); |
300 | ($rho_c, $theta, $z) = spherical_to_cylindrical($rho_s, $theta, $phi); |
301 | |
302 | B<All angles are in radians>. |
303 | |
304 | =head2 COORDINATE SYSTEMS |
305 | |
306 | B<Cartesian> coordinates are the usual rectangular I<(x, y, |
307 | z)>-coordinates. |
308 | |
309 | Spherical coordinates, I<(rho, theta, pi)>, are three-dimensional |
310 | coordinates which define a point in three-dimensional space. They are |
311 | based on a sphere surface. The radius of the sphere is B<rho>, also |
312 | known as the I<radial> coordinate. The angle in the I<xy>-plane |
313 | (around the I<z>-axis) is B<theta>, also known as the I<azimuthal> |
314 | coordinate. The angle from the I<z>-axis is B<phi>, also known as the |
315 | I<polar> coordinate. The `North Pole' is therefore I<0, 0, rho>, and |
316 | the `Bay of Guinea' (think of the missing big chunk of Africa) I<0, |
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317 | pi/2, rho>. In geographical terms I<phi> is latitude (northward |
318 | positive, southward negative) and I<theta> is longitude (eastward |
319 | positive, westward negative). |
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320 | |
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321 | B<BEWARE>: some texts define I<theta> and I<phi> the other way round, |
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322 | some texts define the I<phi> to start from the horizontal plane, some |
323 | texts use I<r> in place of I<rho>. |
324 | |
325 | Cylindrical coordinates, I<(rho, theta, z)>, are three-dimensional |
326 | coordinates which define a point in three-dimensional space. They are |
327 | based on a cylinder surface. The radius of the cylinder is B<rho>, |
328 | also known as the I<radial> coordinate. The angle in the I<xy>-plane |
329 | (around the I<z>-axis) is B<theta>, also known as the I<azimuthal> |
330 | coordinate. The third coordinate is the I<z>, pointing up from the |
331 | B<theta>-plane. |
332 | |
333 | =head2 3-D ANGLE CONVERSIONS |
334 | |
335 | Conversions to and from spherical and cylindrical coordinates are |
336 | available. Please notice that the conversions are not necessarily |
337 | reversible because of the equalities like I<pi> angles being equal to |
338 | I<-pi> angles. |
339 | |
340 | =over 4 |
341 | |
342 | =item cartesian_to_cylindrical |
343 | |
344 | ($rho, $theta, $z) = cartesian_to_cylindrical($x, $y, $z); |
345 | |
346 | =item cartesian_to_spherical |
347 | |
348 | ($rho, $theta, $phi) = cartesian_to_spherical($x, $y, $z); |
349 | |
350 | =item cylindrical_to_cartesian |
351 | |
352 | ($x, $y, $z) = cylindrical_to_cartesian($rho, $theta, $z); |
353 | |
354 | =item cylindrical_to_spherical |
355 | |
356 | ($rho_s, $theta, $phi) = cylindrical_to_spherical($rho_c, $theta, $z); |
357 | |
358 | Notice that when C<$z> is not 0 C<$rho_s> is not equal to C<$rho_c>. |
359 | |
360 | =item spherical_to_cartesian |
361 | |
362 | ($x, $y, $z) = spherical_to_cartesian($rho, $theta, $phi); |
363 | |
364 | =item spherical_to_cylindrical |
365 | |
366 | ($rho_c, $theta, $z) = spherical_to_cylindrical($rho_s, $theta, $phi); |
367 | |
368 | Notice that when C<$z> is not 0 C<$rho_c> is not equal to C<$rho_s>. |
369 | |
370 | =back |
371 | |
372 | =head1 GREAT CIRCLE DISTANCES |
373 | |
374 | You can compute spherical distances, called B<great circle distances>, |
375 | by importing the C<great_circle_distance> function: |
376 | |
377 | use Math::Trig 'great_circle_distance' |
378 | |
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379 | $distance = great_circle_distance($theta0, $phi0, $theta1, $phi1, [, $rho]); |
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380 | |
381 | The I<great circle distance> is the shortest distance between two |
382 | points on a sphere. The distance is in C<$rho> units. The C<$rho> is |
383 | optional, it defaults to 1 (the unit sphere), therefore the distance |
384 | defaults to radians. |
385 | |
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386 | If you think geographically the I<theta> are longitudes: zero at the |
387 | Greenwhich meridian, eastward positive, westward negative--and the |
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388 | I<phi> are latitudes: zero at the North Pole, northward positive, |
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389 | southward negative. B<NOTE>: this formula thinks in mathematics, not |
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390 | geographically: the I<phi> zero is at the North Pole, not at the |
391 | Equator on the west coast of Africa (Bay of Guinea). You need to |
392 | subtract your geographical coordinates from I<pi/2> (also known as 90 |
393 | degrees). |
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394 | |
395 | $distance = great_circle_distance($lon0, pi/2 - $lat0, |
396 | $lon1, pi/2 - $lat1, $rho); |
397 | |
51301382 |
398 | =head1 EXAMPLES |
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399 | |
400 | To calculate the distance between London (51.3N 0.5W) and Tokyo (35.7N |
401 | 139.8E) in kilometers: |
402 | |
403 | use Math::Trig qw(great_circle_distance deg2rad); |
404 | |
405 | # Notice the 90 - latitude: phi zero is at the North Pole. |
406 | @L = (deg2rad(-0.5), deg2rad(90 - 51.3)); |
407 | @T = (deg2rad(139.8),deg2rad(90 - 35.7)); |
408 | |
409 | $km = great_circle_distance(@L, @T, 6378); |
410 | |
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411 | The answer may be off by few percentages because of the irregular |
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412 | (slightly aspherical) form of the Earth. The used formula |
413 | |
414 | lat0 = 90 degrees - phi0 |
415 | lat1 = 90 degrees - phi1 |
416 | d = R * arccos(cos(lat0) * cos(lat1) * cos(lon1 - lon01) + |
417 | sin(lat0) * sin(lat1)) |
418 | |
419 | is also somewhat unreliable for small distances (for locations |
420 | separated less than about five degrees) because it uses arc cosine |
421 | which is rather ill-conditioned for values close to zero. |
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422 | |
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423 | =head1 BUGS |
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424 | |
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425 | Saying C<use Math::Trig;> exports many mathematical routines in the |
426 | caller environment and even overrides some (C<sin>, C<cos>). This is |
427 | construed as a feature by the Authors, actually... ;-) |
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428 | |
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429 | The code is not optimized for speed, especially because we use |
430 | C<Math::Complex> and thus go quite near complex numbers while doing |
431 | the computations even when the arguments are not. This, however, |
432 | cannot be completely avoided if we want things like C<asin(2)> to give |
433 | an answer instead of giving a fatal runtime error. |
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434 | |
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435 | =head1 AUTHORS |
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436 | |
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437 | Jarkko Hietaniemi <F<jhi@iki.fi>> and |
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438 | Raphael Manfredi <F<Raphael_Manfredi@pobox.com>>. |
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439 | |
440 | =cut |
441 | |
442 | # eof |