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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 5.005_64; |
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11 | use strict; |
12 | |
13 | use Math::Complex qw(:trig); |
14 | |
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15 | our($VERSION, $PACKAGE, @ISA, @EXPORT, @EXPORT_OK, %EXPORT_TAGS); |
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16 | |
17 | @ISA = qw(Exporter); |
18 | |
19 | $VERSION = 1.00; |
20 | |
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21 | my @angcnv = qw(rad2deg rad2grad |
22 | deg2rad deg2grad |
23 | grad2rad grad2deg); |
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24 | |
25 | @EXPORT = (@{$Math::Complex::EXPORT_TAGS{'trig'}}, |
26 | @angcnv); |
27 | |
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28 | my @rdlcnv = qw(cartesian_to_cylindrical |
29 | cartesian_to_spherical |
30 | cylindrical_to_cartesian |
31 | cylindrical_to_spherical |
32 | spherical_to_cartesian |
33 | spherical_to_cylindrical); |
34 | |
35 | @EXPORT_OK = (@rdlcnv, 'great_circle_distance'); |
36 | |
37 | %EXPORT_TAGS = ('radial' => [ @rdlcnv ]); |
38 | |
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39 | sub pi2 () { 2 * pi } # use constant generates warning |
40 | sub pip2 () { pi / 2 } # use constant generates warning |
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41 | use constant DR => pi2/360; |
42 | use constant RD => 360/pi2; |
43 | use constant DG => 400/360; |
44 | use constant GD => 360/400; |
45 | use constant RG => 400/pi2; |
46 | use constant GR => pi2/400; |
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47 | |
48 | # |
49 | # Truncating remainder. |
50 | # |
51 | |
52 | sub remt ($$) { |
53 | # Oh yes, POSIX::fmod() would be faster. Possibly. If it is available. |
54 | $_[0] - $_[1] * int($_[0] / $_[1]); |
55 | } |
56 | |
57 | # |
58 | # Angle conversions. |
59 | # |
60 | |
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61 | sub rad2deg ($) { remt(RD * $_[0], 360) } |
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62 | |
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63 | sub deg2rad ($) { remt(DR * $_[0], pi2) } |
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64 | |
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65 | sub grad2deg ($) { remt(GD * $_[0], 360) } |
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66 | |
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67 | sub deg2grad ($) { remt(DG * $_[0], 400) } |
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68 | |
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69 | sub rad2grad ($) { remt(RG * $_[0], 400) } |
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70 | |
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71 | sub grad2rad ($) { remt(GR * $_[0], pi2) } |
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72 | |
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73 | sub cartesian_to_spherical { |
74 | my ( $x, $y, $z ) = @_; |
75 | |
76 | my $rho = sqrt( $x * $x + $y * $y + $z * $z ); |
77 | |
78 | return ( $rho, |
79 | atan2( $y, $x ), |
80 | $rho ? acos( $z / $rho ) : 0 ); |
81 | } |
82 | |
83 | sub spherical_to_cartesian { |
84 | my ( $rho, $theta, $phi ) = @_; |
85 | |
86 | return ( $rho * cos( $theta ) * sin( $phi ), |
87 | $rho * sin( $theta ) * sin( $phi ), |
88 | $rho * cos( $phi ) ); |
89 | } |
90 | |
91 | sub spherical_to_cylindrical { |
92 | my ( $x, $y, $z ) = spherical_to_cartesian( @_ ); |
93 | |
94 | return ( sqrt( $x * $x + $y * $y ), $_[1], $z ); |
95 | } |
96 | |
97 | sub cartesian_to_cylindrical { |
98 | my ( $x, $y, $z ) = @_; |
99 | |
100 | return ( sqrt( $x * $x + $y * $y ), atan2( $y, $x ), $z ); |
101 | } |
102 | |
103 | sub cylindrical_to_cartesian { |
104 | my ( $rho, $theta, $z ) = @_; |
105 | |
106 | return ( $rho * cos( $theta ), $rho * sin( $theta ), $z ); |
107 | } |
108 | |
109 | sub cylindrical_to_spherical { |
110 | return ( cartesian_to_spherical( cylindrical_to_cartesian( @_ ) ) ); |
111 | } |
112 | |
113 | sub great_circle_distance { |
114 | my ( $theta0, $phi0, $theta1, $phi1, $rho ) = @_; |
115 | |
116 | $rho = 1 unless defined $rho; # Default to the unit sphere. |
117 | |
118 | my $lat0 = pip2 - $phi0; |
119 | my $lat1 = pip2 - $phi1; |
120 | |
121 | return $rho * |
122 | acos(cos( $lat0 ) * cos( $lat1 ) * cos( $theta0 - $theta1 ) + |
123 | sin( $lat0 ) * sin( $lat1 ) ); |
124 | } |
125 | |
126 | =pod |
127 | |
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128 | =head1 NAME |
129 | |
130 | Math::Trig - trigonometric functions |
131 | |
132 | =head1 SYNOPSIS |
133 | |
134 | use Math::Trig; |
135 | |
136 | $x = tan(0.9); |
137 | $y = acos(3.7); |
138 | $z = asin(2.4); |
139 | |
140 | $halfpi = pi/2; |
141 | |
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142 | $rad = deg2rad(120); |
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143 | |
144 | =head1 DESCRIPTION |
145 | |
146 | C<Math::Trig> defines many trigonometric functions not defined by the |
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147 | core Perl which defines only the C<sin()> and C<cos()>. The constant |
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148 | B<pi> is also defined as are a few convenience functions for angle |
149 | conversions. |
150 | |
151 | =head1 TRIGONOMETRIC FUNCTIONS |
152 | |
153 | The tangent |
154 | |
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155 | =over 4 |
156 | |
157 | =item B<tan> |
158 | |
159 | =back |
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160 | |
161 | The cofunctions of the sine, cosine, and tangent (cosec/csc and cotan/cot |
162 | are aliases) |
163 | |
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164 | B<csc>, B<cosec>, B<sec>, B<sec>, B<cot>, B<cotan> |
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165 | |
166 | The arcus (also known as the inverse) functions of the sine, cosine, |
167 | and tangent |
168 | |
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169 | B<asin>, B<acos>, B<atan> |
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170 | |
171 | The principal value of the arc tangent of y/x |
172 | |
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173 | B<atan2>(y, x) |
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174 | |
175 | The arcus cofunctions of the sine, cosine, and tangent (acosec/acsc |
176 | and acotan/acot are aliases) |
177 | |
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178 | B<acsc>, B<acosec>, B<asec>, B<acot>, B<acotan> |
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179 | |
180 | The hyperbolic sine, cosine, and tangent |
181 | |
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182 | B<sinh>, B<cosh>, B<tanh> |
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183 | |
184 | The cofunctions of the hyperbolic sine, cosine, and tangent (cosech/csch |
185 | and cotanh/coth are aliases) |
186 | |
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187 | B<csch>, B<cosech>, B<sech>, B<coth>, B<cotanh> |
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188 | |
189 | The arcus (also known as the inverse) functions of the hyperbolic |
190 | sine, cosine, and tangent |
191 | |
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192 | B<asinh>, B<acosh>, B<atanh> |
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193 | |
194 | The arcus cofunctions of the hyperbolic sine, cosine, and tangent |
195 | (acsch/acosech and acoth/acotanh are aliases) |
196 | |
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197 | B<acsch>, B<acosech>, B<asech>, B<acoth>, B<acotanh> |
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198 | |
199 | The trigonometric constant B<pi> is also defined. |
200 | |
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201 | $pi2 = 2 * B<pi>; |
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202 | |
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203 | =head2 ERRORS DUE TO DIVISION BY ZERO |
204 | |
205 | The following functions |
206 | |
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207 | acoth |
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208 | acsc |
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209 | acsch |
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210 | asec |
211 | asech |
212 | atanh |
213 | cot |
214 | coth |
215 | csc |
216 | csch |
217 | sec |
218 | sech |
219 | tan |
220 | tanh |
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221 | |
222 | cannot be computed for all arguments because that would mean dividing |
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223 | by zero or taking logarithm of zero. These situations cause fatal |
224 | runtime errors looking like this |
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225 | |
226 | cot(0): Division by zero. |
227 | (Because in the definition of cot(0), the divisor sin(0) is 0) |
228 | Died at ... |
229 | |
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230 | or |
231 | |
232 | atanh(-1): Logarithm of zero. |
233 | Died at... |
234 | |
235 | For the C<csc>, C<cot>, C<asec>, C<acsc>, C<acot>, C<csch>, C<coth>, |
236 | C<asech>, C<acsch>, the argument cannot be C<0> (zero). For the |
237 | C<atanh>, C<acoth>, the argument cannot be C<1> (one). For the |
238 | C<atanh>, C<acoth>, the argument cannot be C<-1> (minus one). For the |
239 | C<tan>, C<sec>, C<tanh>, C<sech>, the argument cannot be I<pi/2 + k * |
240 | pi>, where I<k> is any integer. |
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241 | |
242 | =head2 SIMPLE (REAL) ARGUMENTS, COMPLEX RESULTS |
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243 | |
244 | Please note that some of the trigonometric functions can break out |
245 | from the B<real axis> into the B<complex plane>. For example |
246 | C<asin(2)> has no definition for plain real numbers but it has |
247 | definition for complex numbers. |
248 | |
249 | In Perl terms this means that supplying the usual Perl numbers (also |
250 | known as scalars, please see L<perldata>) as input for the |
251 | trigonometric functions might produce as output results that no more |
252 | are simple real numbers: instead they are complex numbers. |
253 | |
254 | The C<Math::Trig> handles this by using the C<Math::Complex> package |
255 | which knows how to handle complex numbers, please see L<Math::Complex> |
256 | for more information. In practice you need not to worry about getting |
257 | complex numbers as results because the C<Math::Complex> takes care of |
258 | details like for example how to display complex numbers. For example: |
259 | |
260 | print asin(2), "\n"; |
261 | |
262 | should produce something like this (take or leave few last decimals): |
263 | |
264 | 1.5707963267949-1.31695789692482i |
265 | |
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266 | That is, a complex number with the real part of approximately C<1.571> |
267 | and the imaginary part of approximately C<-1.317>. |
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268 | |
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269 | =head1 PLANE ANGLE CONVERSIONS |
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270 | |
271 | (Plane, 2-dimensional) angles may be converted with the following functions. |
272 | |
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273 | $radians = deg2rad($degrees); |
274 | $radians = grad2rad($gradians); |
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275 | |
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276 | $degrees = rad2deg($radians); |
277 | $degrees = grad2deg($gradians); |
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278 | |
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279 | $gradians = deg2grad($degrees); |
280 | $gradians = rad2grad($radians); |
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281 | |
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282 | The full circle is 2 I<pi> radians or I<360> degrees or I<400> gradians. |
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283 | |
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284 | =head1 RADIAL COORDINATE CONVERSIONS |
285 | |
286 | B<Radial coordinate systems> are the B<spherical> and the B<cylindrical> |
287 | systems, explained shortly in more detail. |
288 | |
289 | You can import radial coordinate conversion functions by using the |
290 | C<:radial> tag: |
291 | |
292 | use Math::Trig ':radial'; |
293 | |
294 | ($rho, $theta, $z) = cartesian_to_cylindrical($x, $y, $z); |
295 | ($rho, $theta, $phi) = cartesian_to_spherical($x, $y, $z); |
296 | ($x, $y, $z) = cylindrical_to_cartesian($rho, $theta, $z); |
297 | ($rho_s, $theta, $phi) = cylindrical_to_spherical($rho_c, $theta, $z); |
298 | ($x, $y, $z) = spherical_to_cartesian($rho, $theta, $phi); |
299 | ($rho_c, $theta, $z) = spherical_to_cylindrical($rho_s, $theta, $phi); |
300 | |
301 | B<All angles are in radians>. |
302 | |
303 | =head2 COORDINATE SYSTEMS |
304 | |
305 | B<Cartesian> coordinates are the usual rectangular I<(x, y, |
306 | z)>-coordinates. |
307 | |
308 | Spherical coordinates, I<(rho, theta, pi)>, are three-dimensional |
309 | coordinates which define a point in three-dimensional space. They are |
310 | based on a sphere surface. The radius of the sphere is B<rho>, also |
311 | known as the I<radial> coordinate. The angle in the I<xy>-plane |
312 | (around the I<z>-axis) is B<theta>, also known as the I<azimuthal> |
313 | coordinate. The angle from the I<z>-axis is B<phi>, also known as the |
314 | I<polar> coordinate. The `North Pole' is therefore I<0, 0, rho>, and |
315 | the `Bay of Guinea' (think of the missing big chunk of Africa) I<0, |
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316 | pi/2, rho>. In geographical terms I<phi> is latitude (northward |
317 | positive, southward negative) and I<theta> is longitude (eastward |
318 | positive, westward negative). |
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319 | |
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320 | B<BEWARE>: some texts define I<theta> and I<phi> the other way round, |
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321 | some texts define the I<phi> to start from the horizontal plane, some |
322 | texts use I<r> in place of I<rho>. |
323 | |
324 | Cylindrical coordinates, I<(rho, theta, z)>, are three-dimensional |
325 | coordinates which define a point in three-dimensional space. They are |
326 | based on a cylinder surface. The radius of the cylinder is B<rho>, |
327 | also known as the I<radial> coordinate. The angle in the I<xy>-plane |
328 | (around the I<z>-axis) is B<theta>, also known as the I<azimuthal> |
329 | coordinate. The third coordinate is the I<z>, pointing up from the |
330 | B<theta>-plane. |
331 | |
332 | =head2 3-D ANGLE CONVERSIONS |
333 | |
334 | Conversions to and from spherical and cylindrical coordinates are |
335 | available. Please notice that the conversions are not necessarily |
336 | reversible because of the equalities like I<pi> angles being equal to |
337 | I<-pi> angles. |
338 | |
339 | =over 4 |
340 | |
341 | =item cartesian_to_cylindrical |
342 | |
343 | ($rho, $theta, $z) = cartesian_to_cylindrical($x, $y, $z); |
344 | |
345 | =item cartesian_to_spherical |
346 | |
347 | ($rho, $theta, $phi) = cartesian_to_spherical($x, $y, $z); |
348 | |
349 | =item cylindrical_to_cartesian |
350 | |
351 | ($x, $y, $z) = cylindrical_to_cartesian($rho, $theta, $z); |
352 | |
353 | =item cylindrical_to_spherical |
354 | |
355 | ($rho_s, $theta, $phi) = cylindrical_to_spherical($rho_c, $theta, $z); |
356 | |
357 | Notice that when C<$z> is not 0 C<$rho_s> is not equal to C<$rho_c>. |
358 | |
359 | =item spherical_to_cartesian |
360 | |
361 | ($x, $y, $z) = spherical_to_cartesian($rho, $theta, $phi); |
362 | |
363 | =item spherical_to_cylindrical |
364 | |
365 | ($rho_c, $theta, $z) = spherical_to_cylindrical($rho_s, $theta, $phi); |
366 | |
367 | Notice that when C<$z> is not 0 C<$rho_c> is not equal to C<$rho_s>. |
368 | |
369 | =back |
370 | |
371 | =head1 GREAT CIRCLE DISTANCES |
372 | |
373 | You can compute spherical distances, called B<great circle distances>, |
374 | by importing the C<great_circle_distance> function: |
375 | |
376 | use Math::Trig 'great_circle_distance' |
377 | |
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378 | $distance = great_circle_distance($theta0, $phi0, $theta1, $phi1, [, $rho]); |
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379 | |
380 | The I<great circle distance> is the shortest distance between two |
381 | points on a sphere. The distance is in C<$rho> units. The C<$rho> is |
382 | optional, it defaults to 1 (the unit sphere), therefore the distance |
383 | defaults to radians. |
384 | |
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385 | If you think geographically the I<theta> are longitudes: zero at the |
386 | Greenwhich meridian, eastward positive, westward negative--and the |
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387 | I<phi> are latitudes: zero at the North Pole, northward positive, |
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388 | southward negative. B<NOTE>: this formula thinks in mathematics, not |
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389 | geographically: the I<phi> zero is at the North Pole, not at the |
390 | Equator on the west coast of Africa (Bay of Guinea). You need to |
391 | subtract your geographical coordinates from I<pi/2> (also known as 90 |
392 | degrees). |
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393 | |
394 | $distance = great_circle_distance($lon0, pi/2 - $lat0, |
395 | $lon1, pi/2 - $lat1, $rho); |
396 | |
51301382 |
397 | =head1 EXAMPLES |
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398 | |
399 | To calculate the distance between London (51.3N 0.5W) and Tokyo (35.7N |
400 | 139.8E) in kilometers: |
401 | |
402 | use Math::Trig qw(great_circle_distance deg2rad); |
403 | |
404 | # Notice the 90 - latitude: phi zero is at the North Pole. |
405 | @L = (deg2rad(-0.5), deg2rad(90 - 51.3)); |
406 | @T = (deg2rad(139.8),deg2rad(90 - 35.7)); |
407 | |
408 | $km = great_circle_distance(@L, @T, 6378); |
409 | |
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410 | The answer may be off by few percentages because of the irregular |
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411 | (slightly aspherical) form of the Earth. The used formula |
412 | |
413 | lat0 = 90 degrees - phi0 |
414 | lat1 = 90 degrees - phi1 |
415 | d = R * arccos(cos(lat0) * cos(lat1) * cos(lon1 - lon01) + |
416 | sin(lat0) * sin(lat1)) |
417 | |
418 | is also somewhat unreliable for small distances (for locations |
419 | separated less than about five degrees) because it uses arc cosine |
420 | which is rather ill-conditioned for values close to zero. |
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421 | |
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422 | =head1 BUGS |
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423 | |
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424 | Saying C<use Math::Trig;> exports many mathematical routines in the |
425 | caller environment and even overrides some (C<sin>, C<cos>). This is |
426 | construed as a feature by the Authors, actually... ;-) |
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427 | |
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428 | The code is not optimized for speed, especially because we use |
429 | C<Math::Complex> and thus go quite near complex numbers while doing |
430 | the computations even when the arguments are not. This, however, |
431 | cannot be completely avoided if we want things like C<asin(2)> to give |
432 | an answer instead of giving a fatal runtime error. |
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433 | |
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434 | =head1 AUTHORS |
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435 | |
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436 | Jarkko Hietaniemi <F<jhi@iki.fi>> and |
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437 | Raphael Manfredi <F<Raphael_Manfredi@pobox.com>>. |
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438 | |
439 | =cut |
440 | |
441 | # eof |