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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 | |
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 | |
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19 | $VERSION = 1.02; |
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20 | |
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21 | my @angcnv = qw(rad2deg rad2grad |
22 | deg2rad deg2grad |
23 | grad2rad grad2deg); |
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24 | |
25 | @EXPORT = (@{$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 | |
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35 | @EXPORT_OK = (@rdlcnv, 'great_circle_distance', 'great_circle_direction'); |
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36 | |
37 | %EXPORT_TAGS = ('radial' => [ @rdlcnv ]); |
38 | |
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39 | sub pi2 () { 2 * pi } |
40 | sub pip2 () { pi / 2 } |
41 | |
42 | sub DR () { pi2/360 } |
43 | sub RD () { 360/pi2 } |
44 | sub DG () { 400/360 } |
45 | sub GD () { 360/400 } |
46 | sub RG () { 400/pi2 } |
47 | sub 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] / $_[1]); |
56 | } |
57 | |
58 | # |
59 | # Angle conversions. |
60 | # |
61 | |
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62 | sub rad2rad($) { remt($_[0], pi2) } |
63 | |
64 | sub deg2deg($) { remt($_[0], 360) } |
65 | |
66 | sub grad2grad($) { remt($_[0], 400) } |
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67 | |
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68 | sub rad2deg ($;$) { my $d = RD * $_[0]; $_[1] ? $d : deg2deg($d) } |
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69 | |
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70 | sub deg2rad ($;$) { my $d = DR * $_[0]; $_[1] ? $d : rad2rad($d) } |
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71 | |
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72 | sub grad2deg ($;$) { my $d = GD * $_[0]; $_[1] ? $d : deg2deg($d) } |
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73 | |
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74 | sub deg2grad ($;$) { my $d = DG * $_[0]; $_[1] ? $d : grad2grad($d) } |
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75 | |
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76 | sub rad2grad ($;$) { my $d = RG * $_[0]; $_[1] ? $d : grad2grad($d) } |
77 | |
78 | sub grad2rad ($;$) { my $d = GR * $_[0]; $_[1] ? $d : rad2rad($d) } |
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79 | |
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80 | sub cartesian_to_spherical { |
81 | my ( $x, $y, $z ) = @_; |
82 | |
83 | my $rho = sqrt( $x * $x + $y * $y + $z * $z ); |
84 | |
85 | return ( $rho, |
86 | atan2( $y, $x ), |
87 | $rho ? acos( $z / $rho ) : 0 ); |
88 | } |
89 | |
90 | sub spherical_to_cartesian { |
91 | my ( $rho, $theta, $phi ) = @_; |
92 | |
93 | return ( $rho * cos( $theta ) * sin( $phi ), |
94 | $rho * sin( $theta ) * sin( $phi ), |
95 | $rho * cos( $phi ) ); |
96 | } |
97 | |
98 | sub spherical_to_cylindrical { |
99 | my ( $x, $y, $z ) = spherical_to_cartesian( @_ ); |
100 | |
101 | return ( sqrt( $x * $x + $y * $y ), $_[1], $z ); |
102 | } |
103 | |
104 | sub cartesian_to_cylindrical { |
105 | my ( $x, $y, $z ) = @_; |
106 | |
107 | return ( sqrt( $x * $x + $y * $y ), atan2( $y, $x ), $z ); |
108 | } |
109 | |
110 | sub cylindrical_to_cartesian { |
111 | my ( $rho, $theta, $z ) = @_; |
112 | |
113 | return ( $rho * cos( $theta ), $rho * sin( $theta ), $z ); |
114 | } |
115 | |
116 | sub cylindrical_to_spherical { |
117 | return ( cartesian_to_spherical( cylindrical_to_cartesian( @_ ) ) ); |
118 | } |
119 | |
120 | sub great_circle_distance { |
121 | my ( $theta0, $phi0, $theta1, $phi1, $rho ) = @_; |
122 | |
123 | $rho = 1 unless defined $rho; # Default to the unit sphere. |
124 | |
125 | my $lat0 = pip2 - $phi0; |
126 | my $lat1 = pip2 - $phi1; |
127 | |
128 | return $rho * |
129 | acos(cos( $lat0 ) * cos( $lat1 ) * cos( $theta0 - $theta1 ) + |
130 | sin( $lat0 ) * sin( $lat1 ) ); |
131 | } |
132 | |
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133 | sub great_circle_direction { |
134 | my ( $theta0, $phi0, $theta1, $phi1 ) = @_; |
135 | |
136 | my $lat0 = pip2 - $phi0; |
137 | my $lat1 = pip2 - $phi1; |
138 | |
139 | my $direction = |
140 | atan2(sin($theta0 - $theta1) * cos($lat1), |
141 | cos($lat0) * sin($lat1) - |
142 | sin($lat0) * cos($lat1) * cos($theta0 - $theta1)); |
143 | |
144 | return rad2rad($direction); |
145 | } |
146 | |
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147 | 1; |
148 | |
149 | __END__ |
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150 | =pod |
151 | |
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152 | =head1 NAME |
153 | |
154 | Math::Trig - trigonometric functions |
155 | |
156 | =head1 SYNOPSIS |
157 | |
158 | use Math::Trig; |
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159 | |
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160 | $x = tan(0.9); |
161 | $y = acos(3.7); |
162 | $z = asin(2.4); |
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163 | |
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164 | $halfpi = pi/2; |
165 | |
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166 | $rad = deg2rad(120); |
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167 | |
168 | =head1 DESCRIPTION |
169 | |
170 | C<Math::Trig> defines many trigonometric functions not defined by the |
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171 | core Perl which defines only the C<sin()> and C<cos()>. The constant |
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172 | B<pi> is also defined as are a few convenience functions for angle |
173 | conversions. |
174 | |
175 | =head1 TRIGONOMETRIC FUNCTIONS |
176 | |
177 | The tangent |
178 | |
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179 | =over 4 |
180 | |
181 | =item B<tan> |
182 | |
183 | =back |
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184 | |
185 | The cofunctions of the sine, cosine, and tangent (cosec/csc and cotan/cot |
186 | are aliases) |
187 | |
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188 | B<csc>, B<cosec>, B<sec>, B<sec>, B<cot>, B<cotan> |
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189 | |
190 | The arcus (also known as the inverse) functions of the sine, cosine, |
191 | and tangent |
192 | |
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193 | B<asin>, B<acos>, B<atan> |
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194 | |
195 | The principal value of the arc tangent of y/x |
196 | |
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197 | B<atan2>(y, x) |
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198 | |
199 | The arcus cofunctions of the sine, cosine, and tangent (acosec/acsc |
200 | and acotan/acot are aliases) |
201 | |
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202 | B<acsc>, B<acosec>, B<asec>, B<acot>, B<acotan> |
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203 | |
204 | The hyperbolic sine, cosine, and tangent |
205 | |
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206 | B<sinh>, B<cosh>, B<tanh> |
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207 | |
208 | The cofunctions of the hyperbolic sine, cosine, and tangent (cosech/csch |
209 | and cotanh/coth are aliases) |
210 | |
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211 | B<csch>, B<cosech>, B<sech>, B<coth>, B<cotanh> |
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212 | |
213 | The arcus (also known as the inverse) functions of the hyperbolic |
214 | sine, cosine, and tangent |
215 | |
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216 | B<asinh>, B<acosh>, B<atanh> |
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217 | |
218 | The arcus cofunctions of the hyperbolic sine, cosine, and tangent |
219 | (acsch/acosech and acoth/acotanh are aliases) |
220 | |
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221 | B<acsch>, B<acosech>, B<asech>, B<acoth>, B<acotanh> |
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222 | |
223 | The trigonometric constant B<pi> is also defined. |
224 | |
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225 | $pi2 = 2 * B<pi>; |
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226 | |
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227 | =head2 ERRORS DUE TO DIVISION BY ZERO |
228 | |
229 | The following functions |
230 | |
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231 | acoth |
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232 | acsc |
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233 | acsch |
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234 | asec |
235 | asech |
236 | atanh |
237 | cot |
238 | coth |
239 | csc |
240 | csch |
241 | sec |
242 | sech |
243 | tan |
244 | tanh |
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245 | |
246 | cannot be computed for all arguments because that would mean dividing |
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247 | by zero or taking logarithm of zero. These situations cause fatal |
248 | runtime errors looking like this |
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249 | |
250 | cot(0): Division by zero. |
251 | (Because in the definition of cot(0), the divisor sin(0) is 0) |
252 | Died at ... |
253 | |
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254 | or |
255 | |
256 | atanh(-1): Logarithm of zero. |
257 | Died at... |
258 | |
259 | For the C<csc>, C<cot>, C<asec>, C<acsc>, C<acot>, C<csch>, C<coth>, |
260 | C<asech>, C<acsch>, the argument cannot be C<0> (zero). For the |
261 | C<atanh>, C<acoth>, the argument cannot be C<1> (one). For the |
262 | C<atanh>, C<acoth>, the argument cannot be C<-1> (minus one). For the |
263 | C<tan>, C<sec>, C<tanh>, C<sech>, the argument cannot be I<pi/2 + k * |
264 | pi>, where I<k> is any integer. |
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265 | |
266 | =head2 SIMPLE (REAL) ARGUMENTS, COMPLEX RESULTS |
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267 | |
268 | Please note that some of the trigonometric functions can break out |
269 | from the B<real axis> into the B<complex plane>. For example |
270 | C<asin(2)> has no definition for plain real numbers but it has |
271 | definition for complex numbers. |
272 | |
273 | In Perl terms this means that supplying the usual Perl numbers (also |
274 | known as scalars, please see L<perldata>) as input for the |
275 | trigonometric functions might produce as output results that no more |
276 | are simple real numbers: instead they are complex numbers. |
277 | |
278 | The C<Math::Trig> handles this by using the C<Math::Complex> package |
279 | which knows how to handle complex numbers, please see L<Math::Complex> |
280 | for more information. In practice you need not to worry about getting |
281 | complex numbers as results because the C<Math::Complex> takes care of |
282 | details like for example how to display complex numbers. For example: |
283 | |
284 | print asin(2), "\n"; |
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285 | |
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286 | should produce something like this (take or leave few last decimals): |
287 | |
288 | 1.5707963267949-1.31695789692482i |
289 | |
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290 | That is, a complex number with the real part of approximately C<1.571> |
291 | and the imaginary part of approximately C<-1.317>. |
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292 | |
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293 | =head1 PLANE ANGLE CONVERSIONS |
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294 | |
295 | (Plane, 2-dimensional) angles may be converted with the following functions. |
296 | |
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297 | $radians = deg2rad($degrees); |
298 | $radians = grad2rad($gradians); |
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299 | |
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300 | $degrees = rad2deg($radians); |
301 | $degrees = grad2deg($gradians); |
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302 | |
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303 | $gradians = deg2grad($degrees); |
304 | $gradians = rad2grad($radians); |
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305 | |
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306 | The full circle is 2 I<pi> radians or I<360> degrees or I<400> gradians. |
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307 | The result is by default wrapped to be inside the [0, {2pi,360,400}[ circle. |
308 | If you don't want this, supply a true second argument: |
309 | |
310 | $zillions_of_radians = deg2rad($zillions_of_degrees, 1); |
311 | $negative_degrees = rad2deg($negative_radians, 1); |
312 | |
313 | You can also do the wrapping explicitly by rad2rad(), deg2deg(), and |
314 | grad2grad(). |
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315 | |
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316 | =head1 RADIAL COORDINATE CONVERSIONS |
317 | |
318 | B<Radial coordinate systems> are the B<spherical> and the B<cylindrical> |
319 | systems, explained shortly in more detail. |
320 | |
321 | You can import radial coordinate conversion functions by using the |
322 | C<:radial> tag: |
323 | |
324 | use Math::Trig ':radial'; |
325 | |
326 | ($rho, $theta, $z) = cartesian_to_cylindrical($x, $y, $z); |
327 | ($rho, $theta, $phi) = cartesian_to_spherical($x, $y, $z); |
328 | ($x, $y, $z) = cylindrical_to_cartesian($rho, $theta, $z); |
329 | ($rho_s, $theta, $phi) = cylindrical_to_spherical($rho_c, $theta, $z); |
330 | ($x, $y, $z) = spherical_to_cartesian($rho, $theta, $phi); |
331 | ($rho_c, $theta, $z) = spherical_to_cylindrical($rho_s, $theta, $phi); |
332 | |
333 | B<All angles are in radians>. |
334 | |
335 | =head2 COORDINATE SYSTEMS |
336 | |
337 | B<Cartesian> coordinates are the usual rectangular I<(x, y, |
338 | z)>-coordinates. |
339 | |
340 | Spherical coordinates, I<(rho, theta, pi)>, are three-dimensional |
341 | coordinates which define a point in three-dimensional space. They are |
342 | based on a sphere surface. The radius of the sphere is B<rho>, also |
343 | known as the I<radial> coordinate. The angle in the I<xy>-plane |
344 | (around the I<z>-axis) is B<theta>, also known as the I<azimuthal> |
345 | coordinate. The angle from the I<z>-axis is B<phi>, also known as the |
346 | I<polar> coordinate. The `North Pole' is therefore I<0, 0, rho>, and |
347 | the `Bay of Guinea' (think of the missing big chunk of Africa) I<0, |
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348 | pi/2, rho>. In geographical terms I<phi> is latitude (northward |
349 | positive, southward negative) and I<theta> is longitude (eastward |
350 | positive, westward negative). |
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351 | |
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352 | B<BEWARE>: some texts define I<theta> and I<phi> the other way round, |
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353 | some texts define the I<phi> to start from the horizontal plane, some |
354 | texts use I<r> in place of I<rho>. |
355 | |
356 | Cylindrical coordinates, I<(rho, theta, z)>, are three-dimensional |
357 | coordinates which define a point in three-dimensional space. They are |
358 | based on a cylinder surface. The radius of the cylinder is B<rho>, |
359 | also known as the I<radial> coordinate. The angle in the I<xy>-plane |
360 | (around the I<z>-axis) is B<theta>, also known as the I<azimuthal> |
361 | coordinate. The third coordinate is the I<z>, pointing up from the |
362 | B<theta>-plane. |
363 | |
364 | =head2 3-D ANGLE CONVERSIONS |
365 | |
366 | Conversions to and from spherical and cylindrical coordinates are |
367 | available. Please notice that the conversions are not necessarily |
368 | reversible because of the equalities like I<pi> angles being equal to |
369 | I<-pi> angles. |
370 | |
371 | =over 4 |
372 | |
373 | =item cartesian_to_cylindrical |
374 | |
375 | ($rho, $theta, $z) = cartesian_to_cylindrical($x, $y, $z); |
376 | |
377 | =item cartesian_to_spherical |
378 | |
379 | ($rho, $theta, $phi) = cartesian_to_spherical($x, $y, $z); |
380 | |
381 | =item cylindrical_to_cartesian |
382 | |
383 | ($x, $y, $z) = cylindrical_to_cartesian($rho, $theta, $z); |
384 | |
385 | =item cylindrical_to_spherical |
386 | |
387 | ($rho_s, $theta, $phi) = cylindrical_to_spherical($rho_c, $theta, $z); |
388 | |
389 | Notice that when C<$z> is not 0 C<$rho_s> is not equal to C<$rho_c>. |
390 | |
391 | =item spherical_to_cartesian |
392 | |
393 | ($x, $y, $z) = spherical_to_cartesian($rho, $theta, $phi); |
394 | |
395 | =item spherical_to_cylindrical |
396 | |
397 | ($rho_c, $theta, $z) = spherical_to_cylindrical($rho_s, $theta, $phi); |
398 | |
399 | Notice that when C<$z> is not 0 C<$rho_c> is not equal to C<$rho_s>. |
400 | |
401 | =back |
402 | |
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403 | =head1 GREAT CIRCLE DISTANCES AND DIRECTIONS |
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404 | |
405 | You can compute spherical distances, called B<great circle distances>, |
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406 | by importing the great_circle_distance() function: |
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407 | |
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408 | use Math::Trig 'great_circle_distance'; |
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409 | |
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410 | $distance = great_circle_distance($theta0, $phi0, $theta1, $phi1, [, $rho]); |
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411 | |
412 | The I<great circle distance> is the shortest distance between two |
413 | points on a sphere. The distance is in C<$rho> units. The C<$rho> is |
414 | optional, it defaults to 1 (the unit sphere), therefore the distance |
415 | defaults to radians. |
416 | |
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417 | If you think geographically the I<theta> are longitudes: zero at the |
418 | Greenwhich meridian, eastward positive, westward negative--and the |
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419 | I<phi> are latitudes: zero at the North Pole, northward positive, |
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420 | southward negative. B<NOTE>: this formula thinks in mathematics, not |
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421 | geographically: the I<phi> zero is at the North Pole, not at the |
422 | Equator on the west coast of Africa (Bay of Guinea). You need to |
423 | subtract your geographical coordinates from I<pi/2> (also known as 90 |
424 | degrees). |
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425 | |
426 | $distance = great_circle_distance($lon0, pi/2 - $lat0, |
427 | $lon1, pi/2 - $lat1, $rho); |
428 | |
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429 | The direction you must follow the great circle can be computed by the |
430 | great_circle_direction() function: |
431 | |
432 | use Math::Trig 'great_circle_direction'; |
433 | |
434 | $direction = great_circle_direction($theta0, $phi0, $theta1, $phi1); |
435 | |
436 | The result is in radians, zero indicating straight north, pi or -pi |
437 | straight south, pi/2 straight west, and -pi/2 straight east. |
438 | |
439 | Notice that the resulting directions might be somewhat surprising if |
440 | you are looking at a flat worldmap: in such map projections the great |
441 | circles quite often do not look like the shortest routes-- but for |
442 | example the shortest possible routes from Europe or North America to |
443 | Asia do often cross the polar regions. |
444 | |
51301382 |
445 | =head1 EXAMPLES |
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446 | |
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447 | To calculate the distance between London (51.3N 0.5W) and Tokyo |
448 | (35.7N 139.8E) in kilometers: |
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449 | |
450 | use Math::Trig qw(great_circle_distance deg2rad); |
451 | |
452 | # Notice the 90 - latitude: phi zero is at the North Pole. |
453 | @L = (deg2rad(-0.5), deg2rad(90 - 51.3)); |
454 | @T = (deg2rad(139.8),deg2rad(90 - 35.7)); |
455 | |
456 | $km = great_circle_distance(@L, @T, 6378); |
457 | |
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458 | The direction you would have to go from London to Tokyo |
459 | |
460 | use Math::Trig qw(great_circle_direction); |
461 | |
462 | $rad = great_circle_direction(@L, @T); |
463 | |
464 | =head2 CAVEAT FOR GREAT CIRCLE FORMULAS |
465 | |
466 | The answers may be off by few percentages because of the irregular |
467 | (slightly aspherical) form of the Earth. The formula used for |
468 | grear circle distances |
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469 | |
470 | lat0 = 90 degrees - phi0 |
471 | lat1 = 90 degrees - phi1 |
472 | d = R * arccos(cos(lat0) * cos(lat1) * cos(lon1 - lon01) + |
473 | sin(lat0) * sin(lat1)) |
474 | |
475 | is also somewhat unreliable for small distances (for locations |
476 | separated less than about five degrees) because it uses arc cosine |
477 | which is rather ill-conditioned for values close to zero. |
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478 | |
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479 | =head1 BUGS |
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480 | |
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481 | Saying C<use Math::Trig;> exports many mathematical routines in the |
482 | caller environment and even overrides some (C<sin>, C<cos>). This is |
483 | construed as a feature by the Authors, actually... ;-) |
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484 | |
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485 | The code is not optimized for speed, especially because we use |
486 | C<Math::Complex> and thus go quite near complex numbers while doing |
487 | the computations even when the arguments are not. This, however, |
488 | cannot be completely avoided if we want things like C<asin(2)> to give |
489 | an answer instead of giving a fatal runtime error. |
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490 | |
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491 | =head1 AUTHORS |
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492 | |
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493 | Jarkko Hietaniemi <F<jhi@iki.fi>> and |
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494 | Raphael Manfredi <F<Raphael_Manfredi@pobox.com>>. |
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495 | |
496 | =cut |
497 | |
498 | # eof |