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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 | |
affad850 |
10 | use 5.005; |
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11 | use strict; |
12 | |
affad850 |
13 | use Math::Complex 1.36; |
14 | use Math::Complex qw(:trig :pi); |
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15 | |
affad850 |
16 | use vars qw($VERSION $PACKAGE @ISA @EXPORT @EXPORT_OK %EXPORT_TAGS); |
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17 | |
18 | @ISA = qw(Exporter); |
19 | |
affad850 |
20 | $VERSION = 1.04; |
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21 | |
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22 | my @angcnv = qw(rad2deg rad2grad |
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23 | deg2rad deg2grad |
24 | grad2rad grad2deg); |
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25 | |
26 | @EXPORT = (@{$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 | |
affad850 |
45 | my @pi = qw(pi pi2 pi4 pip2 pip4); |
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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 | |
affad850 |
57 | sub _DR () { pi2/360 } |
58 | sub _RD () { 360/pi2 } |
59 | sub _DG () { 400/360 } |
60 | sub _GD () { 360/400 } |
61 | sub _RG () { 400/pi2 } |
62 | sub _GR () { pi2/400 } |
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63 | |
64 | # |
65 | # Truncating remainder. |
66 | # |
67 | |
affad850 |
68 | sub _remt ($$) { |
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69 | # Oh yes, POSIX::fmod() would be faster. Possibly. If it is available. |
70 | $_[0] - $_[1] * int($_[0] / $_[1]); |
71 | } |
72 | |
73 | # |
74 | # Angle conversions. |
75 | # |
76 | |
affad850 |
77 | sub rad2rad($) { _remt($_[0], pi2) } |
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78 | |
affad850 |
79 | sub deg2deg($) { _remt($_[0], 360) } |
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80 | |
affad850 |
81 | sub grad2grad($) { _remt($_[0], 400) } |
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82 | |
affad850 |
83 | sub rad2deg ($;$) { my $d = _RD * $_[0]; $_[1] ? $d : deg2deg($d) } |
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84 | |
affad850 |
85 | sub deg2rad ($;$) { my $d = _DR * $_[0]; $_[1] ? $d : rad2rad($d) } |
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86 | |
affad850 |
87 | sub grad2deg ($;$) { my $d = _GD * $_[0]; $_[1] ? $d : deg2deg($d) } |
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88 | |
affad850 |
89 | sub deg2grad ($;$) { my $d = _DG * $_[0]; $_[1] ? $d : grad2grad($d) } |
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90 | |
affad850 |
91 | sub rad2grad ($;$) { my $d = _RG * $_[0]; $_[1] ? $d : grad2grad($d) } |
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92 | |
affad850 |
93 | sub grad2rad ($;$) { my $d = _GR * $_[0]; $_[1] ? $d : rad2rad($d) } |
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94 | |
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95 | sub cartesian_to_spherical { |
96 | my ( $x, $y, $z ) = @_; |
97 | |
98 | my $rho = sqrt( $x * $x + $y * $y + $z * $z ); |
99 | |
100 | return ( $rho, |
101 | atan2( $y, $x ), |
102 | $rho ? acos( $z / $rho ) : 0 ); |
103 | } |
104 | |
105 | sub spherical_to_cartesian { |
106 | my ( $rho, $theta, $phi ) = @_; |
107 | |
108 | return ( $rho * cos( $theta ) * sin( $phi ), |
109 | $rho * sin( $theta ) * sin( $phi ), |
110 | $rho * cos( $phi ) ); |
111 | } |
112 | |
113 | sub spherical_to_cylindrical { |
114 | my ( $x, $y, $z ) = spherical_to_cartesian( @_ ); |
115 | |
116 | return ( sqrt( $x * $x + $y * $y ), $_[1], $z ); |
117 | } |
118 | |
119 | sub cartesian_to_cylindrical { |
120 | my ( $x, $y, $z ) = @_; |
121 | |
122 | return ( sqrt( $x * $x + $y * $y ), atan2( $y, $x ), $z ); |
123 | } |
124 | |
125 | sub cylindrical_to_cartesian { |
126 | my ( $rho, $theta, $z ) = @_; |
127 | |
128 | return ( $rho * cos( $theta ), $rho * sin( $theta ), $z ); |
129 | } |
130 | |
131 | sub cylindrical_to_spherical { |
132 | return ( cartesian_to_spherical( cylindrical_to_cartesian( @_ ) ) ); |
133 | } |
134 | |
135 | sub great_circle_distance { |
136 | my ( $theta0, $phi0, $theta1, $phi1, $rho ) = @_; |
137 | |
138 | $rho = 1 unless defined $rho; # Default to the unit sphere. |
139 | |
140 | my $lat0 = pip2 - $phi0; |
141 | my $lat1 = pip2 - $phi1; |
142 | |
143 | return $rho * |
144 | acos(cos( $lat0 ) * cos( $lat1 ) * cos( $theta0 - $theta1 ) + |
145 | sin( $lat0 ) * sin( $lat1 ) ); |
146 | } |
147 | |
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148 | sub great_circle_direction { |
149 | my ( $theta0, $phi0, $theta1, $phi1 ) = @_; |
150 | |
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151 | my $distance = &great_circle_distance; |
152 | |
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153 | my $lat0 = pip2 - $phi0; |
154 | my $lat1 = pip2 - $phi1; |
155 | |
156 | my $direction = |
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157 | acos((sin($lat1) - sin($lat0) * cos($distance)) / |
158 | (cos($lat0) * sin($distance))); |
159 | |
160 | $direction = pi2 - $direction |
161 | if sin($theta1 - $theta0) < 0; |
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162 | |
163 | return rad2rad($direction); |
164 | } |
165 | |
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166 | *great_circle_bearing = \&great_circle_direction; |
167 | |
168 | sub great_circle_waypoint { |
169 | my ( $theta0, $phi0, $theta1, $phi1, $point ) = @_; |
170 | |
171 | $point = 0.5 unless defined $point; |
172 | |
173 | my $d = great_circle_distance( $theta0, $phi0, $theta1, $phi1 ); |
174 | |
175 | return undef if $d == pi; |
176 | |
177 | my $sd = sin($d); |
178 | |
179 | return ($theta0, $phi0) if $sd == 0; |
180 | |
181 | my $A = sin((1 - $point) * $d) / $sd; |
182 | my $B = sin( $point * $d) / $sd; |
183 | |
184 | my $lat0 = pip2 - $phi0; |
185 | my $lat1 = pip2 - $phi1; |
186 | |
187 | my $x = $A * cos($lat0) * cos($theta0) + $B * cos($lat1) * cos($theta1); |
188 | my $y = $A * cos($lat0) * sin($theta0) + $B * cos($lat1) * sin($theta1); |
189 | my $z = $A * sin($lat0) + $B * sin($lat1); |
190 | |
191 | my $theta = atan2($y, $x); |
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192 | my $phi = acos($z); |
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193 | |
194 | return ($theta, $phi); |
195 | } |
196 | |
197 | sub great_circle_midpoint { |
198 | great_circle_waypoint(@_[0..3], 0.5); |
199 | } |
200 | |
201 | sub great_circle_destination { |
202 | my ( $theta0, $phi0, $dir0, $dst ) = @_; |
203 | |
204 | my $lat0 = pip2 - $phi0; |
205 | |
206 | my $phi1 = asin(sin($lat0)*cos($dst)+cos($lat0)*sin($dst)*cos($dir0)); |
207 | my $theta1 = $theta0 + atan2(sin($dir0)*sin($dst)*cos($lat0), |
208 | cos($dst)-sin($lat0)*sin($phi1)); |
209 | |
210 | my $dir1 = great_circle_bearing($theta1, $phi1, $theta0, $phi0) + pi; |
211 | |
212 | $dir1 -= pi2 if $dir1 > pi2; |
213 | |
214 | return ($theta1, $phi1, $dir1); |
215 | } |
216 | |
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217 | 1; |
218 | |
219 | __END__ |
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220 | =pod |
221 | |
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222 | =head1 NAME |
223 | |
224 | Math::Trig - trigonometric functions |
225 | |
226 | =head1 SYNOPSIS |
227 | |
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228 | use Math::Trig; |
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229 | |
affad850 |
230 | $x = tan(0.9); |
231 | $y = acos(3.7); |
232 | $z = asin(2.4); |
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233 | |
affad850 |
234 | $halfpi = pi/2; |
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235 | |
affad850 |
236 | $rad = deg2rad(120); |
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237 | |
affad850 |
238 | # Import constants pi2, pip2, pip4 (2*pi, pi/2, pi/4). |
239 | use Math::Trig ':pi'; |
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240 | |
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241 | # Import the conversions between cartesian/spherical/cylindrical. |
242 | use Math::Trig ':radial'; |
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243 | |
244 | # Import the great circle formulas. |
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245 | use Math::Trig ':great_circle'; |
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246 | |
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247 | =head1 DESCRIPTION |
248 | |
249 | C<Math::Trig> defines many trigonometric functions not defined by the |
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250 | core Perl which defines only the C<sin()> and C<cos()>. The constant |
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251 | B<pi> is also defined as are a few convenience functions for angle |
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252 | conversions, and I<great circle formulas> for spherical movement. |
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253 | |
254 | =head1 TRIGONOMETRIC FUNCTIONS |
255 | |
256 | The tangent |
257 | |
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258 | =over 4 |
259 | |
260 | =item B<tan> |
261 | |
262 | =back |
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263 | |
264 | The cofunctions of the sine, cosine, and tangent (cosec/csc and cotan/cot |
265 | are aliases) |
266 | |
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267 | B<csc>, B<cosec>, B<sec>, B<sec>, B<cot>, B<cotan> |
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268 | |
269 | The arcus (also known as the inverse) functions of the sine, cosine, |
270 | and tangent |
271 | |
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272 | B<asin>, B<acos>, B<atan> |
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273 | |
274 | The principal value of the arc tangent of y/x |
275 | |
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276 | B<atan2>(y, x) |
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277 | |
278 | The arcus cofunctions of the sine, cosine, and tangent (acosec/acsc |
affad850 |
279 | and acotan/acot are aliases). Note that atan2(0, 0) is not well-defined. |
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280 | |
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281 | B<acsc>, B<acosec>, B<asec>, B<acot>, B<acotan> |
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282 | |
283 | The hyperbolic sine, cosine, and tangent |
284 | |
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285 | B<sinh>, B<cosh>, B<tanh> |
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286 | |
287 | The cofunctions of the hyperbolic sine, cosine, and tangent (cosech/csch |
288 | and cotanh/coth are aliases) |
289 | |
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290 | B<csch>, B<cosech>, B<sech>, B<coth>, B<cotanh> |
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291 | |
292 | The arcus (also known as the inverse) functions of the hyperbolic |
293 | sine, cosine, and tangent |
294 | |
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295 | B<asinh>, B<acosh>, B<atanh> |
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296 | |
297 | The arcus cofunctions of the hyperbolic sine, cosine, and tangent |
298 | (acsch/acosech and acoth/acotanh are aliases) |
299 | |
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300 | B<acsch>, B<acosech>, B<asech>, B<acoth>, B<acotanh> |
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301 | |
affad850 |
302 | The trigonometric constant B<pi> and some of handy multiples |
303 | of it are also defined. |
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304 | |
affad850 |
305 | B<pi, pi2, pi4, pip2, pip4> |
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306 | |
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307 | =head2 ERRORS DUE TO DIVISION BY ZERO |
308 | |
309 | The following functions |
310 | |
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311 | acoth |
312 | acsc |
313 | acsch |
314 | asec |
315 | asech |
316 | atanh |
317 | cot |
318 | coth |
319 | csc |
320 | csch |
321 | sec |
322 | sech |
323 | tan |
324 | tanh |
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325 | |
326 | cannot be computed for all arguments because that would mean dividing |
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327 | by zero or taking logarithm of zero. These situations cause fatal |
328 | runtime errors looking like this |
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329 | |
affad850 |
330 | cot(0): Division by zero. |
331 | (Because in the definition of cot(0), the divisor sin(0) is 0) |
332 | Died at ... |
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333 | |
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334 | or |
335 | |
affad850 |
336 | atanh(-1): Logarithm of zero. |
337 | Died at... |
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338 | |
339 | For the C<csc>, C<cot>, C<asec>, C<acsc>, C<acot>, C<csch>, C<coth>, |
340 | C<asech>, C<acsch>, the argument cannot be C<0> (zero). For the |
341 | C<atanh>, C<acoth>, the argument cannot be C<1> (one). For the |
342 | C<atanh>, C<acoth>, the argument cannot be C<-1> (minus one). For the |
343 | C<tan>, C<sec>, C<tanh>, C<sech>, the argument cannot be I<pi/2 + k * |
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344 | pi>, where I<k> is any integer. |
345 | |
346 | Note that atan2(0, 0) is not well-defined. |
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347 | |
348 | =head2 SIMPLE (REAL) ARGUMENTS, COMPLEX RESULTS |
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349 | |
350 | Please note that some of the trigonometric functions can break out |
351 | from the B<real axis> into the B<complex plane>. For example |
352 | C<asin(2)> has no definition for plain real numbers but it has |
353 | definition for complex numbers. |
354 | |
355 | In Perl terms this means that supplying the usual Perl numbers (also |
356 | known as scalars, please see L<perldata>) as input for the |
357 | trigonometric functions might produce as output results that no more |
358 | are simple real numbers: instead they are complex numbers. |
359 | |
360 | The C<Math::Trig> handles this by using the C<Math::Complex> package |
361 | which knows how to handle complex numbers, please see L<Math::Complex> |
362 | for more information. In practice you need not to worry about getting |
363 | complex numbers as results because the C<Math::Complex> takes care of |
364 | details like for example how to display complex numbers. For example: |
365 | |
affad850 |
366 | print asin(2), "\n"; |
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367 | |
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368 | should produce something like this (take or leave few last decimals): |
369 | |
affad850 |
370 | 1.5707963267949-1.31695789692482i |
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371 | |
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372 | That is, a complex number with the real part of approximately C<1.571> |
373 | and the imaginary part of approximately C<-1.317>. |
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374 | |
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375 | =head1 PLANE ANGLE CONVERSIONS |
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376 | |
377 | (Plane, 2-dimensional) angles may be converted with the following functions. |
378 | |
affad850 |
379 | =over |
380 | |
381 | =item deg2rad |
382 | |
383 | $radians = deg2rad($degrees); |
384 | |
385 | =item grad2rad |
386 | |
387 | $radians = grad2rad($gradians); |
388 | |
389 | =item rad2deg |
390 | |
391 | $degrees = rad2deg($radians); |
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392 | |
affad850 |
393 | =item grad2deg |
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394 | |
affad850 |
395 | $degrees = grad2deg($gradians); |
396 | |
397 | =item deg2grad |
398 | |
399 | $gradians = deg2grad($degrees); |
400 | |
401 | =item rad2grad |
402 | |
403 | $gradians = rad2grad($radians); |
404 | |
405 | =back |
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406 | |
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407 | The full circle is 2 I<pi> radians or I<360> degrees or I<400> gradians. |
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408 | The result is by default wrapped to be inside the [0, {2pi,360,400}[ circle. |
409 | If you don't want this, supply a true second argument: |
410 | |
affad850 |
411 | $zillions_of_radians = deg2rad($zillions_of_degrees, 1); |
412 | $negative_degrees = rad2deg($negative_radians, 1); |
9db5a202 |
413 | |
414 | You can also do the wrapping explicitly by rad2rad(), deg2deg(), and |
415 | grad2grad(). |
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416 | |
affad850 |
417 | =over 4 |
418 | |
419 | =item rad2rad |
420 | |
421 | $radians_wrapped_by_2pi = rad2rad($radians); |
422 | |
423 | =item deg2deg |
424 | |
425 | $degrees_wrapped_by_360 = deg2deg($degrees); |
426 | |
427 | =item grad2grad |
428 | |
429 | $gradians_wrapped_by_400 = grad2grad($gradians); |
430 | |
431 | =back |
432 | |
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433 | =head1 RADIAL COORDINATE CONVERSIONS |
434 | |
435 | B<Radial coordinate systems> are the B<spherical> and the B<cylindrical> |
436 | systems, explained shortly in more detail. |
437 | |
438 | You can import radial coordinate conversion functions by using the |
439 | C<:radial> tag: |
440 | |
441 | use Math::Trig ':radial'; |
442 | |
443 | ($rho, $theta, $z) = cartesian_to_cylindrical($x, $y, $z); |
444 | ($rho, $theta, $phi) = cartesian_to_spherical($x, $y, $z); |
445 | ($x, $y, $z) = cylindrical_to_cartesian($rho, $theta, $z); |
446 | ($rho_s, $theta, $phi) = cylindrical_to_spherical($rho_c, $theta, $z); |
447 | ($x, $y, $z) = spherical_to_cartesian($rho, $theta, $phi); |
448 | ($rho_c, $theta, $z) = spherical_to_cylindrical($rho_s, $theta, $phi); |
449 | |
450 | B<All angles are in radians>. |
451 | |
452 | =head2 COORDINATE SYSTEMS |
453 | |
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454 | B<Cartesian> coordinates are the usual rectangular I<(x, y, z)>-coordinates. |
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455 | |
456 | Spherical coordinates, I<(rho, theta, pi)>, are three-dimensional |
457 | coordinates which define a point in three-dimensional space. They are |
458 | based on a sphere surface. The radius of the sphere is B<rho>, also |
459 | known as the I<radial> coordinate. The angle in the I<xy>-plane |
460 | (around the I<z>-axis) is B<theta>, also known as the I<azimuthal> |
461 | coordinate. The angle from the I<z>-axis is B<phi>, also known as the |
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462 | I<polar> coordinate. The North Pole is therefore I<0, 0, rho>, and |
463 | the Gulf of Guinea (think of the missing big chunk of Africa) I<0, |
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464 | pi/2, rho>. In geographical terms I<phi> is latitude (northward |
465 | positive, southward negative) and I<theta> is longitude (eastward |
466 | positive, westward negative). |
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467 | |
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468 | B<BEWARE>: some texts define I<theta> and I<phi> the other way round, |
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469 | some texts define the I<phi> to start from the horizontal plane, some |
470 | texts use I<r> in place of I<rho>. |
471 | |
472 | Cylindrical coordinates, I<(rho, theta, z)>, are three-dimensional |
473 | coordinates which define a point in three-dimensional space. They are |
474 | based on a cylinder surface. The radius of the cylinder is B<rho>, |
475 | also known as the I<radial> coordinate. The angle in the I<xy>-plane |
476 | (around the I<z>-axis) is B<theta>, also known as the I<azimuthal> |
477 | coordinate. The third coordinate is the I<z>, pointing up from the |
478 | B<theta>-plane. |
479 | |
480 | =head2 3-D ANGLE CONVERSIONS |
481 | |
482 | Conversions to and from spherical and cylindrical coordinates are |
483 | available. Please notice that the conversions are not necessarily |
484 | reversible because of the equalities like I<pi> angles being equal to |
485 | I<-pi> angles. |
486 | |
487 | =over 4 |
488 | |
489 | =item cartesian_to_cylindrical |
490 | |
affad850 |
491 | ($rho, $theta, $z) = cartesian_to_cylindrical($x, $y, $z); |
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492 | |
493 | =item cartesian_to_spherical |
494 | |
affad850 |
495 | ($rho, $theta, $phi) = cartesian_to_spherical($x, $y, $z); |
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496 | |
497 | =item cylindrical_to_cartesian |
498 | |
affad850 |
499 | ($x, $y, $z) = cylindrical_to_cartesian($rho, $theta, $z); |
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500 | |
501 | =item cylindrical_to_spherical |
502 | |
affad850 |
503 | ($rho_s, $theta, $phi) = cylindrical_to_spherical($rho_c, $theta, $z); |
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504 | |
505 | Notice that when C<$z> is not 0 C<$rho_s> is not equal to C<$rho_c>. |
506 | |
507 | =item spherical_to_cartesian |
508 | |
affad850 |
509 | ($x, $y, $z) = spherical_to_cartesian($rho, $theta, $phi); |
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510 | |
511 | =item spherical_to_cylindrical |
512 | |
affad850 |
513 | ($rho_c, $theta, $z) = spherical_to_cylindrical($rho_s, $theta, $phi); |
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514 | |
515 | Notice that when C<$z> is not 0 C<$rho_c> is not equal to C<$rho_s>. |
516 | |
517 | =back |
518 | |
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519 | =head1 GREAT CIRCLE DISTANCES AND DIRECTIONS |
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520 | |
affad850 |
521 | A great circle is section of a circle that contains the circle |
522 | diameter: the shortest distance between two (non-antipodal) points on |
523 | the spherical surface goes along the great circle connecting those two |
524 | points. |
525 | |
526 | =head2 great_circle_distance |
527 | |
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528 | You can compute spherical distances, called B<great circle distances>, |
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529 | by importing the great_circle_distance() function: |
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530 | |
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531 | use Math::Trig 'great_circle_distance'; |
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532 | |
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533 | $distance = great_circle_distance($theta0, $phi0, $theta1, $phi1, [, $rho]); |
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534 | |
535 | The I<great circle distance> is the shortest distance between two |
536 | points on a sphere. The distance is in C<$rho> units. The C<$rho> is |
537 | optional, it defaults to 1 (the unit sphere), therefore the distance |
538 | defaults to radians. |
539 | |
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540 | If you think geographically the I<theta> are longitudes: zero at the |
541 | Greenwhich meridian, eastward positive, westward negative--and the |
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542 | I<phi> are latitudes: zero at the North Pole, northward positive, |
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543 | southward negative. B<NOTE>: this formula thinks in mathematics, not |
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544 | geographically: the I<phi> zero is at the North Pole, not at the |
545 | Equator on the west coast of Africa (Bay of Guinea). You need to |
546 | subtract your geographical coordinates from I<pi/2> (also known as 90 |
547 | degrees). |
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548 | |
549 | $distance = great_circle_distance($lon0, pi/2 - $lat0, |
550 | $lon1, pi/2 - $lat1, $rho); |
551 | |
affad850 |
552 | =head2 great_circle_direction |
553 | |
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554 | The direction you must follow the great circle (also known as I<bearing>) |
555 | can be computed by the great_circle_direction() function: |
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556 | |
557 | use Math::Trig 'great_circle_direction'; |
558 | |
559 | $direction = great_circle_direction($theta0, $phi0, $theta1, $phi1); |
560 | |
affad850 |
561 | =head2 great_circle_bearing |
562 | |
563 | Alias 'great_circle_bearing' for 'great_circle_direction' is also available. |
564 | |
565 | use Math::Trig 'great_circle_bearing'; |
566 | |
567 | $direction = great_circle_bearing($theta0, $phi0, $theta1, $phi1); |
568 | |
569 | The result of great_circle_direction is in radians, zero indicating |
570 | straight north, pi or -pi straight south, pi/2 straight west, and |
571 | -pi/2 straight east. |
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572 | |
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573 | You can inversely compute the destination if you know the |
574 | starting point, direction, and distance: |
575 | |
affad850 |
576 | =head2 great_circle_destination |
577 | |
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578 | use Math::Trig 'great_circle_destination'; |
579 | |
580 | # thetad and phid are the destination coordinates, |
581 | # dird is the final direction at the destination. |
582 | |
583 | ($thetad, $phid, $dird) = |
584 | great_circle_destination($theta, $phi, $direction, $distance); |
585 | |
586 | or the midpoint if you know the end points: |
587 | |
affad850 |
588 | =head2 great_circle_midpoint |
589 | |
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590 | use Math::Trig 'great_circle_midpoint'; |
591 | |
592 | ($thetam, $phim) = |
593 | great_circle_midpoint($theta0, $phi0, $theta1, $phi1); |
594 | |
595 | The great_circle_midpoint() is just a special case of |
596 | |
affad850 |
597 | =head2 great_circle_waypoint |
598 | |
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599 | use Math::Trig 'great_circle_waypoint'; |
600 | |
601 | ($thetai, $phii) = |
602 | great_circle_waypoint($theta0, $phi0, $theta1, $phi1, $way); |
603 | |
604 | Where the $way is a value from zero ($theta0, $phi0) to one ($theta1, |
605 | $phi1). Note that antipodal points (where their distance is I<pi> |
606 | radians) do not have waypoints between them (they would have an an |
607 | "equator" between them), and therefore C<undef> is returned for |
608 | antipodal points. If the points are the same and the distance |
609 | therefore zero and all waypoints therefore identical, the first point |
610 | (either point) is returned. |
611 | |
612 | The thetas, phis, direction, and distance in the above are all in radians. |
613 | |
614 | You can import all the great circle formulas by |
615 | |
616 | use Math::Trig ':great_circle'; |
617 | |
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618 | Notice that the resulting directions might be somewhat surprising if |
619 | you are looking at a flat worldmap: in such map projections the great |
620 | circles quite often do not look like the shortest routes-- but for |
621 | example the shortest possible routes from Europe or North America to |
622 | Asia do often cross the polar regions. |
623 | |
51301382 |
624 | =head1 EXAMPLES |
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625 | |
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626 | To calculate the distance between London (51.3N 0.5W) and Tokyo |
627 | (35.7N 139.8E) in kilometers: |
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628 | |
affad850 |
629 | use Math::Trig qw(great_circle_distance deg2rad); |
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630 | |
affad850 |
631 | # Notice the 90 - latitude: phi zero is at the North Pole. |
632 | sub NESW { deg2rad($_[0]), deg2rad(90 - $_[1]) } |
633 | my @L = NESW( -0.5, 51.3); |
634 | my @T = NESW(139.8, 35.7); |
635 | my $km = great_circle_distance(@L, @T, 6378); # About 9600 km. |
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636 | |
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637 | The direction you would have to go from London to Tokyo (in radians, |
638 | straight north being zero, straight east being pi/2). |
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639 | |
affad850 |
640 | use Math::Trig qw(great_circle_direction); |
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641 | |
affad850 |
642 | my $rad = great_circle_direction(@L, @T); # About 0.547 or 0.174 pi. |
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643 | |
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644 | The midpoint between London and Tokyo being |
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645 | |
affad850 |
646 | use Math::Trig qw(great_circle_midpoint); |
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647 | |
affad850 |
648 | my @M = great_circle_midpoint(@L, @T); |
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649 | |
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650 | or about 89.16N 68.93E, practically at the North Pole. |
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651 | |
bf5f1b4c |
652 | =head2 CAVEAT FOR GREAT CIRCLE FORMULAS |
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653 | |
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654 | The answers may be off by few percentages because of the irregular |
655 | (slightly aspherical) form of the Earth. The errors are at worst |
656 | about 0.55%, but generally below 0.3%. |
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657 | |
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658 | =head1 BUGS |
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659 | |
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660 | Saying C<use Math::Trig;> exports many mathematical routines in the |
661 | caller environment and even overrides some (C<sin>, C<cos>). This is |
662 | construed as a feature by the Authors, actually... ;-) |
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663 | |
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664 | The code is not optimized for speed, especially because we use |
665 | C<Math::Complex> and thus go quite near complex numbers while doing |
666 | the computations even when the arguments are not. This, however, |
667 | cannot be completely avoided if we want things like C<asin(2)> to give |
668 | an answer instead of giving a fatal runtime error. |
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669 | |
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670 | Do not attempt navigation using these formulas. |
671 | |
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672 | =head1 AUTHORS |
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673 | |
affad850 |
674 | Jarkko Hietaniemi <F<jhi!at!iki.fi>> and |
675 | Raphael Manfredi <F<Raphael_Manfredi!at!pobox.com>>. |
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676 | |
677 | =cut |
678 | |
679 | # eof |