File: Trig.pm

package info (click to toggle)
perl 5.24.1-3%2Bdeb9u7
  • links: PTS, VCS
  • area: main
  • in suites: stretch
  • size: 107,108 kB
  • sloc: perl: 559,649; ansic: 293,918; sh: 67,316; pascal: 7,632; cpp: 3,895; makefile: 2,436; xml: 2,410; yacc: 989; sed: 6; lisp: 1
file content (768 lines) | stat: -rw-r--r-- 21,544 bytes parent folder | download | duplicates (4)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
#
# Trigonometric functions, mostly inherited from Math::Complex.
# -- Jarkko Hietaniemi, since April 1997
# -- Raphael Manfredi, September 1996 (indirectly: because of Math::Complex)
#

package Math::Trig;

{ use 5.006; }
use strict;

use Math::Complex 1.59;
use Math::Complex qw(:trig :pi);
require Exporter;

our @ISA = qw(Exporter);

our $VERSION = 1.23;

my @angcnv = qw(rad2deg rad2grad
		deg2rad deg2grad
		grad2rad grad2deg);

my @areal = qw(asin_real acos_real);

our @EXPORT = (@{$Math::Complex::EXPORT_TAGS{'trig'}},
	   @angcnv, @areal);

my @rdlcnv = qw(cartesian_to_cylindrical
		cartesian_to_spherical
		cylindrical_to_cartesian
		cylindrical_to_spherical
		spherical_to_cartesian
		spherical_to_cylindrical);

my @greatcircle = qw(
		     great_circle_distance
		     great_circle_direction
		     great_circle_bearing
		     great_circle_waypoint
		     great_circle_midpoint
		     great_circle_destination
		    );

my @pi = qw(pi pi2 pi4 pip2 pip4);

our @EXPORT_OK = (@rdlcnv, @greatcircle, @pi, 'Inf');

# See e.g. the following pages:
# http://www.movable-type.co.uk/scripts/LatLong.html
# http://williams.best.vwh.net/avform.htm

our %EXPORT_TAGS = ('radial' => [ @rdlcnv ],
	        'great_circle' => [ @greatcircle ],
	        'pi'     => [ @pi ]);

sub _DR  () { pi2/360 }
sub _RD  () { 360/pi2 }
sub _DG  () { 400/360 }
sub _GD  () { 360/400 }
sub _RG  () { 400/pi2 }
sub _GR  () { pi2/400 }

#
# Truncating remainder.
#

sub _remt ($$) {
    # Oh yes, POSIX::fmod() would be faster. Possibly. If it is available.
    $_[0] - $_[1] * int($_[0] / $_[1]);
}

#
# Angle conversions.
#

sub rad2rad($)     { _remt($_[0], pi2) }

sub deg2deg($)     { _remt($_[0], 360) }

sub grad2grad($)   { _remt($_[0], 400) }

sub rad2deg ($;$)  { my $d = _RD * $_[0]; $_[1] ? $d : deg2deg($d) }

sub deg2rad ($;$)  { my $d = _DR * $_[0]; $_[1] ? $d : rad2rad($d) }

sub grad2deg ($;$) { my $d = _GD * $_[0]; $_[1] ? $d : deg2deg($d) }

sub deg2grad ($;$) { my $d = _DG * $_[0]; $_[1] ? $d : grad2grad($d) }

sub rad2grad ($;$) { my $d = _RG * $_[0]; $_[1] ? $d : grad2grad($d) }

sub grad2rad ($;$) { my $d = _GR * $_[0]; $_[1] ? $d : rad2rad($d) }

#
# acos and asin functions which always return a real number
#

sub acos_real {
    return 0  if $_[0] >=  1;
    return pi if $_[0] <= -1;
    return acos($_[0]);
}

sub asin_real {
    return  &pip2 if $_[0] >=  1;
    return -&pip2 if $_[0] <= -1;
    return asin($_[0]);
}

sub cartesian_to_spherical {
    my ( $x, $y, $z ) = @_;

    my $rho = sqrt( $x * $x + $y * $y + $z * $z );

    return ( $rho,
             atan2( $y, $x ),
             $rho ? acos_real( $z / $rho ) : 0 );
}

sub spherical_to_cartesian {
    my ( $rho, $theta, $phi ) = @_;

    return ( $rho * cos( $theta ) * sin( $phi ),
             $rho * sin( $theta ) * sin( $phi ),
             $rho * cos( $phi   ) );
}

sub spherical_to_cylindrical {
    my ( $x, $y, $z ) = spherical_to_cartesian( @_ );

    return ( sqrt( $x * $x + $y * $y ), $_[1], $z );
}

sub cartesian_to_cylindrical {
    my ( $x, $y, $z ) = @_;

    return ( sqrt( $x * $x + $y * $y ), atan2( $y, $x ), $z );
}

sub cylindrical_to_cartesian {
    my ( $rho, $theta, $z ) = @_;

    return ( $rho * cos( $theta ), $rho * sin( $theta ), $z );
}

sub cylindrical_to_spherical {
    return ( cartesian_to_spherical( cylindrical_to_cartesian( @_ ) ) );
}

sub great_circle_distance {
    my ( $theta0, $phi0, $theta1, $phi1, $rho ) = @_;

    $rho = 1 unless defined $rho; # Default to the unit sphere.

    my $lat0 = pip2 - $phi0;
    my $lat1 = pip2 - $phi1;

    return $rho *
	acos_real( cos( $lat0 ) * cos( $lat1 ) * cos( $theta0 - $theta1 ) +
		   sin( $lat0 ) * sin( $lat1 ) );
}

sub great_circle_direction {
    my ( $theta0, $phi0, $theta1, $phi1 ) = @_;

    my $lat0 = pip2 - $phi0;
    my $lat1 = pip2 - $phi1;

    return rad2rad(pi2 -
	atan2(sin($theta0-$theta1) * cos($lat1),
		cos($lat0) * sin($lat1) -
		    sin($lat0) * cos($lat1) * cos($theta0-$theta1)));
}

*great_circle_bearing         = \&great_circle_direction;

sub great_circle_waypoint {
    my ( $theta0, $phi0, $theta1, $phi1, $point ) = @_;

    $point = 0.5 unless defined $point;

    my $d = great_circle_distance( $theta0, $phi0, $theta1, $phi1 );

    return undef if $d == pi;

    my $sd = sin($d);

    return ($theta0, $phi0) if $sd == 0;

    my $A = sin((1 - $point) * $d) / $sd;
    my $B = sin(     $point  * $d) / $sd;

    my $lat0 = pip2 - $phi0;
    my $lat1 = pip2 - $phi1;

    my $x = $A * cos($lat0) * cos($theta0) + $B * cos($lat1) * cos($theta1);
    my $y = $A * cos($lat0) * sin($theta0) + $B * cos($lat1) * sin($theta1);
    my $z = $A * sin($lat0)                + $B * sin($lat1);

    my $theta = atan2($y, $x);
    my $phi   = acos_real($z);

    return ($theta, $phi);
}

sub great_circle_midpoint {
    great_circle_waypoint(@_[0..3], 0.5);
}

sub great_circle_destination {
    my ( $theta0, $phi0, $dir0, $dst ) = @_;

    my $lat0 = pip2 - $phi0;

    my $phi1   = asin_real(sin($lat0)*cos($dst) +
			   cos($lat0)*sin($dst)*cos($dir0));

    my $theta1 = $theta0 + atan2(sin($dir0)*sin($dst)*cos($lat0),
				 cos($dst)-sin($lat0)*sin($phi1));

    my $dir1 = great_circle_bearing($theta1, $phi1, $theta0, $phi0) + pi;

    $dir1 -= pi2 if $dir1 > pi2;

    return ($theta1, $phi1, $dir1);
}

1;

__END__
=pod

=head1 NAME

Math::Trig - trigonometric functions

=head1 SYNOPSIS

    use Math::Trig;

    $x = tan(0.9);
    $y = acos(3.7);
    $z = asin(2.4);

    $halfpi = pi/2;

    $rad = deg2rad(120);

    # Import constants pi2, pip2, pip4 (2*pi, pi/2, pi/4).
    use Math::Trig ':pi';

    # Import the conversions between cartesian/spherical/cylindrical.
    use Math::Trig ':radial';

        # Import the great circle formulas.
    use Math::Trig ':great_circle';

=head1 DESCRIPTION

C<Math::Trig> defines many trigonometric functions not defined by the
core Perl which defines only the C<sin()> and C<cos()>.  The constant
B<pi> is also defined as are a few convenience functions for angle
conversions, and I<great circle formulas> for spherical movement.

=head1 ANGLES

All angles are defined in radians, except where otherwise specified
(for example in the deg/rad conversion functions).

=head1 TRIGONOMETRIC FUNCTIONS

The tangent

=over 4

=item B<tan>

=back

The cofunctions of the sine, cosine, and tangent (cosec/csc and cotan/cot
are aliases)

B<csc>, B<cosec>, B<sec>, B<sec>, B<cot>, B<cotan>

The arcus (also known as the inverse) functions of the sine, cosine,
and tangent

B<asin>, B<acos>, B<atan>

The principal value of the arc tangent of y/x

B<atan2>(y, x)

The arcus cofunctions of the sine, cosine, and tangent (acosec/acsc
and acotan/acot are aliases).  Note that atan2(0, 0) is not well-defined.

B<acsc>, B<acosec>, B<asec>, B<acot>, B<acotan>

The hyperbolic sine, cosine, and tangent

B<sinh>, B<cosh>, B<tanh>

The cofunctions of the hyperbolic sine, cosine, and tangent (cosech/csch
and cotanh/coth are aliases)

B<csch>, B<cosech>, B<sech>, B<coth>, B<cotanh>

The area (also known as the inverse) functions of the hyperbolic
sine, cosine, and tangent

B<asinh>, B<acosh>, B<atanh>

The area cofunctions of the hyperbolic sine, cosine, and tangent
(acsch/acosech and acoth/acotanh are aliases)

B<acsch>, B<acosech>, B<asech>, B<acoth>, B<acotanh>

The trigonometric constant B<pi> and some of handy multiples
of it are also defined.

B<pi, pi2, pi4, pip2, pip4>

=head2 ERRORS DUE TO DIVISION BY ZERO

The following functions

    acoth
    acsc
    acsch
    asec
    asech
    atanh
    cot
    coth
    csc
    csch
    sec
    sech
    tan
    tanh

cannot be computed for all arguments because that would mean dividing
by zero or taking logarithm of zero. These situations cause fatal
runtime errors looking like this

    cot(0): Division by zero.
    (Because in the definition of cot(0), the divisor sin(0) is 0)
    Died at ...

or

    atanh(-1): Logarithm of zero.
    Died at...

For the C<csc>, C<cot>, C<asec>, C<acsc>, C<acot>, C<csch>, C<coth>,
C<asech>, C<acsch>, the argument cannot be C<0> (zero).  For the
C<atanh>, C<acoth>, the argument cannot be C<1> (one).  For the
C<atanh>, C<acoth>, the argument cannot be C<-1> (minus one).  For the
C<tan>, C<sec>, C<tanh>, C<sech>, the argument cannot be I<pi/2 + k *
pi>, where I<k> is any integer.

Note that atan2(0, 0) is not well-defined.

=head2 SIMPLE (REAL) ARGUMENTS, COMPLEX RESULTS

Please note that some of the trigonometric functions can break out
from the B<real axis> into the B<complex plane>. For example
C<asin(2)> has no definition for plain real numbers but it has
definition for complex numbers.

In Perl terms this means that supplying the usual Perl numbers (also
known as scalars, please see L<perldata>) as input for the
trigonometric functions might produce as output results that no more
are simple real numbers: instead they are complex numbers.

The C<Math::Trig> handles this by using the C<Math::Complex> package
which knows how to handle complex numbers, please see L<Math::Complex>
for more information. In practice you need not to worry about getting
complex numbers as results because the C<Math::Complex> takes care of
details like for example how to display complex numbers. For example:

    print asin(2), "\n";

should produce something like this (take or leave few last decimals):

    1.5707963267949-1.31695789692482i

That is, a complex number with the real part of approximately C<1.571>
and the imaginary part of approximately C<-1.317>.

=head1 PLANE ANGLE CONVERSIONS

(Plane, 2-dimensional) angles may be converted with the following functions.

=over

=item deg2rad

    $radians  = deg2rad($degrees);

=item grad2rad

    $radians  = grad2rad($gradians);

=item rad2deg

    $degrees  = rad2deg($radians);

=item grad2deg

    $degrees  = grad2deg($gradians);

=item deg2grad

    $gradians = deg2grad($degrees);

=item rad2grad

    $gradians = rad2grad($radians);

=back

The full circle is 2 I<pi> radians or I<360> degrees or I<400> gradians.
The result is by default wrapped to be inside the [0, {2pi,360,400}[ circle.
If you don't want this, supply a true second argument:

    $zillions_of_radians  = deg2rad($zillions_of_degrees, 1);
    $negative_degrees     = rad2deg($negative_radians, 1);

You can also do the wrapping explicitly by rad2rad(), deg2deg(), and
grad2grad().

=over 4

=item rad2rad

    $radians_wrapped_by_2pi = rad2rad($radians);

=item deg2deg

    $degrees_wrapped_by_360 = deg2deg($degrees);

=item grad2grad

    $gradians_wrapped_by_400 = grad2grad($gradians);

=back

=head1 RADIAL COORDINATE CONVERSIONS

B<Radial coordinate systems> are the B<spherical> and the B<cylindrical>
systems, explained shortly in more detail.

You can import radial coordinate conversion functions by using the
C<:radial> tag:

    use Math::Trig ':radial';

    ($rho, $theta, $z)     = cartesian_to_cylindrical($x, $y, $z);
    ($rho, $theta, $phi)   = cartesian_to_spherical($x, $y, $z);
    ($x, $y, $z)           = cylindrical_to_cartesian($rho, $theta, $z);
    ($rho_s, $theta, $phi) = cylindrical_to_spherical($rho_c, $theta, $z);
    ($x, $y, $z)           = spherical_to_cartesian($rho, $theta, $phi);
    ($rho_c, $theta, $z)   = spherical_to_cylindrical($rho_s, $theta, $phi);

B<All angles are in radians>.

=head2 COORDINATE SYSTEMS

B<Cartesian> coordinates are the usual rectangular I<(x, y, z)>-coordinates.

Spherical coordinates, I<(rho, theta, pi)>, are three-dimensional
coordinates which define a point in three-dimensional space.  They are
based on a sphere surface.  The radius of the sphere is B<rho>, also
known as the I<radial> coordinate.  The angle in the I<xy>-plane
(around the I<z>-axis) is B<theta>, also known as the I<azimuthal>
coordinate.  The angle from the I<z>-axis is B<phi>, also known as the
I<polar> coordinate.  The North Pole is therefore I<0, 0, rho>, and
the Gulf of Guinea (think of the missing big chunk of Africa) I<0,
pi/2, rho>.  In geographical terms I<phi> is latitude (northward
positive, southward negative) and I<theta> is longitude (eastward
positive, westward negative).

B<BEWARE>: some texts define I<theta> and I<phi> the other way round,
some texts define the I<phi> to start from the horizontal plane, some
texts use I<r> in place of I<rho>.

Cylindrical coordinates, I<(rho, theta, z)>, are three-dimensional
coordinates which define a point in three-dimensional space.  They are
based on a cylinder surface.  The radius of the cylinder is B<rho>,
also known as the I<radial> coordinate.  The angle in the I<xy>-plane
(around the I<z>-axis) is B<theta>, also known as the I<azimuthal>
coordinate.  The third coordinate is the I<z>, pointing up from the
B<theta>-plane.

=head2 3-D ANGLE CONVERSIONS

Conversions to and from spherical and cylindrical coordinates are
available.  Please notice that the conversions are not necessarily
reversible because of the equalities like I<pi> angles being equal to
I<-pi> angles.

=over 4

=item cartesian_to_cylindrical

    ($rho, $theta, $z) = cartesian_to_cylindrical($x, $y, $z);

=item cartesian_to_spherical

    ($rho, $theta, $phi) = cartesian_to_spherical($x, $y, $z);

=item cylindrical_to_cartesian

    ($x, $y, $z) = cylindrical_to_cartesian($rho, $theta, $z);

=item cylindrical_to_spherical

    ($rho_s, $theta, $phi) = cylindrical_to_spherical($rho_c, $theta, $z);

Notice that when C<$z> is not 0 C<$rho_s> is not equal to C<$rho_c>.

=item spherical_to_cartesian

    ($x, $y, $z) = spherical_to_cartesian($rho, $theta, $phi);

=item spherical_to_cylindrical

    ($rho_c, $theta, $z) = spherical_to_cylindrical($rho_s, $theta, $phi);

Notice that when C<$z> is not 0 C<$rho_c> is not equal to C<$rho_s>.

=back

=head1 GREAT CIRCLE DISTANCES AND DIRECTIONS

A great circle is section of a circle that contains the circle
diameter: the shortest distance between two (non-antipodal) points on
the spherical surface goes along the great circle connecting those two
points.

=head2 great_circle_distance

You can compute spherical distances, called B<great circle distances>,
by importing the great_circle_distance() function:

  use Math::Trig 'great_circle_distance';

  $distance = great_circle_distance($theta0, $phi0, $theta1, $phi1, [, $rho]);

The I<great circle distance> is the shortest distance between two
points on a sphere.  The distance is in C<$rho> units.  The C<$rho> is
optional, it defaults to 1 (the unit sphere), therefore the distance
defaults to radians.

If you think geographically the I<theta> are longitudes: zero at the
Greenwhich meridian, eastward positive, westward negative -- and the
I<phi> are latitudes: zero at the North Pole, northward positive,
southward negative.  B<NOTE>: this formula thinks in mathematics, not
geographically: the I<phi> zero is at the North Pole, not at the
Equator on the west coast of Africa (Bay of Guinea).  You need to
subtract your geographical coordinates from I<pi/2> (also known as 90
degrees).

  $distance = great_circle_distance($lon0, pi/2 - $lat0,
                                    $lon1, pi/2 - $lat1, $rho);

=head2 great_circle_direction

The direction you must follow the great circle (also known as I<bearing>)
can be computed by the great_circle_direction() function:

  use Math::Trig 'great_circle_direction';

  $direction = great_circle_direction($theta0, $phi0, $theta1, $phi1);

=head2 great_circle_bearing

Alias 'great_circle_bearing' for 'great_circle_direction' is also available.

  use Math::Trig 'great_circle_bearing';

  $direction = great_circle_bearing($theta0, $phi0, $theta1, $phi1);

The result of great_circle_direction is in radians, zero indicating
straight north, pi or -pi straight south, pi/2 straight west, and
-pi/2 straight east.

=head2 great_circle_destination

You can inversely compute the destination if you know the
starting point, direction, and distance:

  use Math::Trig 'great_circle_destination';

  # $diro is the original direction,
  # for example from great_circle_bearing().
  # $distance is the angular distance in radians,
  # for example from great_circle_distance().
  # $thetad and $phid are the destination coordinates,
  # $dird is the final direction at the destination.

  ($thetad, $phid, $dird) =
    great_circle_destination($theta, $phi, $diro, $distance);

or the midpoint if you know the end points:

=head2 great_circle_midpoint

  use Math::Trig 'great_circle_midpoint';

  ($thetam, $phim) =
    great_circle_midpoint($theta0, $phi0, $theta1, $phi1);

The great_circle_midpoint() is just a special case (with $way = 0.5) of

=head2 great_circle_waypoint

  use Math::Trig 'great_circle_waypoint';

  ($thetai, $phii) =
    great_circle_waypoint($theta0, $phi0, $theta1, $phi1, $way);

Where the $way is a value from zero ($theta0, $phi0) to one ($theta1,
$phi1).  Note that antipodal points (where their distance is I<pi>
radians) do not have waypoints between them (they would have an an
"equator" between them), and therefore C<undef> is returned for
antipodal points.  If the points are the same and the distance
therefore zero and all waypoints therefore identical, the first point
(either point) is returned.

The thetas, phis, direction, and distance in the above are all in radians.

You can import all the great circle formulas by

  use Math::Trig ':great_circle';

Notice that the resulting directions might be somewhat surprising if
you are looking at a flat worldmap: in such map projections the great
circles quite often do not look like the shortest routes --  but for
example the shortest possible routes from Europe or North America to
Asia do often cross the polar regions.  (The common Mercator projection
does B<not> show great circles as straight lines: straight lines in the
Mercator projection are lines of constant bearing.)

=head1 EXAMPLES

To calculate the distance between London (51.3N 0.5W) and Tokyo
(35.7N 139.8E) in kilometers:

    use Math::Trig qw(great_circle_distance deg2rad);

    # Notice the 90 - latitude: phi zero is at the North Pole.
    sub NESW { deg2rad($_[0]), deg2rad(90 - $_[1]) }
    my @L = NESW( -0.5, 51.3);
    my @T = NESW(139.8, 35.7);
    my $km = great_circle_distance(@L, @T, 6378); # About 9600 km.

The direction you would have to go from London to Tokyo (in radians,
straight north being zero, straight east being pi/2).

    use Math::Trig qw(great_circle_direction);

    my $rad = great_circle_direction(@L, @T); # About 0.547 or 0.174 pi.

The midpoint between London and Tokyo being

    use Math::Trig qw(great_circle_midpoint);

    my @M = great_circle_midpoint(@L, @T);

or about 69 N 89 E, in the frozen wastes of Siberia.

B<NOTE>: you B<cannot> get from A to B like this:

   Dist = great_circle_distance(A, B)
   Dir  = great_circle_direction(A, B)
   C    = great_circle_destination(A, Dist, Dir)

and expect C to be B, because the bearing constantly changes when
going from A to B (except in some special case like the meridians or
the circles of latitudes) and in great_circle_destination() one gives
a B<constant> bearing to follow.

=head2 CAVEAT FOR GREAT CIRCLE FORMULAS

The answers may be off by few percentages because of the irregular
(slightly aspherical) form of the Earth.  The errors are at worst
about 0.55%, but generally below 0.3%.

=head2 Real-valued asin and acos

For small inputs asin() and acos() may return complex numbers even
when real numbers would be enough and correct, this happens because of
floating-point inaccuracies.  You can see these inaccuracies for
example by trying theses:

  print cos(1e-6)**2+sin(1e-6)**2 - 1,"\n";
  printf "%.20f", cos(1e-6)**2+sin(1e-6)**2,"\n";

which will print something like this

  -1.11022302462516e-16
  0.99999999999999988898

even though the expected results are of course exactly zero and one.
The formulas used to compute asin() and acos() are quite sensitive to
this, and therefore they might accidentally slip into the complex
plane even when they should not.  To counter this there are two
interfaces that are guaranteed to return a real-valued output.

=over 4

=item asin_real

    use Math::Trig qw(asin_real);

    $real_angle = asin_real($input_sin);

Return a real-valued arcus sine if the input is between [-1, 1],
B<inclusive> the endpoints.  For inputs greater than one, pi/2
is returned.  For inputs less than minus one, -pi/2 is returned.

=item acos_real

    use Math::Trig qw(acos_real);

    $real_angle = acos_real($input_cos);

Return a real-valued arcus cosine if the input is between [-1, 1],
B<inclusive> the endpoints.  For inputs greater than one, zero
is returned.  For inputs less than minus one, pi is returned.

=back

=head1 BUGS

Saying C<use Math::Trig;> exports many mathematical routines in the
caller environment and even overrides some (C<sin>, C<cos>).  This is
construed as a feature by the Authors, actually... ;-)

The code is not optimized for speed, especially because we use
C<Math::Complex> and thus go quite near complex numbers while doing
the computations even when the arguments are not. This, however,
cannot be completely avoided if we want things like C<asin(2)> to give
an answer instead of giving a fatal runtime error.

Do not attempt navigation using these formulas.

=head1 SEE ALSO

L<Math::Complex>

=head1 AUTHORS

Jarkko Hietaniemi <F<jhi!at!iki.fi>>,
Raphael Manfredi <F<Raphael_Manfredi!at!pobox.com>>,
Zefram <zefram@fysh.org>

=head1 LICENSE

This library is free software; you can redistribute it and/or modify
it under the same terms as Perl itself. 

=cut

# eof