require Exporter;
package Math::Trig;
+use 5.006;
use strict;
+use Math::Complex 1.35;
use Math::Complex qw(:trig);
-use vars qw($VERSION $PACKAGE
- @ISA
- @EXPORT @EXPORT_OK %EXPORT_TAGS);
+our($VERSION, $PACKAGE, @ISA, @EXPORT, @EXPORT_OK, %EXPORT_TAGS);
@ISA = qw(Exporter);
-$VERSION = 1.00;
+$VERSION = 1.03;
my @angcnv = qw(rad2deg rad2grad
- deg2rad deg2grad
- grad2rad grad2deg);
+ deg2rad deg2grad
+ grad2rad grad2deg);
@EXPORT = (@{$Math::Complex::EXPORT_TAGS{'trig'}},
@angcnv);
spherical_to_cartesian
spherical_to_cylindrical);
-@EXPORT_OK = (@rdlcnv, 'great_circle_distance');
+my @greatcircle = qw(
+ great_circle_distance
+ great_circle_direction
+ great_circle_bearing
+ great_circle_waypoint
+ great_circle_midpoint
+ great_circle_destination
+ );
-%EXPORT_TAGS = ('radial' => [ @rdlcnv ]);
+my @pi = qw(pi2 pip2 pip4);
-use constant pi2 => 2 * pi;
-use constant pip2 => pi / 2;
-use constant DR => pi2/360;
-use constant RD => 360/pi2;
-use constant DG => 400/360;
-use constant GD => 360/400;
-use constant RG => 400/pi2;
-use constant GR => pi2/400;
+@EXPORT_OK = (@rdlcnv, @greatcircle, @pi);
+
+# See e.g. the following pages:
+# http://www.movable-type.co.uk/scripts/LatLong.html
+# http://williams.best.vwh.net/avform.htm
+
+%EXPORT_TAGS = ('radial' => [ @rdlcnv ],
+ 'great_circle' => [ @greatcircle ],
+ 'pi' => [ @pi ]);
+
+sub pi2 () { 2 * pi }
+sub pip2 () { pi / 2 }
+sub pip4 () { pi / 4 }
+
+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.
# Angle conversions.
#
-sub rad2deg ($) { remt(RD * $_[0], 360) }
+sub rad2rad($) { remt($_[0], pi2) }
+
+sub deg2deg($) { remt($_[0], 360) }
-sub deg2rad ($) { remt(DR * $_[0], pi2) }
+sub grad2grad($) { remt($_[0], 400) }
-sub grad2deg ($) { remt(GD * $_[0], 360) }
+sub rad2deg ($;$) { my $d = RD * $_[0]; $_[1] ? $d : deg2deg($d) }
-sub deg2grad ($) { remt(DG * $_[0], 400) }
+sub deg2rad ($;$) { my $d = DR * $_[0]; $_[1] ? $d : rad2rad($d) }
-sub rad2grad ($) { remt(RG * $_[0], 400) }
+sub grad2deg ($;$) { my $d = GD * $_[0]; $_[1] ? $d : deg2deg($d) }
-sub grad2rad ($) { remt(GR * $_[0], pi2) }
+sub deg2grad ($;$) { 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) }
sub cartesian_to_spherical {
my ( $x, $y, $z ) = @_;
sin( $lat0 ) * sin( $lat1 ) );
}
+sub great_circle_direction {
+ my ( $theta0, $phi0, $theta1, $phi1 ) = @_;
+
+ my $distance = &great_circle_distance;
+
+ my $lat0 = pip2 - $phi0;
+ my $lat1 = pip2 - $phi1;
+
+ my $direction =
+ acos((sin($lat1) - sin($lat0) * cos($distance)) /
+ (cos($lat0) * sin($distance)));
+
+ $direction = pi2 - $direction
+ if sin($theta1 - $theta0) < 0;
+
+ return rad2rad($direction);
+}
+
+*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 $phi = atan2($z, sqrt($x*$x + $y*$y));
+
+ 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 $phi1 = asin(sin($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
=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.
+conversions, and I<great circle formulas> for spherical movement.
=head1 TRIGONOMETRIC FUNCTIONS
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.
+pi>, where I<k> is any integer. atan2(0, 0) is undefined.
=head2 SIMPLE (REAL) ARGUMENTS, COMPLEX RESULTS
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
$radians = deg2rad($degrees);
$radians = grad2rad($gradians);
-
+
$degrees = rad2deg($radians);
$degrees = grad2deg($gradians);
-
+
$gradians = deg2grad($degrees);
$gradians = rad2grad($radians);
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().
=head1 RADIAL COORDINATE CONVERSIONS
=head2 COORDINATE SYSTEMS
-B<Cartesian> coordinates are the usual rectangular I<(x, y,
-z)>-coordinates.
+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
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 `Bay of Guinea' (think of the missing big chunk of Africa) I<0,
+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).
=back
-=head1 GREAT CIRCLE DISTANCES
+=head1 GREAT CIRCLE DISTANCES AND DIRECTIONS
You can compute spherical distances, called B<great circle distances>,
-by importing the C<great_circle_distance> function:
+by importing the great_circle_distance() function:
- use Math::Trig 'great_circle_distance'
+ use Math::Trig 'great_circle_distance';
$distance = great_circle_distance($theta0, $phi0, $theta1, $phi1, [, $rho]);
$distance = great_circle_distance($lon0, pi/2 - $lat0,
$lon1, pi/2 - $lat1, $rho);
+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, $phi1);
+
+(Alias 'great_circle_bearing' is also available.)
+The result is in radians, zero indicating straight north, pi or -pi
+straight south, pi/2 straight west, and -pi/2 straight east.
+
+You can inversely compute the destination if you know the
+starting point, direction, and distance:
+
+ use Math::Trig 'great_circle_destination';
+
+ # thetad and phid are the destination coordinates,
+ # dird is the final direction at the destination.
+
+ ($thetad, $phid, $dird) =
+ great_circle_destination($theta, $phi, $direction, $distance);
+
+or the midpoint if you know the end points:
+
+ use Math::Trig 'great_circle_midpoint';
+
+ ($thetam, $phim) =
+ great_circle_midpoint($theta0, $phi0, $theta1, $phi1);
+
+The great_circle_midpoint() is just a special case of
+
+ use Math::Trig 'great_circle_waypoint';
+
+ ($thetai, $phii) =
+ great_circle_waypoint($theta0, $phi0, $theta1, $phi1, $way);
+
+Where the $way is a value from zero ($theta0, $phi0) to one ($theta1,
+$phi1). Note that antipodal points (where their distance is I<pi>
+radians) do not have waypoints between them (they would have an an
+"equator" between them), and therefore C<undef> is returned for
+antipodal points. If the points are the same and the distance
+therefore zero and all waypoints therefore identical, the first point
+(either point) is returned.
+
+The thetas, phis, direction, and distance in the above are all in radians.
+
+You can import all the great circle formulas by
+
+ use Math::Trig ':great_circle';
+
+Notice that the resulting directions might be somewhat surprising if
+you are looking at a flat worldmap: in such map projections the great
+circles quite often do not look like the shortest routes-- but for
+example the shortest possible routes from Europe or North America to
+Asia do often cross the polar regions.
+
=head1 EXAMPLES
-To calculate the distance between London (51.3N 0.5W) and Tokyo (35.7N
-139.8E) in kilometers:
+To calculate the distance between London (51.3N 0.5W) and Tokyo
+(35.7N 139.8E) in kilometers:
use Math::Trig qw(great_circle_distance deg2rad);
# Notice the 90 - latitude: phi zero is at the North Pole.
- @L = (deg2rad(-0.5), deg2rad(90 - 51.3));
- @T = (deg2rad(139.8),deg2rad(90 - 35.7));
+ sub NESW { deg2rad($_[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);
- $km = great_circle_distance(@L, @T, 6378);
+ my $rad = great_circle_direction(@L, @T); # About 0.547 or 0.174 pi.
-The answer may be off by few percentages because of the irregular
-(slightly aspherical) form of the Earth. The used formula
+The midpoint between London and Tokyo being
- lat0 = 90 degrees - phi0
- lat1 = 90 degrees - phi1
- d = R * arccos(cos(lat0) * cos(lat1) * cos(lon1 - lon01) +
- sin(lat0) * sin(lat1))
+ use Math::Trig qw(great_circle_midpoint);
-is also somewhat unreliable for small distances (for locations
-separated less than about five degrees) because it uses arc cosine
-which is rather ill-conditioned for values close to zero.
+ my @M = great_circle_midpoint(@L, @T);
+
+or about 68.11N 24.74E, in the Finnish Lapland.
+
+=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%.
=head1 BUGS
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 AUTHORS
Jarkko Hietaniemi <F<jhi@iki.fi>> and
-Raphael Manfredi <F<Raphael_Manfredi@grenoble.hp.com>>.
+Raphael Manfredi <F<Raphael_Manfredi@pobox.com>>.
=cut
https://perl5.git.perl.org / perl5.git / blobdiff