420 lines
11 KiB
Perl
420 lines
11 KiB
Perl
#
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# Trigonometric functions, mostly inherited from Math::Complex.
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# -- Jarkko Hietaniemi, since April 1997
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# -- Raphael Manfredi, September 1996 (indirectly: because of Math::Complex)
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#
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require Exporter;
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package Math::Trig;
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use strict;
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use Math::Complex qw(:trig);
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use vars qw($VERSION $PACKAGE
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@ISA
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@EXPORT @EXPORT_OK %EXPORT_TAGS);
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@ISA = qw(Exporter);
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$VERSION = 1.00;
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my @angcnv = qw(rad2deg rad2grad
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deg2rad deg2grad
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grad2rad grad2deg);
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@EXPORT = (@{$Math::Complex::EXPORT_TAGS{'trig'}},
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@angcnv);
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my @rdlcnv = qw(cartesian_to_cylindrical
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cartesian_to_spherical
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cylindrical_to_cartesian
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cylindrical_to_spherical
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spherical_to_cartesian
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spherical_to_cylindrical);
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@EXPORT_OK = (@rdlcnv, 'great_circle_distance');
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%EXPORT_TAGS = ('radial' => [ @rdlcnv ]);
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use constant pi2 => 2 * pi;
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use constant pip2 => pi / 2;
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use constant DR => pi2/360;
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use constant RD => 360/pi2;
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use constant DG => 400/360;
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use constant GD => 360/400;
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use constant RG => 400/pi2;
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use constant GR => pi2/400;
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#
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# Truncating remainder.
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#
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sub remt ($$) {
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# Oh yes, POSIX::fmod() would be faster. Possibly. If it is available.
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$_[0] - $_[1] * int($_[0] / $_[1]);
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}
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#
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# Angle conversions.
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#
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sub rad2deg ($) { remt(RD * $_[0], 360) }
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sub deg2rad ($) { remt(DR * $_[0], pi2) }
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sub grad2deg ($) { remt(GD * $_[0], 360) }
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sub deg2grad ($) { remt(DG * $_[0], 400) }
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sub rad2grad ($) { remt(RG * $_[0], 400) }
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sub grad2rad ($) { remt(GR * $_[0], pi2) }
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sub cartesian_to_spherical {
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my ( $x, $y, $z ) = @_;
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my $rho = sqrt( $x * $x + $y * $y + $z * $z );
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return ( $rho,
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atan2( $y, $x ),
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$rho ? acos( $z / $rho ) : 0 );
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}
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sub spherical_to_cartesian {
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my ( $rho, $theta, $phi ) = @_;
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return ( $rho * cos( $theta ) * sin( $phi ),
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$rho * sin( $theta ) * sin( $phi ),
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$rho * cos( $phi ) );
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}
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sub spherical_to_cylindrical {
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my ( $x, $y, $z ) = spherical_to_cartesian( @_ );
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return ( sqrt( $x * $x + $y * $y ), $_[1], $z );
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}
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sub cartesian_to_cylindrical {
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my ( $x, $y, $z ) = @_;
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return ( sqrt( $x * $x + $y * $y ), atan2( $y, $x ), $z );
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}
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sub cylindrical_to_cartesian {
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my ( $rho, $theta, $z ) = @_;
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return ( $rho * cos( $theta ), $rho * sin( $theta ), $z );
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}
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sub cylindrical_to_spherical {
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return ( cartesian_to_spherical( cylindrical_to_cartesian( @_ ) ) );
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}
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sub great_circle_distance {
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my ( $theta0, $phi0, $theta1, $phi1, $rho ) = @_;
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$rho = 1 unless defined $rho; # Default to the unit sphere.
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my $lat0 = pip2 - $phi0;
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my $lat1 = pip2 - $phi1;
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return $rho *
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acos(cos( $lat0 ) * cos( $lat1 ) * cos( $theta0 - $theta1 ) +
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sin( $lat0 ) * sin( $lat1 ) );
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}
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=pod
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=head1 NAME
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Math::Trig - trigonometric functions
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=head1 SYNOPSIS
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use Math::Trig;
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$x = tan(0.9);
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$y = acos(3.7);
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$z = asin(2.4);
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$halfpi = pi/2;
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$rad = deg2rad(120);
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=head1 DESCRIPTION
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C<Math::Trig> defines many trigonometric functions not defined by the
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core Perl which defines only the C<sin()> and C<cos()>. The constant
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B<pi> is also defined as are a few convenience functions for angle
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conversions.
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=head1 TRIGONOMETRIC FUNCTIONS
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The tangent
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=over 4
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=item B<tan>
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=back
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The cofunctions of the sine, cosine, and tangent (cosec/csc and cotan/cot
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are aliases)
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B<csc>, B<cosec>, B<sec>, B<sec>, B<cot>, B<cotan>
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The arcus (also known as the inverse) functions of the sine, cosine,
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and tangent
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B<asin>, B<acos>, B<atan>
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The principal value of the arc tangent of y/x
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B<atan2>(y, x)
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The arcus cofunctions of the sine, cosine, and tangent (acosec/acsc
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and acotan/acot are aliases)
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B<acsc>, B<acosec>, B<asec>, B<acot>, B<acotan>
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The hyperbolic sine, cosine, and tangent
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B<sinh>, B<cosh>, B<tanh>
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The cofunctions of the hyperbolic sine, cosine, and tangent (cosech/csch
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and cotanh/coth are aliases)
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B<csch>, B<cosech>, B<sech>, B<coth>, B<cotanh>
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The arcus (also known as the inverse) functions of the hyperbolic
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sine, cosine, and tangent
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B<asinh>, B<acosh>, B<atanh>
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The arcus cofunctions of the hyperbolic sine, cosine, and tangent
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(acsch/acosech and acoth/acotanh are aliases)
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B<acsch>, B<acosech>, B<asech>, B<acoth>, B<acotanh>
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The trigonometric constant B<pi> is also defined.
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$pi2 = 2 * B<pi>;
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=head2 ERRORS DUE TO DIVISION BY ZERO
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The following functions
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acoth
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acsc
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acsch
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asec
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asech
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atanh
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cot
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coth
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csc
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csch
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sec
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sech
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tan
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tanh
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cannot be computed for all arguments because that would mean dividing
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by zero or taking logarithm of zero. These situations cause fatal
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runtime errors looking like this
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cot(0): Division by zero.
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(Because in the definition of cot(0), the divisor sin(0) is 0)
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Died at ...
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or
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atanh(-1): Logarithm of zero.
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Died at...
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For the C<csc>, C<cot>, C<asec>, C<acsc>, C<acot>, C<csch>, C<coth>,
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C<asech>, C<acsch>, the argument cannot be C<0> (zero). For the
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C<atanh>, C<acoth>, the argument cannot be C<1> (one). For the
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C<atanh>, C<acoth>, the argument cannot be C<-1> (minus one). For the
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C<tan>, C<sec>, C<tanh>, C<sech>, the argument cannot be I<pi/2 + k *
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pi>, where I<k> is any integer.
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=head2 SIMPLE (REAL) ARGUMENTS, COMPLEX RESULTS
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Please note that some of the trigonometric functions can break out
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from the B<real axis> into the B<complex plane>. For example
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C<asin(2)> has no definition for plain real numbers but it has
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definition for complex numbers.
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In Perl terms this means that supplying the usual Perl numbers (also
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known as scalars, please see L<perldata>) as input for the
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trigonometric functions might produce as output results that no more
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are simple real numbers: instead they are complex numbers.
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The C<Math::Trig> handles this by using the C<Math::Complex> package
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which knows how to handle complex numbers, please see L<Math::Complex>
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for more information. In practice you need not to worry about getting
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complex numbers as results because the C<Math::Complex> takes care of
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details like for example how to display complex numbers. For example:
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print asin(2), "\n";
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should produce something like this (take or leave few last decimals):
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1.5707963267949-1.31695789692482i
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That is, a complex number with the real part of approximately C<1.571>
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and the imaginary part of approximately C<-1.317>.
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=head1 PLANE ANGLE CONVERSIONS
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(Plane, 2-dimensional) angles may be converted with the following functions.
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$radians = deg2rad($degrees);
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$radians = grad2rad($gradians);
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$degrees = rad2deg($radians);
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$degrees = grad2deg($gradians);
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$gradians = deg2grad($degrees);
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$gradians = rad2grad($radians);
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The full circle is 2 I<pi> radians or I<360> degrees or I<400> gradians.
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=head1 RADIAL COORDINATE CONVERSIONS
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B<Radial coordinate systems> are the B<spherical> and the B<cylindrical>
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systems, explained shortly in more detail.
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You can import radial coordinate conversion functions by using the
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C<:radial> tag:
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use Math::Trig ':radial';
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($rho, $theta, $z) = cartesian_to_cylindrical($x, $y, $z);
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($rho, $theta, $phi) = cartesian_to_spherical($x, $y, $z);
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($x, $y, $z) = cylindrical_to_cartesian($rho, $theta, $z);
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($rho_s, $theta, $phi) = cylindrical_to_spherical($rho_c, $theta, $z);
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($x, $y, $z) = spherical_to_cartesian($rho, $theta, $phi);
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($rho_c, $theta, $z) = spherical_to_cylindrical($rho_s, $theta, $phi);
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B<All angles are in radians>.
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=head2 COORDINATE SYSTEMS
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B<Cartesian> coordinates are the usual rectangular I<(x, y,
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z)>-coordinates.
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Spherical coordinates, I<(rho, theta, pi)>, are three-dimensional
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coordinates which define a point in three-dimensional space. They are
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based on a sphere surface. The radius of the sphere is B<rho>, also
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known as the I<radial> coordinate. The angle in the I<xy>-plane
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(around the I<z>-axis) is B<theta>, also known as the I<azimuthal>
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coordinate. The angle from the I<z>-axis is B<phi>, also known as the
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I<polar> coordinate. The `North Pole' is therefore I<0, 0, rho>, and
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the `Bay of Guinea' (think of the missing big chunk of Africa) I<0,
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pi/2, rho>.
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B<Beware>: some texts define I<theta> and I<phi> the other way round,
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some texts define the I<phi> to start from the horizontal plane, some
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texts use I<r> in place of I<rho>.
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Cylindrical coordinates, I<(rho, theta, z)>, are three-dimensional
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coordinates which define a point in three-dimensional space. They are
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based on a cylinder surface. The radius of the cylinder is B<rho>,
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also known as the I<radial> coordinate. The angle in the I<xy>-plane
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(around the I<z>-axis) is B<theta>, also known as the I<azimuthal>
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coordinate. The third coordinate is the I<z>, pointing up from the
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B<theta>-plane.
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=head2 3-D ANGLE CONVERSIONS
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Conversions to and from spherical and cylindrical coordinates are
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available. Please notice that the conversions are not necessarily
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reversible because of the equalities like I<pi> angles being equal to
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I<-pi> angles.
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=over 4
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=item cartesian_to_cylindrical
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($rho, $theta, $z) = cartesian_to_cylindrical($x, $y, $z);
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=item cartesian_to_spherical
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($rho, $theta, $phi) = cartesian_to_spherical($x, $y, $z);
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=item cylindrical_to_cartesian
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($x, $y, $z) = cylindrical_to_cartesian($rho, $theta, $z);
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=item cylindrical_to_spherical
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($rho_s, $theta, $phi) = cylindrical_to_spherical($rho_c, $theta, $z);
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Notice that when C<$z> is not 0 C<$rho_s> is not equal to C<$rho_c>.
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=item spherical_to_cartesian
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($x, $y, $z) = spherical_to_cartesian($rho, $theta, $phi);
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=item spherical_to_cylindrical
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($rho_c, $theta, $z) = spherical_to_cylindrical($rho_s, $theta, $phi);
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Notice that when C<$z> is not 0 C<$rho_c> is not equal to C<$rho_s>.
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=back
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=head1 GREAT CIRCLE DISTANCES
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You can compute spherical distances, called B<great circle distances>,
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by importing the C<great_circle_distance> function:
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use Math::Trig 'great_circle_distance'
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$distance = great_circle_distance($theta0, $phi0, $theta1, $phi, [, $rho]);
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The I<great circle distance> is the shortest distance between two
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points on a sphere. The distance is in C<$rho> units. The C<$rho> is
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optional, it defaults to 1 (the unit sphere), therefore the distance
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defaults to radians.
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=head1 EXAMPLES
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To calculate the distance between London (51.3N 0.5W) and Tokyo (35.7N
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139.8E) in kilometers:
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use Math::Trig qw(great_circle_distance deg2rad);
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# Notice the 90 - latitude: phi zero is at the North Pole.
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@L = (deg2rad(-0.5), deg2rad(90 - 51.3));
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@T = (deg2rad(139.8),deg2rad(90 - 35.7));
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$km = great_circle_distance(@L, @T, 6378);
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The answer may be off by up to 0.3% because of the irregular (slightly
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aspherical) form of the Earth.
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=head1 BUGS
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Saying C<use Math::Trig;> exports many mathematical routines in the
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caller environment and even overrides some (C<sin>, C<cos>). This is
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construed as a feature by the Authors, actually... ;-)
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The code is not optimized for speed, especially because we use
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C<Math::Complex> and thus go quite near complex numbers while doing
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the computations even when the arguments are not. This, however,
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cannot be completely avoided if we want things like C<asin(2)> to give
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an answer instead of giving a fatal runtime error.
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=head1 AUTHORS
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Jarkko Hietaniemi <F<jhi@iki.fi>> and
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Raphael Manfredi <F<Raphael_Manfredi@grenoble.hp.com>>.
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=cut
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# eof
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