Provided by: libmath-quaternion-perl_0.07-2_all 
NAME
Math::Quaternion - Perl class to represent quaternions
SYNOPSIS
use Math::Quaternion qw(slerp);
my $q = Math::Quaternion->new; # Make a new unit quaternion
# Make a rotation about the axis (0,1,0)
my $q2 = Math::Quaternion->new({axis=>[0,1,0],angle=>0.1});
my @v = (1,2,3); # A vector.
my @vrotated = $q2->rotate_vector(@v); # Rotate @v about (0,1,0).
my $q3 = Math::Quaternion::rotation(0.7,2,1,4); # A different rotation.
my $q4 = slerp($q2,$q3,0.5); # Interpolated rotation.
my @vinterp = $q4->rotate_vector(@v);
DESCRIPTION
This package lets you create and manipulate quaternions. A quaternion is a mathematical
object developed as a kind of generalization of complex numbers, usually represented by an
array of four real numbers, and is often used to represent rotations in three-dimensional
space.
See, for example, <http://mathworld.wolfram.com/Quaternion.html> for more details on the
mathematics of quaternions.
Quaternions can be added, subtracted, and scaled just like complex numbers or vectors --
they can also be multiplied, but quaternion multiplication DOES NOT COMMUTE. That is to
say, if you have quaternions $q1 and $q2, then in general $q1*$q2 != $q2*$q1. This is
related to their use in representing rotations, which also do not commute.
If you just want to represent rotations and don't care about the internal mathematical
details, this should be all you need to know:
All quaternions have a quantity called the "norm", similar to the length of a vector. A
quaternion with norm equal to 1 is called a "unit quaternion". All quaternions which
represent rotations are unit quaternions.
If you call new() without any arguments, it will give you a unit quaternion which
represents no rotation:
$q = Math::Quaternion->new;
You can make a quaternion which represents a rotation of a given angle (in radians) about
a given axis:
$qrot = Math::Quaternion->new({ angle => 0.1, axis => [ 2,3,4]});
Say you have two rotations, $q1 and $q2, and you want to make a quaternion representing a
rotation of $q1 followed by $q2. Then, you do:
$q3 = $q2 * $q1; # Rotate by $q1, followed by $q2.
Remember that this is NOT the same as $q1 * $q2, which will reverse the order of the
rotations.
If you perform many iterated quaternion operations, the result may not quite be a unit
quaternion due to numerical inaccuracies. You can make sure any quaternion has unit
length, by doing:
$unitquat = $anyquat->normalize;
If you have a rotation quaternion, and you want to find the 3x3 matrix which represents
the corresponding rotation, then:
@matrix = $q->matrix3x3;
Similarly, you can generate a 4x4 matrix of the sort you'd pass to OpenGL:
@glmatrix = $q->matrix4x4;
If you have a vector representing a direction, and you want to rotate the vector by a
quaternion $q:
my @vector = (0,0,1); # Vector pointing in the Z direction.
my @newvec = $q->rotate_vector(@vector); # New direction.
Say you're using quaternions to represent the orientation of a camera, and you have two
quaternions: one to represent a starting orientation, and another to represent a finishing
position. If you want to find all the quaternions representing the orientations in
between, allowing your camera to move smoothly from start to finish, use the slerp()
routine:
use Math::Quaternion qw(slerp);
my ($qstart, $qend) = ... ;
# Set $tween to 9 points between start and end, exclusive.
for my $t (1..9) {
my $tween = slerp($qstart,$qend,0.1*$t);
...
}
METHODS
new
my $q = Math::Quaternion->new; # Make a new unit quaternion.
my $q2 = Math::Quaternion->new(1,2,3,4);# Make a specific quaternion.
my $q3 = Math::Quaternion->new($q2); # Copy an existing quaternion.
my $q4 = Math::Quaternion->new(5.6); # Make the quaternion (5.6,0,0,0)
my $q5 = Math::Quaternion->new(7,8,9); # Make the quaternion (0,7,8,9)
my $q6 = Math::Quaternion->new({ # Make a quaternion corresponding
axis => [ 1,2,3], # to a rotation of 0.2 radians
angle => 0.2, # about the vector (1,2,3).
});
my $q7 = Math::Quaternion->new({ # Make a quaternion which would
'v1' => [ 0,1,2], # rotate the vector (0,1,2) onto
'v2' => [ -1,2,0], # the vector (-1,2,0).
});
If no parameters are given, a unit quaternion is returned. If one non-reference
parameter is given, a "scalar" quaternion is returned. If one parameter is given and it
is a reference to a quaternion or an array of four numbers, the corresponding quaternion
object is returned. If three parameters are given, a "vector" quaternion is returned.
If four parameters are given, the corresponding quaternion is returned.
Rotation quaternions may also be created by passing a hashref with the axis and angle of
rotation, or by specifying two vectors specifying start and finish directions. Bear in
mind that the latter method will take the shortest path between the two vectors, ignoring
the "roll" angle.
unit
Returns a unit quaternion.
my $u = Math::Quaternion->unit; # Returns the quaternion (1,0,0,0).
conjugate
Returns the conjugate of its argument.
my $q = Math::Quaternion->new(1,2,3,4);
my $p = $q->conjugate; # (1,-2,-3,-4)
inverse
Returns the inverse of its argument.
my $q = Math::Quaternion->new(1,2,3,4);
my $qi = $q->inverse;
normalize
Returns its argument, normalized to unit norm.
my $q = Math::Quaternion->new(1,2,3,4);
my $qn = $q->normalize;
modulus
Returns the modulus of its argument, defined as the square root of the scalar obtained by
multiplying the quaternion by its conjugate.
my $q = Math::Quaternion->new(1,2,3,4);
print $q->modulus;
isreal
Returns 1 if the given quaternion is real ,ie has no quaternion part, or else 0.
my $q1 = Math::Quaternion->new(1,2,3,4);
my $q2 = Math::Quaternion->new(5,0,0,0);
print $q1->isreal; # 0;
print $q2->isreal; # 1;
multiply
Performs a quaternion multiplication of its two arguments. If one of the arguments is a
scalar, then performs a scalar multiplication instead.
my $q1 = Math::Quaternion->new(1,2,3,4);
my $q2 = Math::Quaternion->new(5,6,7,8);
my $q3 = Math::Quaternion::multiply($q1,$q2); # (-60 12 30 24)
my $q4 = Math::Quaternion::multiply($q1,$q1->inverse); # (1 0 0 0)
dot
Returns the dot product of two quaternions.
my $q1=Math::Quaternion->new(1,2,3,4);
my $q2=Math::Quaternion->new(2,4,5,6);
my $q3 = Math::Quaternion::dot($q1,$q2);
plus
Performs a quaternion addition of its two arguments.
my $q1 = Math::Quaternion->new(1,2,3,4);
my $q2 = Math::Quaternion->new(5,6,7,8);
my $q3 = Math::Quaternion::plus($q1,$q2); # (6 8 10 12)
minus
Performs a quaternion subtraction of its two arguments.
my $q1 = Math::Quaternion->new(1,2,3,4);
my $q2 = Math::Quaternion->new(5,6,7,8);
my $q3 = Math::Quaternion::minus($q1,$q2); # (-4 -4 -4 -4)
power
Raise a quaternion to a scalar or quaternion power.
my $q1 = Math::Quaternion->new(1,2,3,4);
my $q2 = Math::Quaternion::power($q1,4); # ( 668 -224 -336 -448 )
my $q3 = $q1->power(4); # ( 668 -224 -336 -448 )
my $q4 = $q1**(-1); # Same as $q1->inverse
use Math::Trig;
my $q5 = exp(1)**( Math::Quaternion->new(pi,0,0) ); # approx (-1 0 0 0)
negate
Negates the given quaternion.
my $q = Math::Quaternion->new(1,2,3,4);
my $q1 = $q->negate; # (-1,-2,-3,-4)
squarednorm
Returns the squared norm of its argument.
my $q1 = Math::Quaternion->new(1,2,3,4);
my $sn = $q1->squarednorm; # 30
scale
Performs a scalar multiplication of its two arguments.
my $q = Math::Quaternion->new(1,2,3,4);
my $qq = Math::Quaternion::scale($q,2); # ( 2 4 6 8)
my $qqq= $q->scale(3); # ( 3 6 9 12 )
rotation
Generates a quaternion corresponding to a rotation.
If given three arguments, interprets them as an angle and the three components of an axis
vector.
use Math::Trig; # Define pi. my $theta = pi/2;
# Angle of rotation my $rotquat =
Math::Quaternion::rotation($theta,0,0,1);
# $rotquat now represents a rotation of 90 degrees about Z axis.
my ($x,$y,$z) = (1,0,0); # Unit vector in the X direction.
my ($xx,$yy,$zz) = $rotquat->rotate_vector($x,$y,$z);
# ($xx,$yy,$zz) is now ( 0, 1, 0), to within floating-point error.
rotation() can also be passed a scalar angle and a reference to a vector (in either
order), and will generate the corresponding rotation quaternion.
my @axis = (0,0,1); # Rotate about Z axis
$theta = pi/2;
$rotquat = Math::Quaternion::rotation($theta,\@axis);
If the arguments to rotation() are both references, they are interpreted as two vectors,
and a quaternion is returned which rotates the first vector onto the second.
my @startvec = (0,1,0); # Vector pointing north
my @endvec = (-1,0,0); # Vector pointing west
$rotquat = Math::Quaternion::rotation(\@startvec,\@endvec);
my @newvec = $rotquat->rotate_vector(@startvec); # Same as @endvec
rotation_angle
Returns the angle of rotation represented by the quaternion argument.
my $q = Math::Quaternion::rotation(0.1,2,3,4);
my $theta = $q->rotation_angle; # Returns 0.1 .
rotation_axis
Returns the unit vector representing the axis about which rotations will be performed,
for the rotation represented by the quaternion argument.
my $q = Math::Quaternion::rotation(0.1,1,1,0);
my @v = $q->rotation_axis; # Returns (0.5*sqrt(2),0.5*sqrt(2),0)
rotate_vector
When called as a method on a rotation quaternion, uses this quaternion to perform the
corresponding rotation on the vector argument.
use Math::Trig; # Define pi.
my $theta = pi/2; # Rotate 90 degrees
my $rotquat = Math::Quaternion::rotation($theta,0,0,1); # about Z axis
my ($x,$y,$z) = (1,0,0); # Unit vector in the X direction.
my ($xx,$yy,$zz) = $rotquat->rotate_vector($x,$y,$z)
# ($xx,$yy,$zz) is now ( 0, 1, 0), to within floating-point error.
matrix4x4
Takes one argument: a rotation quaternion. Returns a 16-element array, equal to the
OpenGL matrix which represents the corresponding rotation.
my $rotquat = Math::Quaternion::rotation($theta,@axis); # My rotation.
my @m = $rotquat->matrix4x4;
matrix3x3
Takes one argument: a rotation quaternion. Returns a 9-element array, equal to the 3x3
matrix which represents the corresponding rotation.
my $rotquat = Math::Quaternion::rotation($theta,@axis); # My rotation.
my @m = $rotquat->matrix3x3;
matrix4x4andinverse
Similar to matrix4x4, but returns a list of two array references. The first is a
reference to the rotation matrix; the second is a reference to its inverse. This may be
useful when rendering sprites, since you can multiply by the rotation matrix for the
viewer position, perform some translations, and then multiply by the inverse: any
resulting rectangles drawn will always face the viewer.
my $rotquat = Math::Quaternion::rotation($theta,@axis); # My rotation.
my ($matref,$invref) = $rotquat->matrix4x4andinverse;
stringify
Returns a string representation of the quaternion. This is used to overload the '""'
operator, so that quaternions may be freely interpolated in strings.
my $q = Math::Quaternion->new(1,2,3,4);
print $q->stringify; # "( 1 2 3 4 )"
print "$q"; # "( 1 2 3 4 )"
slerp
Takes two quaternion arguments and one scalar; performs spherical linear interpolation
between the two quaternions. The quaternion arguments are assumed to be unit quaternions,
and the scalar is assumed to lie between 0 and 1: a scalar argument of zero will return
the first quaternion argument, and a scalar argument of one will return the second.
use Math::Trig;
my @axis = (0,0,1);
my $rq1 = Math::Quaternion::rotation(pi/2,\@axis); # 90 degs about Z
my $rq2 = Math::Quaternion::rotation(pi,\@axis); # 180 degs about Z
my $interp = Math::Quaternion::slerp($rq1,$rq2,0.5); # 135 degs about Z
exp
Exponential operator e^q. Any quaternion q can be written as x+uy, where x is a real
number, and u is a unit pure quaternion. Then, exp(q) == exp(x) * ( cos(y) + u sin(y) ).
my $q = Math::Quaternion->new(1,2,3,4);
print Math::Quaternion::exp($q);
log
Returns the logarithm of its argument. The logarithm of a negative real quaternion can
take any value of them form (log(-q0),u*pi) for any unit vector u. In these cases, u is
chosen to be (1,0,0).
my $q = Math::Quaternion->new(1,2,3,4);
print Math::Quaternion::log($q);
AUTHOR
Jonathan Chin, <jon-quaternion.pm@earth.li>
ACKNOWLEDGEMENTS
Thanks to Rene Uittenbogaard and Daniel Connelly for useful suggestions, and Luc Vereecken
and Bruce Gray for patches.
SEE ALSO
<http://mathworld.wolfram.com/Quaternion.html>
<http://sjbaker.org/steve/omniv/eulers_are_evil.html>
Acts 12:4
COPYRIGHT AND LICENSE
Copyright 2003 by Jonathan Chin
This library is free software; you can redistribute it and/or modify it under the same
terms as Perl itself.