NAME
Math::Quaternion - Perl class to represent quaternions
SYNOPSIS
use Math::Quaternion;
my $q = Math::Quaternion->new; # Make a new unit quaternion
# Make a rotation about the axis (0,1,0)
my $q2 = Math::Quaternion::rotation(0.1,0,1,0);
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 $q4 = 0.5*($q2+$q3); # 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({ axis => 0.1, angle => [ 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; # 1;
print $q2->isreal; # 0;
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,inverse($q1)); # (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 returnes 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,
ACKNOWLEDGEMENTS
Thanks to Rene Uittenbogaard for useful suggestions.
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.