man Math::Quaternion (3): 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 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, <[email protected]>

ACKNOWLEDGEMENTS

Thanks to Rene Uittenbogaard and Daniel Connelly for useful suggestions, and Luc Vereecken and Bruce Gray for patches.

COPYRIGHT AND LICENSE