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Math::Quaternion - Perl class to represent quaternions

• NAME
• SYNOPSIS
• DESCRIPTION
• METHODS
• AUTHOR
• ACKNOWLEDGEMENTS
• COPYRIGHT AND LICENSE

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({ 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,$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, <jon-quaternion.pm@earth.li>

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.

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