esve::engine::dim3::Transform Class Reference

Three-dimensional transforms with rotation, translation, and scaling components. More...

#include <esve/engine/dim3/Transform.hxx>

Inheritance diagram for esve::engine::dim3::Transform:

List of all members.

Public Member Functions
Matrix elements
See the description of Transform.
virtual const types::quat &	rotation () const =0
	Rotation state.
virtual void	rotation (const types::quat &a)=0
	Rotation state.
virtual const types::pure &	translation () const =0
	Translation state.
virtual void	translation (const types::pure &a)=0
	Translation state.
virtual types::real	scaling () const =0
	Scaling state.
virtual void	scaling (types::real a)=0
	Scaling state.
Actions
The basis argument is similarity-conjugated: *this := basis*action*(basis)^(-1)*(*this) basis == IDENTITY means left action. basis == *this means right action.
virtual void	left_act (const Transform &a)=0
	*this := a * (*this)
virtual void	right_act (const Transform &a)=0
	*this := (*this) * a
virtual void	act (const Transform &action, const Transform &basis)=0
	General differential action.
virtual void	rotate (types::real angle, const types::pure &axis, const Transform &basis=IDENTITY)=0
	Rotate around this axis by this angle (in radians). The axis must have unit length.
virtual void	translate (const types::pure &dt, const Transform &basis=IDENTITY)=0
	*this := (*this) + dt
virtual void	scalate (types::real ds, const Transform &basis=IDENTITY)=0
	*this := (*this)*ds
Other operations
virtual void	identity ()=0
	*this := 1
virtual void	invert ()=0
	*this := (*this)^(-1)
virtual void	assign (const Transform &a)=0
	*this := a
virtual void	normalize ()=0
	Snap to closest orthogonal transform. The translation component is retained.
virtual bool	operator== (const Transform &) const =0
	Equal by value.
virtual bool	operator!= (const Transform &) const =0
	Unequal by value.
Static Public Attributes
const Transform &	IDENTITY
	The group identity element.

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Detailed Description

Three-dimensional transforms with rotation, translation, and scaling components.

Transforms are implemented as upper-triangular Moebius transformations of the form

    [ s  0 ][ 1  t ][ r  0 ] = [ s*r  s*t*r ]
    [ 0  1 ][ 0  1 ][ 0  r ]   [  0     r   ]

where s is real, t is pure, and r is unitary.

Concatenation gives

    [ s1*r1  s1*t1*r1 ][ s2*r2  s2*t2*r2 ]
    [   0       r1    ][   0       r2    ]
   
        = [ (s1*s2)(r1*r2)  (s1*s2)*(r1*t2*conj(r1) + (1/s2)*t1)*(r1*r2) ]
          [       0                            r1*r2                     ]

and the inverse is

    [ conj(r)     0    ][ 1  -t ][ 1/s  0 ]
    [    0     conj(r) ][ 0   1 ][  0   1 ]
   
        = [ (1/s)*conj(r)  (1/s)*(-s*conj(r)*t*r)*conj(r) ]
          [       0                    conj(r)            ]

Thus we have found a subgroup of Moebius transformations consisting of operations within R^3.

In the public interface, r is rotation(), t is translation(), and s is scaling().

The order in which these compositional matrices are applied matches that of the 4x4 projective transformations used by OpenGL. In the real-projective (column-major) representation, s corresponds to the reciprocal of the lower-right entry, t corresponds to the 3x1 upper-right column, and r corresponds to the 3x3 upper-left minor. (Of course, there is a bijective correspondence between unit quaternions mod sign and real (unitary) orthogonal 3x3 matrices, both being representations of SO(3).)

    +-----------+ +---+     +-----------+ +---+     +-----------+ +---+
    |           | |   |     |           | |   |     |           | |   |
    |           | |   |     |           | |   |     |           | |   |
    |     1     | | 0 |     |     1     | | t |     |     r     | | 0 |
    |           | |   |  *  |           | |   |  *  |           | |   |
    |           | |   |     |           | |   |     |           | |   |
    +-----------+ +---+     +-----------+ +---+     +-----------+ +---+
    +-----------+ +---+     +-----------+ +---+     +-----------+ +---+
    |     0     | |1/s|     |     0     | | 1 |     |     0     | | 1 |
    +-----------+ +---+     +-----------+ +---+     +-----------+ +---+
   
   
             +-----------+ +---+
             |           | |   |
             |           | |   |
         =   |     r     | | t |
             |           | |   |
             |           | |   |
             +-----------+ +---+
             +-----------+ +---+
             |     0     | |1/s|
             +-----------+ +---+
   

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Member Function Documentation

virtual const types::quat& esve::engine::dim3::Transform::rotation (   )  const [pure virtual]

Rotation state. Precondition: unit length Implemented in esve::engine::dim3::impl::Transform.

virtual void esve::engine::dim3::Transform::rotation (  const types::quat &  a  )  [pure virtual]

Rotation state. Precondition: unit length Implemented in esve::engine::dim3::impl::Transform.

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