esve::engine::dim4::Transform Class Reference

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

#include <esve/engine/dim4/Transform.hxx>

Inheritance diagram for esve::engine::dim4::Transform:

List of all members.

Public Member Functions
Matrix elements
f(z) = (a*z + b)*inv(c*z + d) See the description of Transform.
virtual const types::quat &	a () const =0
	Upper-left.
virtual void	a (const types::quat &a)=0
	Upper-left.
virtual const types::quat &	b () const =0
	Upper-right.
virtual void	b (const types::quat &b)=0
	Upper-right.
virtual const types::quat &	c () const =0
	Lower-left.
virtual void	c (const types::quat &c)=0
	Lower-left.
virtual const types::quat &	d () const =0
	Lower-right.
virtual void	d (const types::quat &d)=0
	Lower-right.
Actions
Relative actions on the current state. 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 &left)=0
	*this := left * (*this)
virtual void	right_act (const Transform &right)=0
	*this := (*this) * right
virtual void	act (const Transform &action, const Transform &basis=IDENTITY)=0
	General differential action.
virtual void	rotate (types::real alpha, types::real beta, const types::quat &r, const types::quat &s, const Transform &basis=IDENTITY)=0
	Rotate by angle alpha in the (r, s) plane; rotate by angle beta in the (r, s)-complement plane.
virtual void	rotate (types::real alpha, types::real beta, const types::pure &u, const Transform &basis=IDENTITY)=0
	Rotate by angle alpha in the (1, u) plane; rotate by angle beta in the (1, u)-complement plane.
virtual void	translate (const types::quat &r, const Transform &basis=IDENTITY)=0
	Move the local origin from g to g' = g + r.
virtual void	invertate (const types::quat &r, const Transform &basis=IDENTITY)=0
	Move the local infinity from h to h' per the relation (h')^(-1) = h^(-1) + r.
virtual void	scalate (types::real gamma, const Transform &basis=IDENTITY)=0
	Scale from the current state by a factor of gamma.
Other operations
virtual void	identity ()=0
	*this := 1
virtual void	invert ()=0
	*this := (*this)^(-1)
virtual void	assign (const Transform &t)=0
	*this := t
virtual void	normalize ()=0
	Snap to upper-triangular with unitary diagonal elements.
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

Four-dimensional transforms with rotation, translation, inversion, and scaling components.

These are Moebius transformations, which have the form

 f(z) = (a*z + b)*inv(c*z + d)

and are isomorphic to projective 2x2 quaternion matrix transformations

      T [ z1 ] = [ a  b ] [ z1 ]
        [ z2 ]   [ c  d ] [ z2 ]

Just as real 4x4 projective matrices represent 3D transformations which carry a local rotation, origin, and a scaling factor, these quaternionic 2x2 projective matrices represent 4D transformations which carry a local rotation, origin, and a scaling factor --- as well as a local infinity.

Moebius transformations blur the distinction between spheres and planes, continuously transforming one into the other. A plane is merely a sphere with infinite radius, and a line is merely a circle with infinite radius.

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Member Function Documentation

virtual void esve::engine::dim4::Transform::rotate (  types::real  alpha, types::real  beta, const types::quat &  r, const types::quat &  s, const Transform &  basis = IDENTITY )  [pure virtual]

Rotate by angle alpha in the (r, s) plane; rotate by angle beta in the (r, s)-complement plane. Orientation is positive with r rotating toward s. Precondition: r, s must be orthonormal.

virtual void esve::engine::dim4::Transform::rotate (  types::real  alpha, types::real  beta, const types::pure &  u, const Transform &  basis = IDENTITY )  [pure virtual]

Rotate by angle alpha in the (1, u) plane; rotate by angle beta in the (1, u)-complement plane. Orientation is positive with 1 rotating toward u. Precondition: u is unit length.

virtual void esve::engine::dim4::Transform::normalize (   )  [pure virtual]

Snap to upper-triangular with unitary diagonal elements. In other words, remove inversion and scaling, leaving only the rotation group with translations. After normalize(), the transformation will be of the form f(z) = a*z*conj(d) + b*conj(d) where a and d are unitary. Implemented in esve::engine::dim4::impl::Transform.

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