Eagle::STA::CartesianChart4D< real >::bivector Class Reference

A four-dimensional Bi-Vector which is span by two vectors to define a plane. More...

#include <elementary/eagle/STA.hpp>

Inheritance diagram for Eagle::STA::CartesianChart4D< real >::bivector:

Public Member Functions
template<char CompType>
	bivector (const bivector &l, const bivector &r, const Operator< CompType > &C)
	Generic Computational Constructor for operations on two bi-vectors.
template<char CompType>
	bivector (const bivector &l, const double &scalar, const Operator< CompType > &C)
	Generic Computational Constructor for operations on bi-vector and scalar.
template<char CompType>
	bivector (const double &scalar, const bivector &l, const Operator< CompType > &C)
	Generic Computational Constructor for operations on scalar and bi-vector.
	bivector (const vector &a, const vector &b)
	Construct bivector from two vectors.
bivector	inv () const
	Compute the inverse bivector.
vector	operator() (const vector &v) const
	Compute UvU, which is a 90 degree rotation of the vector v along the plane U.
bivector	operator* () const
	Star operator, yields the bivector that is dual to this bi-vector.
bivector	operator+ (const bivector &r) const
	Addition operator.
bivector	operator- () const
	Unary minus.
bivector	operator- (const bivector &r) const
	Subtraction operator.
bivector	unit () const
	Yield a bivector of unit area.

Friends
double	dot (const bivector &a, const bivector &b)
	Inner product of two bivectors (note: this actually involves an implicit metric!)
double	norm (const bivector &v)
	The norm of a bivector.
double	norm2 (const bivector &v)
	Compute the squared norm of a bivector, which is the area span by the plane.
bivector	operator* (const bivector &v, double r)
	Scale bivector.
bivector	operator* (double r, const bivector &b)
	Scale bivector.
bivector	operator/ (const bivector &b, double r)
	Divide bivector by scalar.

Detailed Description

template<class real>
class Eagle::STA::CartesianChart4D< real >::bivector

A four-dimensional Bi-Vector which is span by two vectors to define a plane.

It is represented by six numbers in the STA.

Bi-vectors are useful specify rotations. The sum of a bivector and a scalar yields a rotor, which can be applied to any object such as a vector (sweeping a direction) or another bivector (sweeping a plane).

A bi-vector is constructed from two vectors:

vector A, B;
bivector AB = bivector(A,B);

The wedge operator of two vectors also yields a bi-vector and is the same operation:

vector A, B;
bivector& AB = A^B;

Alternatively the bivector can be setup by explicit specification of its component, which corresponds to a normal vector. Given a bivector, the associated normal vector can be retrieved using the star operator:

bivector A;
vector normalA = *A;

Member Function Documentation

operator()()

template<class real >

vector Eagle::STA::CartesianChart4D< real >::bivector::operator()

(

const vector &

v

)

const

Compute UvU, which is a 90 degree rotation of the vector v along the plane U.

Parameters

this	The bivector U. vector v; bivector U; vector v_rotated = U(v);

operator*()

template<class real >

bivector Eagle::STA::CartesianChart4D< real >::bivector::operator* ( ) const

inline

Star operator, yields the bivector that is dual to this bi-vector.

@TODO CHECK SIGNS and http://mathworld.wolfram.com/DualBivector.html