Geometric algebra of the physical 3D-space, by which we mean the Clifford algebara over Euclidean, three-dimensional space. More...

Classes
struct	AABB
	Simplistic class for Axis-Aligned Bounding Box. More...
struct	AABB_2D
	Axis-aligned bounding box in 2D dimensions out of 3D, allowing for projections other than XY. More...
struct	AnalyticFunction
	Abstract base class for analytical functions that are evaluated on a physical space. More...
class	bivector
	A three-dimensional Bi-Vector which is span by two vectors to define a plane. More...
struct	Frustum
	A frustum and operations on it, as described by either a set of corner points or six planes. More...
struct	HessianNormalPlane
class	MultiVector
	A full multivector in 3D consists of components. More...
class	OddMultiVector
	The combination of a vector and a trivector yields an odd multivector, which is dual to a rotor. More...
struct	Plane
	Normal form of a plane. More...
class	point
	A point in physical 3D space. More...
class	rotor
	A rotor is the sum of scalar with a bivector. More...
struct	Span
struct	Span< 1 >
	A one-dimensional span, also a known as a Ray. More...
struct	Span< 2 >
	A two-dimensional span, also a known as the parametric form of a plane. More...
struct	Span< 3 >
class	trivector
	A Tri-Vector, or oriented volume in 3D space. More...
class	vector
	3-dimensional vector. More...
struct	Viewport
	A helper class describing an axis-aligned rectangle. More...

Typedefs
typedef rotor	EvenMultiVector
	A rotor is an even multivector.

Functions
double	computeIntersectionDistance (const Plane &P, const Ray &R)
	Compute the intersection parameter $\lambda$ for a Ray hitting a Plane.
vector	dot (const bivector &a, const vector &b)
	Inner (dot) product of a bivector and a vector.
vector	dot (const trivector &a, const bivector &b)
	xyz ( xy + yz + zx ) = xyz xy + xyz yz + xyz zx = -xzy xy - xyz zy - yxz zx = zxy xy = -zyx xy - xyz zy - yxz zx z x y
vector	dot (const vector &a, const bivector &b)
	Inner (dot) product of a vector and bivector.
double	dot (const vector &a, const vector &b)
	Euclidan Inner (dot) product of two vectors.
Metric	dyadic (const vector &a, const vector &b)
	Tensor product or dyadic product of two vectors.
rotor	exp (const bivector &i)
	Compute the exponential of a bivector, which implements a rotation as given by the norm of the bivector.
rotor	exp (const bivector &U, double phi)
	Implement a rotation of phi degrees along the bivector U.
rotor	exp (const rotor &s)
	Compute the exponential of a rotor (by Andrew Hamilton).
void	glTranslate (const Eagle::PhysicalSpace::point &origin)
	Perform an OpenGL glTranslate() call on a point given in physical space.
void	glVertex (const Eagle::PhysicalSpace::point &P)
	Perform an OpenGL glVertex() call on a point given in physical space.
rotor	ln (const rotor &s)
	Compute the natural logarithm of a rotor (by Andrew Hamilton).
Ray	makeRay (const point &A, const point &B)
	Make a ray from two points, starting from the first one.
vector	operator% (const vector &l, const vector &r)
	Compute the three-dimensional cross product.
bivector	operator() (const bivector &V) const
	Rotate a bivector.
OddMultiVector	operator* (const bivector &a, const vector &b)
	Geometric product of bivector and vector yields an odd multivector, the sum of a vector and trivector.
OddMultiVector	operator* (const rotor &R, const vector &v)
rotor	operator* (const vector &l, const vector &r)
	The multiplication operation on two vectors implements the geometric product , thus generating a clifford algebra.
rotor	operator+ (const bivector &U, double r)
	Create a rotor as sum of bi-vector and scalar.
rotor	operator+ (const rotor &U, double r)
	Add a scalar value to a rotor.
rotor	operator+ (double r, const bivector &U)
	Create a rotor as sum of scalar and bi-vector.
rotor	operator+ (double r, const rotor &U)
	Add a scalar value to a rotor.
point	operator- (const point &P, const vector &v)
rotor	operator- (const rotor &U, double r)
	Add a scalar value to a rotor.
rotor	operator- (double r, const bivector &U)
	Create a rotor as difference of scalar and bi-vector.
rotor	operator- (double r, const rotor &U)
	Add a scalar value to a rotor.
rotor	operator/ (const rotor &l, const rotor &r)
	Note that $l/r := lr^{-1} \neq r^{-1} l$ .
rotor	operator/ (const vector &l, const vector &r)
	Divide two vectors, yielding the rotation operation to sweep the second vector to the first one.
trivector	operator^ (const bivector &a, const vector &b)
	Compute outer product of bivector and vector.
trivector	operator^ (const vector &a, const bivector &b)
	Compute outer product of vector and bivector.
bivector	operator^ (const vector &a, const vector &b)
	Wedge product of two vectors yields a bivector.
rotor	operator~ (const rotor &r)
	Compute the inverse rotor.
rotor	pow (const rotor &r, double t)
	Compute the power of a rotor.
Matrix< 4, 4, double >	rotmatrix (const rotor &r)
	Compute the classical 4x4 rotation Matrix.
rotor	slerp (const rotor &r1, const rotor &r2, double t)
	Interpolate between q1 (t = 0) and q2 (t = 1): q = (q2/q1)^t q1 .

Detailed Description

Geometric algebra of the physical 3D-space, by which we mean the Clifford algebara over Euclidean, three-dimensional space.

It can reflect the properties of a flat, 3-dimensional manifold or the tangential space of a curved, 3-dimensional manifold.

Note: The classes in PhysicalSpace are currently implemented to operate on the double type. It were easy to change this to a generic template instead, in case the need arises. For now, this appears to be an overhead to always write <> in an instance, even though a typedef could fix that easily as well. If there is a real need to instantiate geometric algebra over other types than doubles, we can do it then.

Typedef Documentation

◆ EvenMultiVector

typedef rotor Eagle::PhysicalSpace::EvenMultiVector

A rotor is an even multivector.

A rotor is an even MultiVector.

Function Documentation

◆ computeIntersectionDistance()

double Eagle::PhysicalSpace::computeIntersectionDistance	(	const Plane &	P,
		const Ray &	R )

inline

Compute the intersection parameter $\lambda$ for a Ray hitting a Plane.

The actual intersection point can be computed as R( $\lambda$ ) .

References dot().

◆ dot() [1/2]

vector Eagle::PhysicalSpace::dot	(	const bivector &	a,
		const vector &	b )

inline

Inner (dot) product of a bivector and a vector.

(A xy + B yz + C zx)|(ax+ by+ cz) = ( -Aa y + Ca z + Ab x - Bb z + Bc y - Cc x) = (Ab-Cc) x + (Bc - Aa) y + ( Ca - Bb) z

◆ dot() [2/2]

double Eagle::PhysicalSpace::dot	(	const vector &	a,
		const vector &	b )

inline

Euclidan Inner (dot) product of two vectors.

Compute the Euclidean dot product (inner product) of two vectors.

Referenced by computeIntersectionDistance(), Eagle::PhysicalSpace::rotor::operator()(), operator*(), and Eagle::PhysicalSpace::rotor::operator*().

◆ dyadic()

Metric Eagle::PhysicalSpace::dyadic	(	const vector &	a,
		const vector &	b )

inline

Tensor product or dyadic product of two vectors.

Creates a tensor of order two.

◆ operator*() [1/3]

OddMultiVector Eagle::PhysicalSpace::operator*	(	const bivector &	a,
		const vector &	b )

inline

Geometric product of bivector and vector yields an odd multivector, the sum of a vector and trivector.

An odd multivector is dual to a rotor, an even multivector.

References dot().

◆ operator*() [2/3]

OddMultiVector Eagle::PhysicalSpace::operator*	(	const rotor &	R,
		const vector &	v )

inline

Use its vec() member function to retrieve the vectorial part.

◆ operator-()

point Eagle::PhysicalSpace::operator-	(	const point &	P,
		const vector &	v )

inline

Todo: Use computational constructor for '-'

Classes

Typedefs

Functions

Detailed Description

Typedef Documentation

◆ EvenMultiVector

Function Documentation

◆ computeIntersectionDistance()

◆ dot() [1/2]

◆ dot() [2/2]

◆ dyadic()

◆ operator*() [1/3]

◆ operator*() [2/3]

◆ operator-()