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

Classes

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...
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...
class point
A point in physical 3D space. More...
class quaternion
class rotor
A rotor is the sum of scalar with a bivector. More...
class trivector
A Tri-Vector, or oriented volume in 3D space. More...
class vector
3-dimensional vector. More...

Typedefs

double dot (const vector &a, const vector &b)
Euclidan Inner (dot) product of two vectors.
vector dot (const bivector &a, const vector &b)
Inner (dot) product of a bivector and a vector.
vector dot (const vector &a, const bivector &b)
Inner (dot) product of a vector and bivector.
vector dot (const trivector &a, const bivector &b)
xyz ( xy + yz + zx ) = xyz xy + xyz yz + xyz zx =
rotor exp (const rotor &r)
Compute the exponential of a rotor.
rotor ln (const rotor &r)
compute the natural logarithm of a rotor
rotor operator* (const rotor &l, const rotor &r)
rotor operator* (const vector &l, const vector &r)
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.
point operator+ (const point &P, const vector &v)
point operator+ (const vector &v, const point &P)
OddMultiVector operator+ (const trivector &V, const vector &v)
OddMultiVector operator+ (const vector &v, const trivector &V)
vector operator- (const point &a, const point &b)
point operator- (const point &P, const vector &v)
bivector operator^ (const vector &a, const vector &b)
Wedge product of two vectors yields a bivector.
trivector operator^ (const bivector &a, const vector &b)
Compute outer product of bivector an vector.

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.

A rotor is an even multivector.

A rotor is an even MultiVector.

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.

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

vector Eagle::PhysicalSpace::dot	(	const trivector &	a,
		const bivector &	b
	)		`[inline]`

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

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.

point Eagle::PhysicalSpace::operator-	(	const point &	P,
		const vector &	v
	)		`[inline]`