A rotor is the sum of scalar with a bivector. More...

#include <elementary/eagle/rotor3.hpp>

Inheritance diagram for Eagle::PhysicalSpace::rotor:

Public Member Functions
bivector &	bivec ()
	Retrieve bivector component, read-only.

const bivector &	bivec () const
	Retrieve bivector component, read-only.

rotor	conj () const
	Compute the conjugate rotor.

rotor	inv () const
	Compute the inverse rotor.

bivector	operator() (const bivector &V) const
	Rotate a bivector.

vector	operator() (const vector &v) const
	The sandwich product: Compute Rv ~R, which is the rotation of the vector v by the current rotor.

OddMultiVector	operator* () const
	Applying the hodge star operator.

	rotor ()
	Default constructor.

	rotor (const bivector &b, double r)
	Construct rotor from bivector and scalar.

	rotor (const double s)
	Create rotor with just scalar component, bivector is zero.

	rotor (const rotor &b)
	Copy constructor.

	rotor (const rotor &b, const Operator<'~'> &)
	Computational constructor computing the conjugate (inverse) rotor.

	rotor (const Vector< double, 4 > &V)
	Explicit construction from double.

	rotor (double r, const bivector &b)
	Construct rotor from scalar and bivector.

	rotor (double r, double b1, double b2, double b3)
	Construct rotor from four doubles.

double &	scalar ()
	Retrieve scalar component, modifyable.

double	scalar () const
	Retrieve scalar component, read-only.

rotor	unit () const
	Normalize to a unit rotor (added by mr).

Public Member Functions inherited from Eagle::Vector< double, 4 >
Base_t &	fixed_array ()
	Provide access to the base class, writable.

const Base_t &	fixed_array () const
	Provide access to the base class, read-only.

Vector &	*operator=** (const double &value)
	Self-multiplication.

Vector &	operator+= (const Vector &v)
	Self-addition.

Vector	operator- () const
	Unary minus.

Vector &	operator-= (const Vector &v)
	Self-subtraction.

Vector &	operator/= (const double &value)
	Self-division.

Assignment< Vector, 0 >	operator= (const scalar &x)
	Assign first element with scalar and use consecutive comma expressions for assignment of further elements As such a Vector may be assigned as.

Vector &	operator= (const Vector &src)
	Assignment operator.

void	set (const V &value)
	Assign all values to the same scalar.

constexpr	Vector ()
	Default constructor.

	Vector (const double &a)
	Convenience Constructor: initialize the first element.

	Vector (const double &a, const double &b)
	Convenience Constructor: initialize the first two elements.

	Vector (const double &a, const double &b, const double &c)
	Convenience Constructor: initialize the first three elements.

	Vector (const double &a, const double &b, const double &c, const double &d)
	Convenience Constructor: initialize the first four elements.

	Vector (const FixedArray< AType, n > &l, const FixedArray< BType, n > &r, const Operator< CompType > &Op)
	Computational constructor for binary operations on two fixed arrays, calls the Operator's ternary() member function for each element.

	Vector (const FixedArray< AType, n > &l, const Operator< CompType > &, const ScalarType &scalar)
	Computational constructor from vector with scalar, calls the Operator's ternary() member function for each element and the scalar as its 3rd argument.

	Vector (const FixedArray< InputType, SIZE > &V)
	Type conversion constructor.

	Vector (const Indicable &I, const AnyType &)
	Construct from indicable object, i.e.

	Vector (const Operator< CompType > &, const ScalarType &scalar, const FixedArray< AType, n > &v)
	Computational constructor from scalar with vector calls the Operator's ternary() member function for each element and the scalar as its 2nd argument.

constexpr	Vector (const std::initializer_list< InputType > &Init)
	Construct from a initialization array.

	Vector (const Vector &l, const Vector &r, const Operator< CompType > &)
	Computational constructor for binary operations on two vectors calls the Operator's ternary() member function for each element.

	Vector (const Vector &NV, const Sub &, const Sub &)
	Computational constructor for unary minus.

constexpr	Vector (const Vector &v)
	Copy constructor.

	Vector (const Vector &v, const Operator<'-'> &)
	Unitary minus computational constructor Applies the unitary minus to each element.

	Vector (const Vector< InputType, SIZE > &V)
	Type conversion constructor.

Public Member Functions inherited from Eagle::FixedArray< value, N >
void	expandMinMax (FixedArray &MinValues, FixedArray &MaxValues) const
	Expand a range given by MinValues,MaxValues to include the values of this array, done per component.

constexpr	FixedArray () noexcept
	Default constructor.

template<class AType , class BType , OperatorID_t CompType>
constexpr	FixedArray (const FixedArray< AType, N > &l, const FixedArray< BType, N > &r, const Operator< CompType > &)
	Computational constructor for binary operations on two fixed arrays, calls the Operator's ternary() member function for each element.

template<typename T , dimension_t M>
constexpr	FixedArray (const FixedArray< T, M > &F, const T &DefaultValue)
	Construct array from another array type of variable length.

template<typename T >
constexpr	FixedArray (const FixedArray< T, N > &F)
	Automatic type conversion function.

template<typename T , typename ConversionFunctor >
constexpr	FixedArray (const FixedArray< T, N > &F, const ConversionFunctor &CF, const FA_CONVERT &)
	Construct array from another array type of same length.

template<class InitType >
constexpr	FixedArray (const std::initializer_list< InitType > &Init)
	Construct from a initialization array.

constexpr	FixedArray (const value &a)
	Convenience Constructor: initialize the first element.

constexpr	FixedArray (const value &a, const value &b)
	Convenience Constructor: initialize the first two elements.

	FixedArray (const value &a, const value &b, const value &c)
	Convenience Constructor: initialize the first three elements.

constexpr	FixedArray (const value &a, const value &b, const value &c, const value &d)
	Convenience Constructor: initialize the first four elements.

constexpr	FixedArray (const value &a, const value &b, const value &c, const value &d, const value &e)
	Convenience Constructor: initialize the first five elements.

constexpr value &	get (int i)
	writable element access

constexpr const value &	get (int i) const
	read-only element access

template<int subsize, int offset>
constexpr const FixedArray< value, subsize > *	get_subarray () const
	Helper class for retrieve a subset of the current array.

constexpr const FixedArray< value, SIZE-1 > &	get_subdim () const
	Mainly introduced because of Interpolate::operator[].

const FixedArray_t &	getFixedArray () const noexcept
	convenience function for base classes

FixedArray_t &	getFixedArray () noexcept
	convenience function for base classes

constexpr value	getMax () const
	Compute the maximum element value of this array.

constexpr value	getMin () const
	Compute the maximum element value of this array.

template<typename T >
constexpr bool	operator!= (const FixedArray< T, N > &F) const
	Compare two arrays.

template<typename T >
constexpr bool	operator== (const FixedArray< T, N > &F) const
	Compare two arrays.

constexpr value &	operator[] (int i)
	writeable element access operator

constexpr const value &	operator[] (int i) const
	read-only element access operator

constexpr value *	ptr (int i=0) noexcept
	Return a pointer to the ith element, no range checking.

template<class Value >
constexpr void	set (const Value &x)
	Set all array elements to the same value.

void	setFromArray (const value *x, int length=SIZE, int offset=0)
	Set all array elements to an array's element value.

void	sort (FixedArray< int, N > &I) const
	Sort the elements of an array into the index array.

Friends
eagle_API rotor	operator* (const rotor &, const rotor &)
	Multiply rotors, which is a concatenation operation of rotations.

OddMultiVector	operator* (const rotor &, const vector &)
	Multiply rotor and vector, yielding an odd multivector, the dual of a rotor.

bivector &	operator*= (bivector &V, const rotor &R)
	Rotate a certain bivector by a rotor "in place".

vector &	*operator=** (vector &v, const rotor &R)
	Rotate a certain vector by a rotor "in place".

Related Symbols
(Note that these are not member symbols.)
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)

rotor	ln (const rotor &s)
	Compute the natural logarithm of a rotor (by Andrew Hamilton)

vector	operator% (const vector &l, const vector &r)
	Compute the three-dimensional cross product.

bivector	operator() (const bivector &V) const
	Rotate a bivector.

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.

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.

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 .

Additional Inherited Members
Public Types inherited from Eagle::Vector< double, 4 >
enum
	The number of elements in this array.

Public Types inherited from Eagle::FixedArray< value, N >
typedef FlattenedArray_t::value_type	Element_t
	The maximally reduced type of elements stored here.

typedef GetFixedArrayType< value >::result_t	ElementAsFixedArray_t
	Element type as fixed array, if it is convertible to a fixed array.

typedef FixedArray	FixedArray_t
	Exporting the type definition of this array cass type.

typedef FixedArrayFlattener< FixedArrayOfElements_t >	FixedArrayFlattener_t
	Internal helper class.

typedef FixedArray< ElementAsFixedArray_t, N >	FixedArrayOfElements_t
	If the value is derived from a fixed array type itself, then the FixedArrayOfElements_t is an FixedArray<FixedArray<>> type.

typedef FixedArrayFlattener_t::result_t	FlattenedArray_t
	Reduction of this array type, if it is nested, to an flattened type which contains elementary types.

typedef value	value_type
	Export the type of the respective element value type.

Static Public Member Functions inherited from Eagle::FixedArray< value, N >
template<typename T , typename ConversionFunctor >
static constexpr FixedArray	convert (const FixedArray< T, N > &F, const ConversionFunctor &CF)
	Perform a type conversion on a given fixed array into another basis type:

static constexpr dimension_t	size ()
	The number of elements here.

Protected Attributes inherited from Eagle::FixedArray< value, N >
value	data [SIZE]
	The actual data values, intentional no-initialization for data member, which increases performance on vector creation.

Detailed Description

A rotor is the sum of scalar with a bivector.

It represents the even subalgebra (for instance grade 0 and 2) of the geometric algebra of euclidean 3-space and implements the same algebraic properties as a Pauli spinor. There are 3+1 components in a rotor in 3-space.

To use a rotor to rotate a vector v around a bivector u look at rotor::operator()

Todo:: Need to implement optimization for unit rotors, such that the norm2() function always returns 1.0 without computation.

Member Function Documentation

◆ operator()()

vector Eagle::PhysicalSpace::rotor::operator() ( const vector & v ) const

inline

The sandwich product: Compute Rv ~R, which is the rotation of the vector v by the current rotor.

R(v) = R v ~R = (s+U) v (s-U) = (s v + U v) (s-U) = s v s + U v s - s v U - U v U = s s v + Uv s - s vU - U(v) = ss v + 2 Uv s - U(v)

To rotate a vector v with angle phi around a bivector u you may use:

bivector u;
vector v;
double phi;
rotor R = exp(u * phi);   // with ||u||=1 (use unit() ) and [phi] in rad
vector rotated_v = R(v);

Todo:: Hmm, isn't there a sign error in the 2 s vU part? Plz verify!!

References bivec(), Eagle::PhysicalSpace::dot(), and scalar().

Friends And Related Symbol Documentation

◆ exp()

rotor exp	(	const bivector &	U,
		double	phi
	)

If U is not unit, then the resulting rotor will also scale a given vector.

Parameters

phi	Rotation in radians.

◆ ln()

rotor ln ( const rotor & s )

important: maybe error of ln with rotor r = (1,0,0,0); must be tested

Eagle::PhysicalSpace::rotor::ln

rotor ln(const rotor &s)

Compute the natural logarithm of a rotor (by Andrew Hamilton)

Definition rotor3.hpp:378

References bivec(), Eagle::norm2(), and unit().

◆ operator()()

bivector operator() ( const bivector & V ) const

This operator is equivalent to $ *R(*V) $ .

Note: Let's hope the compiler optimizes this function sufficiently.

◆ operator* [1/2]

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

friend

Multiply rotor and vector, yielding an odd multivector, the dual of a rotor.

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

◆ operator*() [2/2]

rotor operator*	(	const vector &	l,
		const vector &	r
	)

Geometrically it means the operation to sweep one vector r to a vector l, which results in a rotor.

References Eagle::PhysicalSpace::dot().

◆ operator*=

bivector & operator*=	(	bivector &	V,
		const rotor &	R
	)

friend

Rotate a certain bivector by a rotor "in place".

Note: This is numerically not more efficient than assignment.

◆ operator/()

rotor operator/	(	const vector &	l,
		const vector &	r
	)

This operation is identical to (l r^-1).

References Eagle::PhysicalSpace::vector::inv().

◆ operator~()

rotor operator~ ( const rotor & r )

Note: This is probably only correct for unit rotors...

References bivec(), and scalar().

◆ rotmatrix()

Matrix< 4, 4, double > rotmatrix ( const rotor & r )

E.g. for use with openGls: glMultMatrixd like: glMultMatrixd( RotMat.ptr() );

References Eagle::Matrix< R, C, Value >::column().

Public Member Functions

Friends

Related Symbols

Additional Inherited Members

Detailed Description

Member Function Documentation

◆ operator()()

Friends And Related Symbol Documentation

◆ exp()

◆ ln()

◆ operator()()

◆ operator* [1/2]

◆ operator*() [2/2]

◆ operator*=

◆ operator/()

◆ operator~()

◆ rotmatrix()