This class represents the rotate-and-shift transform which gives the location and orientation of a new frame F in a base (reference) frame B. More...

Public Member Functions
	Transform_ ()
	Default constructor gives an identity transform. More...

	Transform_ (const Rotation_< P > &R, const Vec< 3, P > &p)
	Combine a rotation and a translation into a transform. More...

	Transform_ (const Rotation_< P > &R)
	Construct or default-convert a rotation into a transform containing that rotation and zero translation. More...

	Transform_ (const Vec< 3, P > &p)
	Construct or default-convert a translation (expressed as a Vec3) into a transform with that translation and a zero rotation. More...

Transform_ &	operator= (const InverseTransform_< P > &X)
	Assignment from InverseTransform. More...

template<int S>
Transform_ &	operator+= (const Vec< 3, P, S > &offset_B)
	Add an offset to the position vector in this Transform. More...

template<int S>
Transform_ &	operator-= (const Vec< 3, P, S > &offset_B)
	Subtract an offset from the position vector in this Transform. More...

Transform_ &	set (const Rotation_< P > &R, const Vec< 3, P > &p)
	Assign a new value to this transform, explicitly providing the rotation and translation separately. More...

Transform_ &	setToZero ()
	By zero we mean "zero transform", i.e., an identity rotation and zero translation. More...

Transform_ &	setToNaN ()
	This fills both the rotation and translation with NaNs. More...

const InverseTransform_< P > &	invert () const
	Return a read-only inverse of the current Transform_. More...

InverseTransform_< P > &	updInvert ()
	Return a writable (lvalue) inverse of the current transform, simply by casting it to the InverseTransform_. More...

const InverseTransform_< P > &	operator~ () const
	Overload transpose operator to mean inversion. More...

InverseTransform_< P > &	operator~ ()
	Overload transpose operator to mean inversion. More...

Transform_	compose (const Transform_ &X_FY) const
	Compose the current transform (X_BF) with the given one. More...

Transform_	compose (const InverseTransform_< P > &X_FY) const
	Compose the current transform (X_BF) with one that is supplied as an InverseTransform_ (typically as a result of applying the "~" operator to a transform). More...

Vec< 3, P >	xformFrameVecToBase (const Vec< 3, P > &vF) const
	Transform a vector expressed in our "F" frame to our "B" frame. More...

Vec< 3, P >	xformBaseVecToFrame (const Vec< 3, P > &vB) const
	Transform a vector expressed in our "B" frame to our "F" frame. More...

Vec< 3, P >	shiftFrameStationToBase (const Vec< 3, P > &sF) const
	Transform a point (station) measured from and expressed in our "F" frame to that same point but measured from and expressed in our "B" frame. More...

Vec< 3, P >	shiftBaseStationToFrame (const Vec< 3, P > &sB) const
	Transform a point (station) measured from and expressed in our "B" frame to that same point but measured from and expressed in our "F" frame. More...

const Rotation_< P > &	R () const
	Return a read-only reference to the contained rotation R_BF. More...

Rotation_< P > &	updR ()
	Return a writable (lvalue) reference to the contained rotation R_BF. More...

const Rotation_< P >::ColType &	x () const
	Return a read-only reference to the x direction (unit vector) of the F frame, expressed in the B frame. More...

const Rotation_< P >::ColType &	y () const
	Return a read-only reference to the y direction (unit vector) of the F frame, expressed in the B frame. More...

const Rotation_< P >::ColType &	z () const
	Return a read-only reference to the z direction (unit vector) of the F frame, expressed in the B frame. More...

const InverseRotation_< P > &	RInv () const
	Return a read-only reference to the inverse (transpose) of our contained rotation, that is R_FB. More...

InverseRotation_< P > &	updRInv ()
	Return a writable (lvalue) reference to the inverse (transpose) of our contained rotation, that is R_FB. More...

const Vec< 3, P > &	p () const
	Return a read-only reference to our translation vector p_BF. More...

Vec< 3, P > &	updP ()
	Return a writable (lvalue) reference to our translation vector p_BF. More...

Transform_< P > &	setP (const Vec< 3, P > &p)
	Assign a new value to our translation vector. More...

Vec< 3, P >	pInv () const
	Calculate the inverse of the translation vector in this transform. More...

Transform_< P > &	setPInv (const Vec< 3, P > &p_FB)
	Assign a value to the inverse of our translation vector. More...

const Mat< 3, 4, P > &	asMat34 () const
	Recast this transform as a read-only 3x4 matrix. More...

Mat< 3, 4, P >	toMat34 () const
	Less efficient version of asMat34(); copies into return variable. More...

Mat< 4, 4, P >	toMat44 () const
	Return the equivalent 4x4 transformation matrix. More...

const Vec< 3, P > &	T () const

Vec< 3, P > &	updT ()

Related Functions
(Note that these are not member functions.)
template<class P , int S>
Vec< 3, P >	operator* (const Transform_< P > &X_BF, const Vec< 3, P, S > &s_F)
	If we multiply a transform or inverse transform by a 3-vector, we treat the vector as though it had a 4th element "1" appended, that is, it is treated as a station rather than a vector. More...

template<class P , int S>
Transform_< P >	operator+ (const Transform_< P > &X_BF, const Vec< 3, P, S > &offset_B)
	Adding a 3-vector to a Transform produces a new shifted transform. More...

template<class P , int S>
Transform_< P >	operator+ (const Vec< 3, P, S > &offset_B, const Transform_< P > &X_BF)
	Adding a 3-vector to a Transform produces a new shifted transform. More...

template<class P , int S>
Transform_< P >	operator- (const Transform_< P > &X_BF, const Vec< 3, P, S > &offset_B)
	Subtracting a 3-vector from a Transform produces a new shifted transform. More...

template<class P , int S>
Vec< 4, P >	operator* (const Transform_< P > &X_BF, const Vec< 4, P, S > &a_F)
	If we multiply a transform or inverse transform by an augmented 4-vector, we use the 4th element to decide how to treat it. More...

template<class P , class E >
Vector_< E >	operator* (const Transform_< P > &X, const VectorBase< E > &v)
	Multiplying a matrix or vector by a Transform_. More...

template<class P >
Transform_< P >	operator* (const Transform_< P > &X1, const Transform_< P > &X2)
	Composition of transforms. More...

template<class P >
bool	operator== (const Transform_< P > &X1, const Transform_< P > &X2)
	Comparison operators return true only if the two transforms are bit identical; that's not too useful. More...

template<class P >
std::ostream &	operator<< (std::ostream &, const Transform_< P > &)
	Generate formatted output of a Transform to an output stream. More...

Detailed Description

template<class P>
class SimTK::Transform_

This class represents the rotate-and-shift transform which gives the location and orientation of a new frame F in a base (reference) frame B.

A frame is an orthogonal, right-handed set of three axes, and an origin point. A transform X from frame B to F consists of 3 perpendicular unit vectors defining F's axes as viewed from B (that is, as expressed in the basis formed by B's axes), and a vector from B's origin point Bo to F's origin point Fo. Note that the meaning of "B" comes from the context in which the transform is used. We use the phrase "frame F is in frame B" to describe the above relationship, that is, "in" means both measured from and expressed in.

The axis vectors constitute a Rotation_. They are ordered 1-2-3 or x-y-z as you prefer, with z = x X y, making a right-handed set. These axes are arranged as columns of a 3x3 rotation matrix R_BF = [ x y z ] which is a direction cosine (rotation) matrix useful for conversions between frame B and F. (The columns of R_BF are F's coordinate axes, expressed in B.) For example, given a vector vF expressed in the F frame, that same vector re-expressed in B is given by vB = R_BF*vF. F's origin point OF is stored as the translation vector p_BF=(Fo-Bo) and expressed in B.

Transform is designed to behave as much as possible like the computer graphics 4x4 affine transform X which would be arranged like this:

        [       |   ]
    X = [   R   | p ]    R is a 3x3 orthogonal rotation matrix
        [.......|...]    p is a 3x1 translation vector
        [ 0 0 0   1 ]

These can be composed directly by matrix multiplication, but more importantly they have a particularly simple inverse:

   -1   [       |    ]
  X   = [  ~R   | p* ]   ~R is R transpose, p* = ~R(-p).
        [.......|....]
        [ 0 0 0   1  ]

This inverse is so simple that we compute it simply by defining another type, InverseTransform_, which is identical to Transform_ in memory but behaves as though it contains the inverse. That way we invert just by changing point of view (recasting) rather than computing.

This is a "POD" (plain old data) class with a well-defined memory layout on which a client of this class may depend: There are exactly 4 consecutive, packed 3-vectors in the order x,y,z,p. That is, this class is equivalent to an array of 12 Reals with the order x1,x2,x3,y1,y2,y3,z1,z2,z3,p1,p2,p3. It is expressly allowed to reinterpret Transform objects in any appropriate manner that depends on this memory layout.

Constructor & Destructor Documentation

◆ Transform_() [1/4]

template<class P>

SimTK::Transform_::Transform_ ( )

inline

Default constructor gives an identity transform.

◆ Transform_() [2/4]

template<class P>

SimTK::Transform_< P >::Transform_	(	const Rotation_< P > &	R,
		const Vec< 3, P > &	p
	)

inline

Combine a rotation and a translation into a transform.

◆ Transform_() [3/4]

template<class P>

SimTK::Transform_::Transform_ ( const Rotation_ & R )

inline

Construct or default-convert a rotation into a transform containing that rotation and zero translation.

◆ Transform_() [4/4]

template<class P>

SimTK::Transform_::Transform_ ( const Vec< 3, P > & p )

inline

Construct or default-convert a translation (expressed as a Vec3) into a transform with that translation and a zero rotation.

Member Function Documentation

◆ operator=()

template<class P>

Transform_ & SimTK::Transform_::operator= ( const InverseTransform_ & X )

inline

Assignment from InverseTransform.

This means that the transform we're assigning to must end up with the same meaning as the inverse transform X has, so we'll need to end up with:

p == X.p()
R == X.R()

Cost: one frame conversion and a negation, 18 flops.

◆ operator+=()

template<class P>

template<int S>

Transform_& SimTK::Transform_::operator+= ( const Vec< 3, P, S > & offset_B )

inline

Add an offset to the position vector in this Transform.

Cost is 3 flops.

◆ operator-=()

template<class P>

template<int S>

Transform_& SimTK::Transform_::operator-= ( const Vec< 3, P, S > & offset_B )

inline

Subtract an offset from the position vector in this Transform.

Cost is 3 flops.

◆ set()

template<class P>

Transform_& SimTK::Transform_< P >::set	(	const Rotation_< P > &	R,
		const Vec< 3, P > &	p
	)

inline

Assign a new value to this transform, explicitly providing the rotation and translation separately.

We return a reference to the now-modified transform as though this were an assignment operator.

◆ setToZero()

template<class P>

Transform_& SimTK::Transform_::setToZero ( )

inline

By zero we mean "zero transform", i.e., an identity rotation and zero translation.

We return a reference to the now-modified transform as though this were an assignment operator.

◆ setToNaN()

template<class P>

Transform_& SimTK::Transform_::setToNaN ( )

inline

This fills both the rotation and translation with NaNs.

Note: this is not the same as a default-constructed transform, which is a legitimate identity transform instead. We return a reference to the now-modified transform as though this were an assignment operator.

◆ invert()

template<class P>

const InverseTransform_& SimTK::Transform_::invert ( ) const

inline

Return a read-only inverse of the current Transform_.

, simply by casting it to the InverseTransform_

type. Zero cost.

◆ updInvert()

template<class P>

InverseTransform_& SimTK::Transform_::updInvert ( )

inline

Return a writable (lvalue) inverse of the current transform, simply by casting it to the InverseTransform_.

type. That is, this is an lvalue. Zero cost.

◆ operator~() [1/2]

template<class P>

const InverseTransform_& SimTK::Transform_::operator~ ( ) const

inline

Overload transpose operator to mean inversion.

See also: invert

◆ operator~() [2/2]

template<class P>

InverseTransform_& SimTK::Transform_::operator~ ( )

inline

Overload transpose operator to mean inversion.

See also: updInvert

template<class P>

Mat<3,4,P> SimTK::Transform_::toMat34 ( ) const

inline

Less efficient version of asMat34(); copies into return variable.

◆ toMat44()

template<class P>

Mat<4,4,P> SimTK::Transform_::toMat44 ( ) const

inline

Return the equivalent 4x4 transformation matrix.

◆ T()

template<class P>

const Vec<3,P>& SimTK::Transform_::T ( ) const

inline

◆ updT()

template<class P>

Vec<3,P>& SimTK::Transform_::updT ( )

inline

Friends And Related Function Documentation

◆ operator*() [1/4]

template<class P , int S>

Vec< 3, P > operator*	(	const Transform_< P > &	X_BF,
		const Vec< 3, P, S > &	s_F
	)

This way we use both the rotational and translational components of the transform.

◆ operator+() [1/2]

template<class P , int S>

Transform_< P > operator+	(	const Transform_< P > &	X_BF,
		const Vec< 3, P, S > &	offset_B
	)

◆ operator+() [2/2]

template<class P , int S>

Transform_< P > operator+	(	const Vec< 3, P, S > &	offset_B,
		const Transform_< P > &	X_BF
	)

◆ operator-()

template<class P , int S>

Transform_< P > operator-	(	const Transform_< P > &	X_BF,
		const Vec< 3, P, S > &	offset_B
	)

◆ operator*() [2/4]

template<class P , int S>

Vec< 4, P > operator*	(	const Transform_< P > &	X_BF,
		const Vec< 4, P, S > &	a_F
	)

The 4th element must be 0 or 1. If 0 it is treated as a vector only and the translation is ignored. If 1 it is treated as a station and rotated & shifted.

◆ operator*() [3/4]

template<class P , class E >

Vector_< E > operator*	(	const Transform_< P > &	X,
		const VectorBase< E > &	v
	)

applies it to each element individually.

◆ operator*() [4/4]

template<class P >

Transform_< P > operator*	(	const Transform_< P > &	X1,
		const Transform_< P > &	X2
	)

Operators are provided for all the combinations of transform and inverse transform.

◆ operator==()

template<class P >

bool operator==	(	const Transform_< P > &	X1,
		const Transform_< P > &	X2
	)

◆ operator<<()

template<class P >

std::ostream & operator<<	(	std::ostream &	,
		const Transform_< P > &
	)

The documentation for this class was generated from the following file:

Transform.h

Public Member Functions

Related Functions

Detailed Description

template<class P> class SimTK::Transform_< P >

Constructor & Destructor Documentation

◆ Transform_() [1/4]

◆ Transform_() [2/4]

◆ Transform_() [3/4]

◆ Transform_() [4/4]

Member Function Documentation

◆ operator=()

◆ operator+=()

◆ operator-=()

◆ set()

◆ setToZero()

◆ setToNaN()

◆ invert()

◆ updInvert()

◆ operator~() [1/2]

◆ operator~() [2/2]

◆ compose() [1/2]

◆ compose() [2/2]

◆ xformFrameVecToBase()

◆ xformBaseVecToFrame()

◆ shiftFrameStationToBase()

◆ shiftBaseStationToFrame()

◆ R()

◆ updR()

◆ x()

◆ y()

◆ z()

◆ RInv()

◆ updRInv()

◆ p()

◆ updP()

◆ setP()

◆ pInv()

◆ setPInv()

◆ asMat34()

◆ toMat34()

◆ toMat44()

◆ T()

◆ updT()

Friends And Related Function Documentation

◆ operator*() [1/4]

◆ operator+() [1/2]

◆ operator+() [2/2]

◆ operator-()

◆ operator*() [2/4]

◆ operator*() [3/4]

◆ operator*() [4/4]

◆ operator==()

◆ operator<<()

template<class P>
class SimTK::Transform_< P >