Simbody
3.6

This class represents the rotateandshift 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 defaultconvert a rotation into a transform containing that rotation and zero translation. More...  
Transform_ (const Vec< 3, P > &p)  
Construct or defaultconvert 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 readonly 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 readonly 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 readonly 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 readonly 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 readonly reference to the z direction (unit vector) of the F frame, expressed in the B frame. More...  
const InverseRotation_< P > &  RInv () const 
Return a readonly 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 readonly 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 readonly 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 3vector, 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 3vector 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 3vector 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 3vector 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 4vector, 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...  
This class represents the rotateandshift transform which gives the location and orientation of a new frame F in a base (reference) frame B.
A frame is an orthogonal, righthanded 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 123 or xyz as you prefer, with z = x X y, making a righthanded 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 reexpressed in B is given by vB = R_BF*vF. F's origin point OF is stored as the translation vector p_BF=(FoBo) 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 welldefined memory layout on which a client of this class may depend: There are exactly 4 consecutive, packed 3vectors 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.

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Default constructor gives an identity transform.

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Combine a rotation and a translation into a transform.

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Construct or defaultconvert a rotation into a transform containing that rotation and zero translation.

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

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

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Add an offset to the position vector in this Transform.
Cost is 3 flops.

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Subtract an offset from the position vector in this Transform.
Cost is 3 flops.

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Assign a new value to this transform, explicitly providing the rotation and translation separately.
We return a reference to the nowmodified transform as though this were an assignment operator.

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By zero we mean "zero transform", i.e., an identity rotation and zero translation.
We return a reference to the nowmodified transform as though this were an assignment operator.

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This fills both the rotation and translation with NaNs.
Note: this is not the same as a defaultconstructed transform, which is a legitimate identity transform instead. We return a reference to the nowmodified transform as though this were an assignment operator.

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Return a readonly inverse of the current Transform_.
, simply by casting it to the InverseTransform_
type. Zero cost.

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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.

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Overload transpose operator to mean inversion.

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Overload transpose operator to mean inversion.

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Compose the current transform (X_BF) with the given one.
That is, return X_BY=X_BF*X_FY. Cost is 63 flops.

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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).
That is, return X_BY=X_BF*X_FY, but now X_FY is represented as ~X_YF. Cost is an extra 18 flops to calculate X_FY.p(), total 81 flops.

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Transform a vector expressed in our "F" frame to our "B" frame.
Note that this involves rotation only; it is independent of the translation stored in this transform. Cost is 15 flops.

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Transform a vector expressed in our "B" frame to our "F" frame.
Note that this involves rotation only; it is independent of the translation stored in this transform. Cost is 15 flops.

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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.
Cost is 18 flops.

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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.
Cost is 18 flops.

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Return a readonly reference to the contained rotation R_BF.

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Return a writable (lvalue) reference to the contained rotation R_BF.

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Return a readonly reference to the x direction (unit vector) of the F frame, expressed in the B frame.

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Return a readonly reference to the y direction (unit vector) of the F frame, expressed in the B frame.

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Return a readonly reference to the z direction (unit vector) of the F frame, expressed in the B frame.

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Return a readonly reference to the inverse (transpose) of our contained rotation, that is R_FB.

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Return a writable (lvalue) reference to the inverse (transpose) of our contained rotation, that is R_FB.

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Return a readonly reference to our translation vector p_BF.

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Return a writable (lvalue) reference to our translation vector p_BF.
Caution: if you write through this reference you update the transform.

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Assign a new value to our translation vector.
We expect the supplied vector p
to be expressed in our B frame. A reference to the nowmodified transform is returned as though this were an assignment operator.

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Calculate the inverse of the translation vector in this transform.
The returned vector will be the negative of the original and will be expressed in the F frame rather than our B frame. Cost is 18 flops.

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Assign a value to the inverse of our translation vector.
That is, we're given a vector in F which we invert and reexpress in B to store it in p, so that we get the original argument back if we ask for the inverse of p. Sorry, can't update pInv as an lvalue, but here we want (~R_BF*p_BF)=p_FB => p_BF=(R_BF*p_FB) so we can calculate it in 18 flops. A reference to the nowmodified transform is returned as though this were an assignment operator.

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Recast this transform as a readonly 3x4 matrix.
This is a zerocost reinterpretation of the data; the first three columns are the columns of the rotation and the last column is the translation.

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

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Return the equivalent 4x4 transformation matrix.

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related 
If we multiply a transform or inverse transform by a 3vector, 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.
This way we use both the rotational and translational components of the transform.

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Adding a 3vector to a Transform produces a new shifted transform.

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Adding a 3vector to a Transform produces a new shifted transform.

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Subtracting a 3vector from a Transform produces a new shifted transform.

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If we multiply a transform or inverse transform by an augmented 4vector, we use the 4th element to decide how to treat it.
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.

related 
Multiplying a matrix or vector by a Transform_.
applies it to each element individually.

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

related 
Comparison operators return true only if the two transforms are bit identical; that's not too useful.

related 
Generate formatted output of a Transform to an output stream.