hom_mat3d_rotateT_hom_mat3d_rotatehom_mat3d_rotateHomMat3dRotateHomMat3dRotate (Operator)

Name

hom_mat3d_rotateT_hom_mat3d_rotatehom_mat3d_rotateHomMat3dRotateHomMat3dRotate — Add a rotation to a homogeneous 3D transformation matrix.

Signature

Description

hom_mat3d_rotatehom_mat3d_rotatehom_mat3d_rotateHomMat3dRotateHomMat3dRotate adds a rotation by the angle PhiPhiPhiPhiphi around the axis passed in the parameter AxisAxisAxisAxisaxis to the homogeneous 3D transformation matrix HomMat3DHomMat3DHomMat3DHomMat3DhomMat3D and returns the resulting matrix in HomMat3DRotateHomMat3DRotateHomMat3DRotateHomMat3DRotatehomMat3DRotate. The axis can by specified by passing the strings 'x', 'y', or 'z', or by passing a vector [x,y,z] as a tuple.

The rotation is decribed by a 3x3 rotation matrix R. It is performed relative to the global (i.e., fixed) coordinate system; this corresponds to the following chain of transformation matrices:

Axis = 'x':

                   /        0 \                     / 1    0         0     \
  HomMat3DRotate = |  Rx    0 | * HomMat3D     Rx = | 0 cos(Phi) -sin(Phi) |
                   |        0 |                     \ 0 sin(Phi)  cos(Phi) /
                   \ 0 0 0  1 /

Axis = 'y':

                   /        0 \                     /  cos(Phi) 0 sin(Phi) \
  HomMat3DRotate = |  Ry    0 | * HomMat3D     Ry = |     0     1    0     |
                   |        0 |                     \ -sin(Phi) 0 cos(Phi) /
                   \ 0 0 0  1 /

Axis = 'z':

                   /        0 \                     / cos(Phi) -sin(Phi) 0 \
  HomMat3DRotate = |  Rz    0 | * HomMat3D     Rz = | sin(Phi)  cos(Phi) 0 |
                   |        0 |                     \    0         0     1 /
                   \ 0 0 0  1 /

Axis = [x,y,z]:

                   /        0 \
  HomMat3DRotate = |  Ra    0 | * HomMat3D
                   |        0 |
                   \ 0 0 0  1 /

                       T                  T
               Ra = u*u + cos(Phi)*( I-u*u ) + sin(Phi)*S

                      AxisAxisAxisAxisaxis       / x' \
               u  = --------  =  | y' |
                    ||AxisAxisAxisAxisaxis||     \ z' /

                   / 1 0 0 \         /  0  -z'  y' \
               I = | 0 1 0 |     S = |  z'  0  -x' |
                   \ 0 0 1 /         \ -y'  x'  0  /

The point (PxPxPxPxpx,PyPyPyPypy,PzPzPzPzpz) is the fixed point of the transformation, i.e., this point remains unchanged when transformed using HomMat3DRotateHomMat3DRotateHomMat3DRotateHomMat3DRotatehomMat3DRotate. To obtain this behavior, first a translation is added to the input transformation matrix that moves the fixed point onto the origin of the global coordinate system. Then, the rotation is added, and finally a translation that moves the fixed point back to its original position. This corresponds to the following chain of transformations:

                   / 1 0 0 +Px \   /        0 \   / 1 0 0 -Px \
  HomMat3DRotate = | 0 1 0 +Py | * |   R    0 | * | 0 1 0 -Py | * HomMat3D
                   | 0 0 1 +Pz |   |        0 |   | 0 0 1 -Pz |
                   \ 0 0 0  1  /   \ 0 0 0  1 /   \ 0 0 0  1  /

To perform the transformation in the local coordinate system, i.e., the one described by HomMat3DHomMat3DHomMat3DHomMat3DhomMat3D, use hom_mat3d_rotate_localhom_mat3d_rotate_localhom_mat3d_rotate_localHomMat3dRotateLocalHomMat3dRotateLocal.

Attention

Note that homogeneous matrices are stored row-by-row as a tuple; the last row is not stored because it is identical for all homogeneous matrices that describe an affine transformation. For example, the homogeneous matrix

    / ra rb rc td \
    | re rf rg th |
    | ri rj rk tl |
    \ 0  0  0  1  /

is stored as the tuple [ra, rb, rc, td, re, rf, rg, th, ri, rj, rk, tl].

Parallelization

• Multithreading type: reentrant (runs in parallel with non-exclusive operators).
• Multithreading scope: global (may be called from any thread).
• Processed without parallelization.

Parameters

Result

If the parameters are valid, the operator hom_mat3d_rotatehom_mat3d_rotatehom_mat3d_rotateHomMat3dRotateHomMat3dRotate returns 2 (H_MSG_TRUE). If necessary, an exception is raised.

Possible Predecessors

hom_mat3d_identityhom_mat3d_identityhom_mat3d_identityHomMat3dIdentityHomMat3dIdentity, hom_mat3d_translatehom_mat3d_translatehom_mat3d_translateHomMat3dTranslateHomMat3dTranslate, hom_mat3d_scalehom_mat3d_scalehom_mat3d_scaleHomMat3dScaleHomMat3dScale, hom_mat3d_rotatehom_mat3d_rotatehom_mat3d_rotateHomMat3dRotateHomMat3dRotate

Possible Successors

hom_mat3d_translatehom_mat3d_translatehom_mat3d_translateHomMat3dTranslateHomMat3dTranslate, hom_mat3d_scalehom_mat3d_scalehom_mat3d_scaleHomMat3dScaleHomMat3dScale, hom_mat3d_rotatehom_mat3d_rotatehom_mat3d_rotateHomMat3dRotateHomMat3dRotate

See also

hom_mat3d_inverthom_mat3d_inverthom_mat3d_invertHomMat3dInvertHomMat3dInvert, hom_mat3d_identityhom_mat3d_identityhom_mat3d_identityHomMat3dIdentityHomMat3dIdentity, hom_mat3d_rotate_localhom_mat3d_rotate_localhom_mat3d_rotate_localHomMat3dRotateLocalHomMat3dRotateLocal, pose_to_hom_mat3dpose_to_hom_mat3dpose_to_hom_mat3dPoseToHomMat3dPoseToHomMat3d, hom_mat3d_to_posehom_mat3d_to_posehom_mat3d_to_poseHomMat3dToPoseHomMat3dToPose, hom_mat3d_composehom_mat3d_composehom_mat3d_composeHomMat3dComposeHomMat3dCompose

Module

Foundation