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ContentAffine Spatial Transformation
Affine Spatial TransformationAffine spatial transformation matrices are used to represent the orientation and position of a global 3 dimensional coordinate system.𝑂=
1000 0100 0010 0001 Orientation and Position of an ObjectThe orientation and position of an object 𝑃 at (𝑥,𝑦,𝑧) can be represented by a 3D affine transformation matrice.𝑃=
𝑎11𝑎12𝑎13𝑥 𝑎21𝑎22𝑎23𝑦 𝑎31𝑎32𝑎33𝑧 0001 Direction Cosine MatrixDirection cosine matrix is the upper left 3x3 area of the affine spatial transformation matrix. The direction cosine matrix (DCM) is a transformation matrix used to represent the orientation of the object with respect to the original coordinate reference frame.DCM=
𝑎11𝑎12𝑎13 𝑎21𝑎22𝑎23 𝑎31𝑎32𝑎33 Coordinate Reference FrameThe orientation of the object is refered to the coordinate reference frame represented by unit vectors obtained by the direction cosine matrix.
Position VectorPosition vector is the upper right 3x1 area of the affine spatial transformation matrix. The position vector is a vector used to specify the position of the object with respect to the original position.𝑟= 𝑥 𝑦 𝑧 The 4th RowThe 4th row is always [0, 0, 0, 1] in forming a affine spatial transformation matrices and is used to maintain the 4x4 transformation matrix format.3D Affine Transformation MatricesThe 3D Affine Transformation, translation, rotations, scalings, reflections and shears can be combined in a single 4x4 affine transformation matrix𝐴=
The transformation of an object 𝑃 is applied by matrix transformation multiplication. The transformation matrix
𝑃′=𝐴𝑃
𝑎11𝑎12𝑎13𝑎14 𝑎21𝑎22𝑎23𝑎24 𝑎31𝑎32𝑎33𝑎34 0001 TranslationA translation moves an object along one or more of the three axes.Translation MatrixA translation matrix is used to translate an object with the specified translations, 𝑑𝑥, 𝑑𝑦, 𝑑𝑧, along the three axes.𝐴=
100𝑑𝑥 010𝑑𝑦 001𝑑𝑧 0001 ExamplesExamples𝑃′=𝐴𝑃=
100𝑑𝑥 010𝑑𝑦 001𝑑𝑧 0001 100𝑥 010𝑦 001𝑧 0001 100𝑥+𝑑𝑥 010𝑦+𝑑𝑦 001𝑧+𝑑𝑧 0001 Examples𝑃′=𝐴𝑃=
100𝑑𝑥 010𝑑𝑦 001𝑑𝑧 0001 𝑥 𝑦 𝑧 1 𝑥+𝑑𝑥 𝑦+𝑑𝑦 𝑧+𝑑𝑧 1 ScalingA scaling changes the size of an object along one or more of the three axes.Scaling MatrixA scaling matrix is used to change the size of an object with the specified scales, 𝑠𝑥, 𝑠𝑦, 𝑠𝑧, along the three axes.𝐴=
𝑠𝑥000 0𝑠𝑦00 00𝑠𝑧0 0001 ExamplesExamples𝑃′=𝐴𝑃=
𝑠𝑥000 0𝑠𝑦00 00𝑠𝑧0 0001 100𝑥 010𝑦 001𝑧 0001 𝑠𝑥00𝑠𝑥𝑥 0𝑠𝑦0𝑠𝑦𝑦 00𝑠𝑧𝑠𝑧𝑧 0001 Examples𝑃′=𝐴𝑃=
𝑠𝑥000 0𝑠𝑦00 00𝑠𝑧0 0001 𝑥 𝑦 𝑧 1 𝑠𝑥𝑥 𝑠𝑦𝑦 𝑠𝑧𝑧 1 RotationA rotation changes the orientation of an object along one of the three axes, or any arbitrary vector.Rotation MatrixA scaling matrix is used to change the orientation of an object with the specified angles in radian according to the right handed rule. The most common way is to specify arbitrary rotations with a sequence of simple rotation along one the the three cardinal axes.Rotation About 𝑋𝑅𝑥=
1000 0 Rotation About 𝑌𝑅𝑦=
Rotation About 𝑍𝑅𝑧=
Sources and References
©sideway ID: 220100014 Last Updated: 1/14/2022 Revision: 0 Ref: References
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