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GeometryCoordinate GeometryGeometric Transformation

Affine Spatial Transformation

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Affine Spatial Transformation
 Orientation and Position of an Object
  Direction Cosine Matrix
   Coordinate Reference Frame
  Position Vector
  The 4th Row
 3D Affine Transformation Matrices
 Translation
  Translation Matrix
  Examples
   Examples
   Examples
 Scaling
  Scaling Matrix
  Examples
   Examples
   Examples
 Rotation
  Rotation Matrix
   Rotation About 𝑋
   Rotation About 𝑌
   Rotation About 𝑍
 Sources and References

Affine Spatial Transformation

Affine 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 Object

The 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 Matrix

Direction 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 Frame

The orientation of the object is refered to the coordinate reference frame represented by unit vectors obtained by the direction cosine matrix.
  • unit vector 𝑥′ : 𝑎11𝑥+𝑎12𝑦+𝑎13𝑧
  • unit vector 𝑦′ : 𝑎21𝑥+𝑎22𝑦+𝑎23𝑧
  • unit vector 𝑧′ : 𝑎31𝑥+𝑎32𝑦+𝑎33𝑧

Position Vector

Position 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 Row

The 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 Matrices

The 3D Affine Transformation, translation, rotations, scalings, reflections and shears can be combined in a single 4x4 affine transformation matrix
𝐴= 𝑎11𝑎12𝑎13𝑎14 𝑎21𝑎22𝑎23𝑎24 𝑎31𝑎32𝑎33𝑎34 0001
The transformation of an object 𝑃 is applied by matrix transformation multiplication. The transformation matrix 𝑃′=𝐴𝑃

Translation

A translation moves an object along one or more of the three axes.

Translation Matrix

A translation matrix is used to translate an object with the specified translations, 𝑑𝑥, 𝑑𝑦, 𝑑𝑧, along the three axes.
𝐴= 100𝑑𝑥 010𝑑𝑦 001𝑑𝑧 0001

Examples

Examples

𝑃′=𝐴𝑃= 100𝑑𝑥 010𝑑𝑦 001𝑑𝑧 0001 100𝑥 010𝑦 001𝑧 0001 = 100𝑥+𝑑𝑥 010𝑦+𝑑𝑦 001𝑧+𝑑𝑧 0001

Examples

𝑃′=𝐴𝑃= 100𝑑𝑥 010𝑑𝑦 001𝑑𝑧 0001 𝑥 𝑦 𝑧 1 = 𝑥+𝑑𝑥 𝑦+𝑑𝑦 𝑧+𝑑𝑧 1

Scaling

A scaling changes the size of an object along one or more of the three axes.

Scaling Matrix

A 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

Examples

Examples

𝑃′=𝐴𝑃= 𝑠𝑥000 0𝑠𝑦00 00𝑠𝑧0 0001 100𝑥 010𝑦 001𝑧 0001 = 𝑠𝑥00𝑠𝑥𝑥 0𝑠𝑦0𝑠𝑦𝑦 00𝑠𝑧𝑠𝑧𝑧 0001

Examples

𝑃′=𝐴𝑃= 𝑠𝑥000 0𝑠𝑦00 00𝑠𝑧0 0001 𝑥 𝑦 𝑧 1 = 𝑠𝑥𝑥 𝑠𝑦𝑦 𝑠𝑧𝑧 1

Rotation

A rotation changes the orientation of an object along one of the three axes, or any arbitrary vector.

Rotation Matrix

A 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 𝑋

𝑅𝑥= 1000 0cos𝜃sin𝜃0 0sin𝜃cos𝜃0 0001

Rotation About 𝑌

𝑅𝑦= cos𝜃0sin𝜃0 0100 sin𝜃0cos𝜃0 0001

Rotation About 𝑍

𝑅𝑧= cos𝜃sin𝜃00 sin𝜃cos𝜃00 0010 0001

Sources and References

  • https://en.wikipedia.org/wiki/Direction_cosine
  • https://en.wikiversity.org/wiki/PlanetPhysics/Direction_Cosine_Matrix

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ID: 220100014 Last Updated: 1/14/2022 Revision: 0 Ref:

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References

  1. Hilbert, D. (translated by Townsend E.J.), 1902, The Foundations of Geometry
  2. Moore, E.H., 1902, On the projective axioms of geometry
  3. Fitzpatrick R. (translated), Heiberg J.L. (Greek Text), Euclid (Author), 2008, Euclid's Elements of Geometry
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