Affine Spatial Transformation

Affine spatial transformation matrices are used to represent the orientation and position of a global 3 dimensional coordinate system.

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.

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.

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.

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 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, 𝑑_𝑥, 𝑑_𝑦, 𝑑_𝑧, along the three axes.

Examples

Examples

Examples

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.

Examples

Examples

Examples

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

Rotation About 𝑌

Rotation About 𝑍

Sources and References

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