3D Transformation Matrix Summary and Derivations

Translate

Derivation

Scale

Derivation

Rotate Around z-axis

Counter-Clockwise

Derivation

Clockwise

Derivation

Rotate Around y-axis

Counter-Clockwise

Derivation

Clockwise

Derivation

Rotate Around x-axis

Counter-Clockwise

Derivation

Clockwise

Derivation

Rotate Around Arbitrary Axis 
—Defined by the unit vector (u, v, w)

Counter-Clockwise

Derivation

Clockwise

Derivation

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Derivation: Translate

Derivation: Scale

Derivation: Rotate Around z-axis (Counter-Clockwise)

Derivation: Rotate Around z-axis (Clockwise)

Derivation: Rotate Around y-axis (Counter-Clockwise)

Derivation: Rotate Around y-axis (Clockwise)

Derivation: Rotate Around x-axis (Counter-Clockwise)

Derivation: Rotate Around x-axis (Clockwise)

Derivation: Rotate Around Arbitrary Axis (Counter-Clockwise)

1. Rotate vector around z-axis (clockwise): R_z
2. Rotate vector around y-axis (clockwise): R_y
3. Rotate around vector (counter-clockwise): R
4. Rotate vector around y-axis (counter-clockwise)*: R_y^-1
5. Rotate vector around z-axis (counter-clockwise)*: R_z^-1

* The inverse of a rotation matrix is the same as the rotation matrix for the same angle in the opposite direction.
Total Rotation Transformation: R_z^-1R_y^-1RR_yR_z
= (R_z^-1R_y^-1)(R)(R_yR_z)
First and Last Rotations (R_z and R_z^-1)


Second and Second-to-Last Rotations (R_y and R_y^-1)


Middle rotation (R)

First two rotations combined (R_yR_z)

Final two rotations combined (R_z^-1R_y^-1)

Final three rotations combined (R_z^-1R_y^-1R)

Entire transformation ((R_z^-1R_y^-1R)(R_yR_z))

Derivation: Rotate Around Arbitrary Axis (Clockwise)

See the previous section, and change the sign of each sin() function.