tform2rotm

Extract rotation matrix from homogeneous transformation

Syntax

rotm = tform2rotm(tform)

Description

rotm = tform2rotm(tform) extracts the rotational component from a homogeneous transformation, tform, and returns it as an orthonormal rotation matrix, rotm. The translational components of tform are ignored. The input homogeneous transformation must be in the pre-multiply form for transformations. When using the rotation matrix, premultiply it with the coordinates to be rotated (as opposed to postmultiplying).

Examples

Convert Homogeneous Transformation to Rotation Matrix

tform = [1 0 0 0; 0 -1 0 0; 0 0 -1 0; 0 0 0 1];
rotm = tform2rotm(tform)

rotm = 3×3

     1     0     0
     0    -1     0
     0     0    -1

Input Arguments

tform — Homogeneous transformation

3-by-3-by-n array | 4-by-4-by-n array

Homogeneous transformation, specified as a 3-by-3-by-n array or 4-by-4-by-n array. n is the number of homogeneous transformations. The input homogeneous transformation must be in the premultiplied form for transformations.

2-D homogeneous transformation matrices are of the form:

$$T=\left[\begin{array}{ccc}{r}_{11}& r_{12}& t_1\\ r_{21}& r_{22}& t_2\\ 0& 0& 1\end{array}\right]$$

3-D homogeneous transformation matrices are of the form:

$$T=\left[\begin{array}{cccc}{r}_{11}& r_{12}& r_{13}& t_1\\ r_{21}& r_{22}& r_{23}& t_2\\ r_{31}& r_{32}& r_{33}& t_3\\ 0& 0& 0& 1\end{array}\right]$$

Example: [0 0 1 0; 0 1 0 0; -1 0 0 0; 0 0 0 1]

Output Arguments

rotm — Rotation matrix

2-by-2-by-n array | 3-by-3-by-n array

Rotation matrix, returned as a 2-by-2-n array or 3-by-3-by-n array containing n rotation matrices. Each rotation matrix in the array has either a size of 2-by-2 or 3-by-3 and is orthonormal. When using the rotation matrix, premultiply it with the coordinates to be rotated (as opposed to postmultiplying).

2-D rotation matrices are of the form:

$$R=\left[\begin{array}{cc}{r}_{11}& r_{12}\\ r_{21}& r_{22}\end{array}\right]$$

3-D rotation matrices are of the form:

$$R=\left[\begin{array}{ccc}{r}_{11}& r_{12}& r_{13}\\ r_{11}& r_{22}& r_{23}\\ r_{31}& r_{32}& r_{33}\end{array}\right]$$

Example: [0 0 1; 0 1 0; -1 0 0]

More About

Homogeneous Transformation Matrices

Homogeneous transformation matrices consist of both an orthogonal rotation and a translation.

2-D Transformations

2-D transformations have a rotation θ about the z-axis:

$$R_z(\theta )=\left[\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right]$$

, and a translation along the x and y axis:

$$t=\left[\begin{array}{c}x\\ y\end{array}\right]$$

, resulting in the 2-D transformation matrix of the form:

$$T=\left[\begin{array}{cc}R& t\\ 0_{1\times 2}& 1\end{array}\right]=\left[\begin{array}{cc}{I}_2& t\\ 0_{1\times 2}& 1\end{array}\right]·\left[\begin{array}{cc}R& 0\\ 0_{1\times 2}& 1\end{array}\right]$$

3-D Transformations

3-D transformations contain information about three rotations about the x-, y-, and z-axes:

$$R_x(\varphi )=\left[\begin{array}{ccc}1& 0& 0\\ 0& \cos \varphi & \sin \varphi \\ 0& -\sin \varphi & \cos \varphi \end{array}\right],R_y(\psi )=\left[\begin{array}{ccc}\cos \psi & 0& \sin \psi \\ 0& 1& 0\\ -\sin \psi & 0& \cos \psi \end{array}\right],R_z(\theta )=\left[\begin{array}{ccc}\cos \theta & -\sin \theta & 0\\ \sin \theta & \cos \theta & 0\\ 0& 0& 1\end{array}\right]$$

and after multiplying become the rotation about the xyz-axes:

$$\begin{array}{l}{R}_{xyz}=R_x(\varphi )R_y(\psi )R_z(\theta )=\\ \left[\begin{array}{ccc}\cos \varphi \cos \psi \cos \theta -\sin \varphi \sin \theta & -\cos \varphi \cos \psi \sin \theta -\sin \varphi \cos \theta & \cos \varphi \sin \psi \\ \sin \varphi \cos \psi \cos \theta +\cos \varphi \sin \theta & -\sin \varphi \cos \psi \sin \theta +\cos \varphi \cos \theta & \sin \varphi \sin \psi \\ -\sin \psi \cos \theta & \sin \psi \sin \theta & \cos \psi \end{array}\right]\end{array}$$

and a translation along the x-, y-, and z-axis:

$$t=\left[\begin{array}{c}x\\ y\\ z\end{array}\right]$$

, resulting in the 3-D transformation matrix of the form:

$$T=\left[\begin{array}{cc}R& t\\ 0_{1x3}& 1\end{array}\right]=\left[\begin{array}{cc}{I}_3& t\\ 0_{1x3}& 1\end{array}\right]·\left[\begin{array}{cc}R& 0\\ 0_{1x3}& 1\end{array}\right]$$

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

Version History

Introduced in R2015a

R2023a: tform2rotm Supports 2-D Homogeneous Transformation Matrices

The tform argument now accepts 2-D homogeneous transformation matrices as a 3-by-3-by-n array and tform2rotm outputs 2-D rotation matrices 2-by-2-by-n array.