tform2trvec

R2026b

Extract translation vector from homogeneous transformation

Syntax

trvec = tform2trvec(tform)

Description

trvec = tform2trvec(tform) extracts the Cartesian representation of the translation vector trvec from the homogeneous transformation tform. The rotational components of tform are ignored. The input homogeneous transformation must be in the premultiplied form for transformations.

Examples

Extract Translation Vector from Homogeneous Transformation

tform = [1 0 0 0.5; 0 -1 0 5; 0 0 -1 -1.2; 0 0 0 1];
trvec = tform2trvec(tform)

trvec = 1×3

    0.5000    5.0000   -1.2000

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

trvec — Cartesian representation of translation vector

n-by-2 matrix | n-by-3 matrix

Cartesian representation of a translation vector, returned as an n-by-2 matrix if tform is a 3-by-3-by-n array and an n-by-3 matrix if tform is a 4-by-4-by-n array. n is the number of translation vectors. Each vector is of the form [x y] or [x y z].

Example: [0.5 6 100]

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™.

GPU Code Generation
Generate CUDA® code for NVIDIA® GPUs using GPU Coder™.

Version History

Introduced in R2015a

R2023a: tform2trvec 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 tform2trvec outputs a n-by-2 matrix of 2-D translation vectors.