trvec2tform

Convert translation vector to homogeneous transformation

Syntax

tform = trvec2tform(trvec)

Description

tform = trvec2tform(trvec) converts the Cartesian representation of the translation vector trvec to the corresponding homogeneous transformation tform. When using the transformation matrix, premultiply it by the coordinates to be transformed (as opposed to postmultiplying).

Examples

Convert Translation Vector to Homogeneous Transformation

trvec = [0.5 6 100];
tform = trvec2tform(trvec)

tform = 4×4

    1.0000         0         0    0.5000
         0    1.0000         0    6.0000
         0         0    1.0000  100.0000
         0         0         0    1.0000

Input Arguments

trvec — Cartesian representation of translation vector

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

Cartesian representation of a translation vector, specified 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]

Output Arguments

tform — Homogeneous transformation

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

Homogeneous transformation, returned as a 3-by-3-by-n array or 4-by-4-by-n array. n is the number of homogeneous transformations. When using the rotation matrix, premultiply it with the coordinates to be rotated (as opposed to postmultiplying).

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

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:

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

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: trvec2tform Supports 2-D Translation Vectors

The trvec argument now accepts 2-D translation vectors as a n-by-2 matrix and trvec2tform outputs 2-D homogeneous transformation matrices as a 3-by-3-by-n array.