so2

SO(2) rotation

Since R2022b

Description

The so2 object represents an SO(2) rotation in 2-D.

For more information, see the 2-D Orthonormal Rotation Matrix section.

This object acts like a numerical matrix, enabling you to compose rotations using multiplication and division.

Creation

Syntax

rotation = so2

rotation = so2(rotation)

rotation = so2(transformation)

rotation = so2(angle,"theta")

Description

rotation = so2 creates an SO(2) rotation representing an identity rotation with no translation.

$$rotation=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$$

rotation = so2(rotation) creates an SO(2) rotation rotation representing a pure rotation defined by the orthonormal rotation rotation.

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

rotation = so2(transformation) creates an SO(2) rotation from the SE(2) transformation transformation.

rotation = so2(angle,"theta") creates an SO(2) rotation rotation from a rotation angle about the z-axis angle.

Note

If any inputs contain more than one rotation, the output rotation is an N-element array of so2 objects corresponding to each of the N input rotations.

Input Arguments

rotation — Orthonormal rotation

2-by-2 matrix | 2-by-2-by-N matrix | so2 object | N-element array of so2 objects

Orthonormal rotation, specified as a 2-by-2 matrix, a 2-by-2-by-N matrix, a scalar so2 object, or an N-element array of so2 objects. N is the total number of rotations.

If rotation is an array, the resulting number of created so2 objects in the output array is equal to N.

Example: eye(2)

Data Types: single | double

transformation — Homogeneous transformation

3-by-3 matrix | 3-by-3-by-N array | se2 object | N-element array of se2 objects

Homogeneous transformation, specified as a 3-by-3 matrix, a 3-by-3-by-N array, a scalar se3 object, or an N-element array of se2 objects. N is the total number of transformations specified.

If transformation is an array, the resulting number of created so2 objects is equal to N.

Example: eye(3)

Data Types: single | double

angle — z-axis rotation angle

N-by-M matrix

z-axis rotation angle, specified as an N-by-M matrix. Each element of the matrix is an angle, in radians, about the z-axis. The so2 object creates an so2 object for each angle.

If angle is an N-by-M matrix, the resulting number of created so2 objects is equal to N.

The rotation angle is counterclockwise positive when you look along the axis toward the origin.

Data Types: single | double

Object Functions

Mathematical Operations

mtimes, *	Transformation or rotation multiplication
mrdivide, /	Transformation or rotation right division
rdivide, ./	Element-wise transformation or rotation right division
times, .*	Element-wise transformation or rotation multiplication

Utilities

interp	Interpolate between transformations
dist	Calculate distance between transformations
normalize	Normalize transformation or rotation matrix
transform	Apply rigid body transformation to points

Numerical Conversions

rotm	Extract rotation matrix
trvec	Extract translation vector
tform	Extract homogeneous transformation
theta	Convert transformation or rotation to 2-D rotation angle
xytheta	Convert transformation or rotation to compact 2-D pose representation

Object Conversions

so3	SO(3) rotation

Examples

Create SO(2) Rotation Using Angle

Define an angle rotation of pi/4 and a xy translation of [6 4].

angle = pi/4;

Create an SO(2) rotation using the angle.

R = so2(angle,"theta")

R = so2
    0.7071   -0.7071
    0.7071    0.7071

Algorithms

2-D Orthonormal Rotation Matrix

SO(2) rotation matrices are 2-by-2 orthonormal matrices that represent a rotation about a single axis 2-D Euclidean space. SO(2) rotations have many special properties. For example, SO(2) rotation matrices are in the 2-D special orthogonal group, so the product of two SO(2) rotation matrices is an SO(2) rotation matrix. This enables you to compose rotations from multiple rotations.

This is a 2-D orthonormal rotation matrix that describes a rotation θ about the z-axis:

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

These are other properties of SO(2) rotations:

• Each column is both orthogonal and a unit vector, meaning that none of the columns are multiples of each other.

• The determinant of the matrix is positive 1.

• The inverse of the matrix is the same as the transpose of the matrix: R^-1 = R^T.

Extended Capabilities

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

Version History

Introduced in R2022b

R2023a: New methods and syntaxes

so2 supports new methods and syntaxes for converting to and from other transformations and rotations.

The new methods are:

• so3

• theta

• tform

• trvec

• xytheta