6.3.1. Rotation of a Point

See also

This YouTube Video by Peter Corke discusses rotation of points.

Rotation of points is an important topic for both machine vision and robotics. Pixels in an image might be rotated to align objects with a model. After describing the rotation of a point, we can extend the matrix rotation concept to transformations combining rotation and translation. We then consider transformations of coordinate frames used to describe the pose of robots and moving parts.

Here, we only consider rotating points about the origin. Rotation about other points is an extension of rotation about the origin.

As shown in figure Fig. 6.12, point \(\mathbf{p} = (a, b)\) is rotated by an angle \(\theta\) about the origin to point \(\mathbf{p}' = (a', b')\).

Fig. 6.12 Point \(\mathbf{p} = (a, b)\) is rotated by \(\theta\) to \(\mathbf{p}' = (a', b')\). To facilitate the discussion, the point \(\mathbf{p}\) is defined in terms of unit vectors \(\hat{\bm{x}} = (1, 0)\) and \(\hat{\bm{y}} = (0, 1)\). As shown in figure Fig. 6.13, the new location, \(\mathbf{p}'\) is then defined by unit vectors \(\hat{\bm{x'}}\) and \(\hat{\bm{y'}}\) formed by rotating \(\hat{\bm{x}}\) and \(\hat{\bm{y}}\) by the angle \(\theta\).

Fig. 6.13 Rotating the axis allows us to see the coordinates of the rotated point as scalar multiples of the rotated unit vectors \(\hat{\bm{x}}'\) and \(\hat{\bm{y}}'\).

\[\begin{split}\begin{array}{rl} \mathbf{p} &= a\,\hat{\bm{x}} + b\,\hat{\bm{y}} \\ \hfill \\ \mathbf{p}'&= a\,\hat{\bm{x'}} + b\,\hat{\bm{y'}} \end{array}\end{split}\]

\[\begin{split}\begin{aligned} \hat{\bm{x'}} &= \hat{\bm{x}}\,\cos\theta + \hat{\bm{y}}\,\sin\theta \\ \\ \hat{\bm{y'}} &= -\hat{\bm{x}}\,\sin\theta + \hat{\bm{y}}\,\cos\theta \end{aligned}\end{split}\]

(6.8)\[\mathbf{p}' = \hat{\bm{x}}(\,a\,\cos\theta - b\,\sin\theta\,) + \hat{\bm{y}}(\,a\,\sin\theta + b\,\cos\theta\,)\]

Equation (6.8) expressed in matrix notation becomes:

\[\begin{split}\begin{array}{rl} \mathbf{p}' &= \begin{bmatrix} \hat{\bm{x}} & \hat{\bm{y}} \end{bmatrix} \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix} \begin{bmatrix} a \\ b \end{bmatrix} \\ \\ &= \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix} \begin{bmatrix} a \\ b \end{bmatrix} \\ \\ &= \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix} \begin{bmatrix} a \\ b \end{bmatrix} \end{array}\end{split}\]

Thus, we have a \(2{\times}2\) rotation matrix, which yields the coordinates of the rotated point when multiplied by the coordinates of a point.

\[\begin{split}\mathbf{R}(\theta) = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}\end{split}\]

The rotation matrix has the following special properties.

1. The columns define basis vectors for the rotated coordinate frame.

2. The matrix is orthogonal (square with unit length, orthogonal columns).

3. \(\mathbf{R}(\theta)^{-1} = \mathbf{R}(-\theta) = \mathbf{R}^{T}(\theta)\).

4. The determinant, \(det(\mathbf{R}(\theta)) = 1, \; \forall \theta\), thus \(\bf{R}\) is never singular.