On-Line Computer Graphics Notes
DEVELOPMENT OF THE ROTATION MATRIX

Overview

The 3-dimensional rotation matrix, when rotating about one of the coordinate axes is quite similar to the $3 \times 3$ rotation matrix developed for rotation in two-dimensions. Here rotation is much simpler to describe, as rotation is about a point in two-dimensions. Here we develop the rotation matrix in two-dimensions that rotates a point about the origin in the Cartesian frame.

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In Two Dimensions

In two dimensions, one rotates about a point. We will rotate about the origin, and will consider our 2-d frame to be the two-dimensional Cartesian frame. Consider the following figure

where we depict a rotation of $\theta$ units about the origin and the point $(x,y)$ is rotated into the point $(x',y')$. By considering the following figure,

we note that $(x,y)$ can be written in polar coordinates as

$$(x,y) \: = \: (r\cos\phi, r\sin\phi)$$

and also that $(x',y')$ can be written in polar coordinates as

$$(x',y') \: = \: (r\cos(\phi+\theta), r\sin(\phi+\theta))$$

Expanding the description of $(x',y')$, we obtain

$$(x',y') = (r\cos(\phi+\theta), r\sin(\phi+\theta))$$

$$= (r\cos\phi\cos\theta - r\sin\phi\sin\theta, r\sin\phi\cos\theta + r\cos\phi\sin\theta )$$

$$= (x\cos\theta - y\sin\theta, x\sin\theta + y\cos\theta )$$

which can be written in matrix form as

$$\left[ \begin{array}{ccc} x & y & 1 \end{array} \right] \left[ \b... ...in\theta & 0 \\ -\sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \end{array} \right]$$

So in 2-dimensions, rotation is implemented as a $3 \times 3$ matrix given by

$$\left[ \begin{array}{ccc} \cos\theta & \sin\theta & 0 \\ -\sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \end{array} \right]$$

Summary

Rotation about the origin in two-dimensions is given by a simple $3 \times 3$ matrix written in terms of the cosine and sine of the angle of rotation. This development can be applied directly to develop the rotation matrices for the three-dimensional rotations about the X, Y and Z axes.

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