On-Line Computer Graphics Notes
DEVELOPING A GENERAL ROTATION
BY USING FRAMES

Overview

An axis in space is specified by a point ${\bf P}$ and a vector direction ${\vec t}$. Suppose that we wish to rotate an object about this arbitrary axis.

We know how to do this in the cases that the axis is the x axis, the y axis, or the z axis in the Cartesian Frame, which are just generalizations of the two-dimensional rotations, but the general case is more difficult.

We present two methods to approach this problem. Here we present a solution based upon frame conversion. One can also approach the problem directly by using translation and rotation transformations

Specifying a Frame on the Axis.

The general idea here is to establish a frame on the axis. That is, define a frame ${\cal F} =( {\vec u} , {\vec v} , {\vec w} , {\bf O} )$ where the point ${\bf P}$ is the origin ${\bf O}$ of the frame, and the vector ${\vec t}$ is one of the vectors defining the frame. Then transform this frame to the Cartesian frame, do the rotation in the Cartesian frame about the appropriate axis, and transform back

More specifically, we establish a frame ${\cal F} =( {\vec u} , {\vec v} , {\vec w} , {\bf O} )$ as follows:

• Let ${\vec w} = {\vec t}$ and let ${\bf O} = {\bf P}$
• define any other vector ${\vec a }$ that is not in the same direction as ${\vec t}$ or $- {\vec t}$ (i.e. ${\vec a }\not = c {\vec t}$, for some constant $c$).
• Let ${\vec v} = {\vec a }\times {\vec t}$.
• Let ${\vec u} = {\vec v} \times {\vec t}$.

We note that this frame is orthogonal - that is, all vectors are mutually perpendicular.

Now let $F$ be the frame-to-Cartesian-frame, conversion transformation. Then we can perform the general rotation by the following transformation

$$F R_{z;\alpha} F^{-1}$$

which transforms the constructed frame to the Cartesian frame, rotates about the $z$ axis and then transforms back to the constructed frame. The resulting matrix will perform the desired rotation on the object.

What if the Axis was Specified in a Local Coordinate System?

In this case, we just convert the coordinates of the point ${\bf P}$ and vector ${\vec t}$ defining the axis to Cartesian coordinates using the frame-to-Cartesian-frame transformation, do the above operations, and then use the Cartesian-frame-to-frame to convert the resulting coordinates back to the local system.

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