The relationship between commutation of matrices and motion control is well illustrated by looking at rotation and translation of a car on a flat road. We note that expansion, contraction, shear and reflection are not particularly relevant for describing the motion of a car that has a fixed size and shape. We will therefore have to deal with translations and rotations only. In order to make our computation job a little easier, we shall look at the whole motion a little differently from before.

So far, we have always assumed that the coordinate frame was fixed once and for all. When we looked at the diver, our coordinate frame was fixed, with x-axis parallel with the long side of the pool, for instance, and y-axis vertical. For the rock climber in the previous homework, the origin lay on one of the cliff walls, and one axis was the ground, the other was vertical, along a cliff. But it may be convenient to change that point of view.

If you are driving in a car, moving forwards or backwards, then the translations that the car undergoes, as you drive it, depend on the direction in which the car is pointing: for the fixed frame given by an overhead view of the street grid, your "move forward" could be in the x-direction, or in the y-direction, depending on whether you are pointed East or North, or some other direction in between if you are on an oblique street. Yet for you, the driver in the car, it is always "forward". So you could imagine defining all motion with respect to your sense of forward, backwards, right, left, as you sit in the car. Your move forward then just means that with respect to you the whole world moves backwards. Your turn to the left means that the whole world is turning to the right with respect to you. (It is similar to, when reading a map, turning the map as you navigate for the driver next to you, so that the street on which you drive is always pointing "straight ahead" on your lap.)

The advantage of looking at things this way is that, if you put the x-axis along the center of the car, pointing forward, then a "move forward" or a "move backward" is ALWAYS in the x-direction. Turning is now a rotation with respect to the CENTER of the car, since that is where the origin of the x-y frame is that is "bound" to the car.

Try it out on the interactive car below. You can set an angle of rotation θ, and a translation amount e. The fixed frame that is plotted is just an indication of the world in which the car is moving.

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This applet shows the top view of a car, with the arrow directing the front facing part. Enter theta and e and then press "Transform" To reset the car to its original orientation press the button marked "Reset". The coordinates of a point on the graph can be obtained by clicking anywhere on the graph. The x and y coordinates will be displayed in the lower left hand side of the applet. To zoom in or zoom out, click the appropriate button.