Location and Motion

Any point of the form

P = a + t (b - a) = (1 - t) a + t b

is on the line determined by the two points a and b.

If 0 <= t <= 1 then P is between a and b.
If t < 0 then P is beyond a.
If 1 < t then P is beyond b.

The Java applet below illustrates this. The point represented by the vector a is indicated by a red dot and the point represented by the vector b is represented by a blue dot. If you click anyplace along the line determined by these two points, the corresponding value of t will appear in the "t" box. You can also type a value of t in the t box and click on the move button. Then the corresponding point will be indicated on the line determined by a and b.

The applet below uses exactly the same mathematics but now the two vectors represent two different images and "moving" from one to the other is what moviemakers caled "dissolving."

The mathematics of location and motion is linear algebra and this example illustrates the power of the general language of vector spaces. Exactly the same mathematics is used to "mix" the audio track of a movie.