Rich and General in Contrast to Abstract

The expression

can be used to move smoothly from the vector a to the vector b. The Java applet below illustrates this for two vectors in R². You can click anyplace along the line determined by the two vectors (represented by a red dot and a blue dot) to see the corresponding value of t or you can enter the value of t and click the move button to see the corresponding point.

The applet below illustrates this for two vectors representing images. Click anyplace along the scale at the bottom of the vector to "mix" the two images. The same formula is used to mix images as to move from one point to another.

We can do exactly the same thing to mix sounds.

The Java applet below illustrates the correspondence between points in a triangle whose vertices are represented by the vectors a, b, and c and expressions of the form

where

The same expression works for a triangle in R³ or, as shown in the applet below, for mixing colors represented by vectors.