Weighted Averages

If a+b=1, then

    \[an+bm\]

is a weighted average of n and m.

This simple idea is the basis for the way that a teenager gradually turns into a werewolf. Since (1-t)+t=1, we can take a=1-t and b=t and create a series of pictures of (1-t)F+tW as t increases from 0 to 1 to observe Fred morph into a Werewolf. Note that if t=0, then (1-t)P+tQ=P, and if t=1, then (1-t)P+tQ=Q. If t is somewhere between 0 and 1, then (1-t)P+tQ is some mixture of P and Q.

It n and m are numbers, then it is not very exciting to watch n morph into m. More interesting is to take two pictures P and Q, then create an animation that shows a series of pictures of (1-t)P+tQ as t increases from 0 to 1. Perhaps most interesting of all is to look at the case where P and Q are three-dimensional objects.

In the examples, two objects P and Q are given, the animation is shown, followed by a set of instructions for using Scientific Notebook to actually construct these animations.

  1. Function example.
  2. Polar coordinates example.