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Evolute

We examine the evolute of an ellipse. The evolute is the envelope of the normals ... don't worry it is easy to create, and to understand once you create it!

We'll draw a parametric curve with equation (2*cos(t),sin(t))

Picture 1

We'll get the piece of the curve between t=0 and t=1. Change the upper limit to 6.28 and zoom in a bit to see the whole ellipse

Picture 2

Now create a tangent

Picture 3

And a perpendicular to the tangent (this is the normal)

Picture 4

Now create the envelope of the normal

Picture 5

If you animate t, you can observe.that the envelope is a curve which is tangent to all the normals.

Picture 6

Use the Symbolic Panel to get the equation of the curve.

Picture 7

Can you work out what the x and y intercepts of this curve are? Here is that equation again:

Picture 8 << Back