1.5.3 Circles

Refresh your memory on the standard way to parameterize a circle in the plane.

This can be rewritten this way:

we can shift our circle out of the

plane by finding two non-zero orthogonal vectors in the plane containing our circle. If

is the center of the circle and

and

are two orthogonal vectors in the plane containing our circle, then we have the following parameterization:

The GeoGebra file below illustrates this approach. Have a conversation with a partner about why this works. Why are we dividing by

and

? What happens if

and

are not orthogonal?

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