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Global Theory of Plane Curves

Inversion with Respect to a Circle

Another interesting tool that one uses in exploring plane curves is the inversion of a curve with respect to a circle. This means we take each point on the curve to the point on the other side of the circle such that where the line segment between the image and the preimage points meets the circle, the tangent line to the circle is its perpendicular bisector. The formula for inversion with respect to a circle centered at C with radius r is given by:

    YC(t )=r2( X-C)/|X-C |2 +C

Circular Inversion Demo What can be said about the inverted images of circles and lines in the plane?

    What about the image of an ellipse? Under what circumstances will the resulting curve be convex? What are the conditions on the number of inflection points the curve has? What is the relation between the number of inflection points of the image curve and the eccentricity of the ellipse? This is a hint which explains the Secret Button.

    Under what circumstances will the image curve have a cusp?


Next: The Four-Vertex Theorem