Previous: Reconstruction from Curvature
Global Theory of Plane CurvesInversion with Respect to a CircleAnother 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:
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? |