Previous: Normalized Parallel Curves
2.9: 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? 2.10: The Four-Vertex TheoremGraph of the Curvature
This demo simply takes a curve as input and displays the evolute to that curve
and then, in another window, draws the graph of
The purpose of this demo and the following exercise is to illustrate what is called the four-vertex theorem and will be explicitly stated below. Compare the numbers of cusps of the evolute curve for the ellipse, the epicycloid, and the curves in the cardioid family
Hopefully, the exercise demonstrated the following theorem. The four-vertex
theorem states that for a closed
convex
curve
2.11: Winding Numbers of Plane CurvesFor any point Q not on a curve X , we may determine the number of times the curve In this demo, you may input a curve, in the control panel and then choose a point Q. The demo displays the angular variation of the curve with respect to
the point
Q
. The various lines with integer values
n
represent angles
2n
Investigate how the curve
WQ(t)
changes for various positions of
Q
. Try curves from the cardioid family. |