2.9: 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:

Y_C(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?

2.10: The Four-Vertex Theorem

Graph 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 k_g(t).

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
X_c(t )=((c+cos( t))cos(t ),(c+cos(t))sin(t))
for various values of c .

Hopefully, the exercise demonstrated the following theorem. The four-vertex theorem states that for a closed convex curve k_g(t) has at least two minima and at least two maxima. Furthermore, this implies that the evolute to any closed convex curve has a least four cusps . The result does not say anything about curves that are self-intersecting. Such curves may have just two cusps on their evolute curves.

2.11: Winding Numbers of Plane Curves

For any point Q not on a curve X , we may determine the number of times the curve

W_Q(t)=(X(t) -Q)/|X(t) -Q|

winds around the point Q . Clearly, this curve W_Q(t) is a section of a circle centered at Q. The winding number is then defined as the total angular change in W_Q(t) divided by 2{pi} . This number is also the number of times the curve X(t) winds around Q .

Winding Number Demo

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{pi} . Thus, the function's final destination on the right-hand side of the window corresponds the winding number of the curve around Q .

Investigate how the curve W_Q(t) changes for various positions of Q . Try curves from the cardioid family.

Next: Inversion with Respect to a Circle