2.7: 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. Consider a circle with center C and radius r. For a given point P, the inverse point, P', is the point on ray CP such that |CP|*|CP'| = r2. So, if P were to lie on the circle, for example, it would be its own inverse point (i.e. P = P'). For every other case where P does not lie on the circle, we can deduce from the definition that P and P' must lie on opposite sides of the circle. For a curve X(t), we can find its inverse curve with respect to a circle of radius r centered at C. This curve can be written as

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




This demo allows you to invert a curve X(t) with respect to a circle of radius r and center C. The point C can be moved around using the hotspot in the curve window and the radius r can be changed using the tapedeck in the control panel. The curve X(t) is drawn in cyan and its inverse curve Yc(t) is drawn in orange. You can scroll through the tapedeck controlling t0 to see X(t) and its image Yc(t) traced out.



1. What can be said about the inverted images of circles and lines in the plane? Prove your conjectures.

2. 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?

3. Under what circumstances will the image curve have a cusp?

4. Let P be a point outside of a circle with center C and radius r and consider the line through P that is tangent to the circle. Let Q be the point of tangency and construct the right triangle CPQ. It turns out that the foot of the perpendicular from Q to side CP is P', the inverse of P. The diagram for this construction can be seen in the demonstration by clicking on the checkbox in the control panel labeled "ShowTriangle". Show that this definition of the inverse point of P is consistent with the definition given at the beginning of this section.

The Four-Vertex Theorem

The four-vertex theorem states that for a closed convex curve κg(t) has at least two minima and at least two maxima. Furthermore, this implies that the evolute to any closed convex curve has at 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.



This demo shows you the curve X(t) and its evolute in one window and a graph of the curvature of X(t), plotted against t, in the second window. The purpose of this demo and of the exercise below is to illustrate the four-vertex theorem. You can change the value of c as well as the value of t0 using the two tapedecks in the control panel.



Compare the numbers of cusps of the evolute curve for the ellipse, the epicycloid, and the curves in the cardioid family X(t)=(c+cos(t))*(cos(t), sin(t)) for various values of c.
Winding Numbers of Plane Curves

For any point Q not on a curve X, we may determine the number of times the curve WQ(t) = (X(t)-Q)/|X(t)-Q| winds around the point Q. Clearly, the curve WQ(t) is a section of a unit circle centered at Q. The winding number is then defined as the total angular change in WQ(t) divided by 2π. This number is also the number of times the curve X(t) winds around Q.


This demo plots the curve X(t) in one window, and the curve WQ(t) = (X(t) - Q)/|X(t) - Q| in the second window. You can choose the point Q by moving the hotspot in the curve window. There is a tapedeck that controls the value of c in the equation for the cardioid family. There is also a tapedeck for the parameter u, which lets you to perturb WQ(t) off the unit circle and allows you to see which parts of the unit circle are covered multiple times. Note that a section of the circle may be covered negatively or positively corresponding to whether WQ(t) is traced out in a clockwise or counterclockwise direction. Negative coverings of the circle are colored blue, while positive coverings are colored red.



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

2. Give an argument for why the winding number of a point Q with respect to a closed curve X(t) is always an integer. (Assume that Q does not lie on X(t))
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