The Cauchy-Crofton Formula

The Cauchy-Crofton Formula1 is:

Theorem: Let c be a regular, plane curve; let m be the measure of all lines in the plane that intersect c (counting multiplicity). Then the length L of c is .5m
The goal of this webpage is to describe the proof using images and applets and to allow the user to explore the concepts used in the proof.


What is "the measure of all lines ..."?

Let's start by desribing the lines in as , where p, are as indicated.
We've now described the set as a region of ; we'll take the inherited measure from (area). Therefore the measure we'll take on the set is . It can be shown that this measure is invariant under rigid motions (Euclidean isometries) of the plane. For the details, see (1). We can now state the Cauchy-Crofton theorem more precisely: , where n(p, ) is the number of intersections between the line given by (p, ) and the curve c.

Is there a way to estimate this measure?

Yes, an interesting way.1 Construct a grid of lines paralell to the x-axis at a distance r apart. Rotate this family of curves about the origin by 45o(/4), 90o ( /2), and 135o (3 Pi/4) to get four families of paralell lines.

Now if a line in the grid intersects a curve n times, there might be more intersections if we consider all lines in - we only know about the grid lines. The parameters of these unknown lines can range from 0 to r and from 0 to /4. Therefore we can approximate the length of c by:

Proof of Cauchy-Crofton

The proof is given by 3 steps.

  1. Cauchy-Crofton holds for a line.

    Assume that the curve c is actually a straight line segment. We aim to show that the length of the line is 1/2 times the measure of all lines that intersect c. Following the suggestion of the previous section, use the applet below to explore this concept. Drag the endpoints of the line to desired positions, and hit the "Refine Mesh" button (this better approximates the true measure). You should see the experimental length converging to the actual length.

    Proof of step 1: Without loss of generality, we can position the line c so that the midpoint is at the origin and that c lies on the x-axis. Recalling our parameterization of lines from before, we aim to see for a fixed for which values of p do the lines intersect c. The picture below clearly illustrates that p can range from 0 to . Therefore

  2. Any curve can be approximated by straight lines.

    Explore this using the applet below. Move the curve by dragging the red points to new positions. Then hit the "Refine Approximation" button; convince yourself that the length of the segments gets very close to the length of the curve.

    Proof of step 2: Up to this point, we haven't been very precise about defining length. We'll do so now. Suppose that the curve c is parameterized on the interval [a,b]. Define a partition to be an collection of points that subdivides the interval, say . Define . Then

    It can be shown that this agrees with the intergral definition of length by using the property .

  3. Put them together and what have you got?

    Bippity, boppity, ...

    Proof of step 3: Again, assume c maps [a,b] to . Partition [a,b] into n points and use the new points as the vertices of the polygonal curve in (2). At each point $t_i$, there is a corresponding triangle T, and the measure of lines that intersect the curve between and is less than twice the measure of the lines that intersect T (if a line l intersects the curve, it must intersect the triangle twice).

    As n tends to infinity, notice that goes to zero by the definition of derivative, so the measure of the lines that intersect the triangle tends to twice the measure of the lines that intersect the line segment. Thus the measure of lines that intersect the curve segment goes to the measure of lines that intersect the line segment. Then by (1) and (2), the theorem is proven.

So Who Cares?

Other than the fact that it's an inherently interesting theorem (it uses simple approximation techniques to arrive at a less than obvious result), the Cauchy-Crofton Theorem a couple of other interesting uses. The approximation above gives a way to roughly calculate the length of a given curve without having to know its equation (as illustrated in the last applet). Also, (1) below mentions that it can sometimes be used to given a meaning of "length" to curves which are non-rectifiable, i.e., the usual integral definition (or the partitition definition) does not apply.

Adam G. Weyhaupt2,3
Indiana University
July 25, 2003

1Most of the information for this page comes from Differential Geometry of Curves and Surfaces; DoCarmo, M.; 1976.
2The impetus for this project was the MSRI Summer School in Mathematical Graphics.
3A special thanks to David Austin and Bill Casselman for their invaluable help.