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 .5mThe 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.
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:
The proof is given by 3 steps.
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
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 .
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.
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.
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.