Generalizing Pythagoras and Carnot

Consider a circle of radius r centered on the point Pc, and an arbitrary point P0 at a distance R from Pc. We then draw a line through P0 intersecting the circle in the points P1 and P2 as shown below.

The angles with which the circular locus cuts across the line at the points P1 and P2 are equal and opposite (i.e., complementary), so if we rotate the line incrementally about the point P0, the lengths of the segments P0P1 and P0P2 will change at a rate always proportional to their current lengths, and these rates of change will have opposite signs. Letting L1 and L2 denote these two distances, we have dL1/L1 = dL2/L2, which implies L1dL2 + L2dL1 = d(L1L2) = 0. Therefore, the product of these two distances is constant. To determine the value of this constant, we need only rotate the line so that is passes through Pc as shown below, and then note that the distances from P0 to the points of intersection are Rr and R+r, so their constant product is R2 – r2.

On the other hand, if we rotate the line so that it’s tangent to the circle, the points P1 and P2 coincide, both at a distance L from P0, and the line from Pc to the point of tangency is perpendicular to the tangent line, as shown below. In this case our proposition gives the relation L2 = R2 – r2, which is Pythagoras’ theorem.

Thus the Pythagorean theorem L2 = R2 – r2 can be seen as just a special case of the more general proposition L1L2 = R2 – r2. Incidentally, the same relation applies even if P0 is inside the circle, provided we treat the lengths L1 and L2 as signed quantities depending on their direction from P0. Since the product of the two distances from a point to the circle along any given line is constant, we can immediately infer the “hyperbolic” form of Pythagoras’ theorem, L1L2 = h2, from the figure below, where the line along L1 and L2 is a diagonal of the circle. (By completing the rectangle, we can see that the upper vertex is a right angle.)

In Propositions 35 and 36 of Book III of Euclid’s Elements the proposition L1L2 = R2 – r2 is proved by mean of Pythagoras’ theorem, as shown in the figure below.

We drop the perpendicular h from Pc to the line P1 P2, and then note that the product of the segments P0P1 and P0P2 is (st)(s+t) = s2 – t2. Also, the Pythagorean theorem gives s2 = r2 – h2 and t2 = R2 – h2, so the product of our two line segments is r2  R2. This is a nice proof, but it’s reliance on the Pythaogrean theorem prevented Euclid from placing it in Book I. Compare this with our previous proof, which consisted of the observation that as the line P1P2 rotates about P0 the segments P0P1 and P0P2 vary at rates always proportional to their lengths, one increasing and the other decreasing, so their product is constant. Admittely this is essentially a calculus argument, which Euclid would probably not have considered acceptable, but it bears a strong resemblence to some of the arguments used by Archimedes several centuries later. If Euclid had found a way to prove this proposition with invoking his I.47, he could then have immediately proved I.47 based on this result.

The proposition L1L2 = R2 – r2 represents an algebraic relation between a set of distances r, R, L1, and L2 defined in terms of orthogonal coordinate differences by the basic function

The distance between the points Pi and Pj is defined as f(xixj, yiyj). Assigning the point P0 to the origin of our coordinate system, we can express the proposition as

where P1 and P2 are the intersections of any line through the origin with the locus defined by f(xxc,yyc) = r. Letting n denote the degree of the base polynomial (so n = 2 in this case), we can write this in the form

It turns out (as shown in the article on A Quadrilateral in a Circle) that this proposition is true for any arbitrary homogeneous polynomial f of any degree n in any number of variables. Also, we can convert any polynomial to a homogeneous one in a higher dimension. To illustrate, consider a simple cubic curve

This locus is not expressed in homogeneous form, but if we define X = x/z and Y = y/z we can substitute into this equation to give the homogeneous surface in three dimensions

We now define the function

On the plane w = 1 with u = X and v = Y this is our original cubic curve, so we choose the “center” point xc = yc = 0, zc = 1, and consider the locus of points such that

On the plane z = 0 this locus is our original cubic. We choose to evaluate the intersections of this curve with the line through the origin with the slopes ky = 0 and kz = 0, so we know the intersection points are at x1 = 2, x2 = 3, and x3 = 5. Noting that r = 0, the theorem gives us the relation

Both sides evaluate to 30, confirming the equality. It follows that if we tilt the line through the origin by setting ky to some non-zero value, the product of the f-functions of the coordinates of the three new points of intersection with our original cubic is always 30. Morevoer, we can tilt the line in the z direction by setting kz to some non-zero value, and still the product of the f-function of the coordinates of the three points of insection with the cubic surface is 30, even if we tilt the line to such an extent that the points of intersection have complex coordinates.

In a sense, the “f-function” serves as a kind of “distance” function, but of course it isn’t the literal distance, unless f happens to be the metric function

However, our proposition can still give us information about the relations between the literal distances. Consider the locus of points satisfying f(xxc,yyc,zzc) = r where f is an arbitrary homogeneous polynomial of degree n. Letting P1, P2,… Pn denote the points of intersection of the line given by y = kyx, z = zyx, we can write our basic proposition in the form

In general the left-hand side is not the product of the literal distances, but we can form the product of the literal distances if we simply divide by f(1,ky,kz) and then multiply by the metric function g(1,ky,kz). Thus we have

where L1, L2, … Ln are the literal distances from the origin to the points of intersection P1, P2, …, Pn. Obviously this product is not invariant because it depends on the slopes of the line, but the quantity in the square brackets is invariant for any given focal point in relation to the “center” of the locus. Thus the product of distances from a given focal point P0 to a locus along a line with the slopes , can be factored as F(P0)G(,) where the function F depends only on the focal point and the function G depends only on the slopes of the line.

Now consider a set of m arbitrary points, denoted by Q1, Q2, …, Qm, and the cycle of m lines Q1Q2, Q2Q3, …, QmQ1. Let the slopes of the line from Qj to Qj+1 be denoted by j and j, and let j,k denote the product of the literal distances from Qj to the locus along the line from Qj to Qk. Then we have

Each of the points has another set of distances to the curve, taken along the other line passing through that point, and we can evaluate the product of those products as

Both of the above products consist of the same factors, so we have the relation for the products of the literal distances

This corresponds to Carnot’s theorem on transversals, generalized to cycles of arbitrary length. We can immediately generalize it still further, because the relation applies not just to the products of the literal distances, but to the products of any homogeneous function of the coordinate differences. To see this, simply replace the metric function g with any other function. It also follows that the ratio of any function of the coordinate differences from any two given points to a curve along parallel lines (with arbitrary slopes) is invariant. So, for example, in the figure below, the product of the distances from A to S divided by the product of the distances from B to S is the same, regardless of whether the distances are evaluated along the two solid lines or the two dashed lines (or any other pair of parallel lines through the points A and B).