Curvature Theorems on Polar Curves

PROCEEDINGS OF THE NATIONAL ACADEMY OF SCIENCES Volume 33 March 15, 1947 Number 3

CURVATURE THEOREMS ON POLAR CUR VES* By EDWARD KASNER AND JOHN DE CICCO DEPARTMENT OF MATHEMATICS, COLUMBIA UNIVERSITY, NEW YoRK Communicated February 4, 1947 1. The principle of duality in projective geometry is based on the theory of poles and polars with respect to a conic. For a conic, a point has only one kind of polar, the first polar, or polar straight line. However, for a general algebraic curve of higher degree n, a point has not only the first polar of degree n - 1, but also the second polar of degree n - 2, the rth polar of degree n - r, :. ., the (n - 1) polar of degree 1. This last polar is a straight line. The general polar theory' goes back to Newton, and was developed by Bobillier, Cayley, Salmon, Clebsch, Aronhold, Clifford and Mayer. For a given curve Cn of degree n, the first polar of any point 0 of the plane is a curve C.-1 of degree n - 1. If 0 is a point of C, it is well known that the polar curve C,,- passes through 0 and touches the given curve Cn at 0. However, the two curves do not have the same curvature. We find that the ratio pi of the curvature of C,- i to that of Cn is Pi = (n -. 2)/(n - 1). For example, if the given curve is a cubic curve C3, the first polar is a conic C2, and at an ordinary point 0 of C3, the ratio of the curvatures is 1/2. These simple theorems appear to be hew. We generalize this for all the higher polars. All the higher polars go through the given point 0 -of the curve Cn and touch C. at 0. In particular, the last polar coincides with the tangent line at 0. The curvatures at 0 all have different values (the last one being obviously zero), and we determine all these curvatures In the above discussion, we assumed that 0 is an ordinary point of the algebraic curve Cn. We conclude by discussing also the theory .at a singular point 0 of C,,; for example, inflections and cusps. Although our new theorems are stated in metric terms, they are essen- tially theorems of differential projective geometry. This is a consequence of the theorem of Mehmke-Segre which states that if any two differ- Downloaded by guest on September 30, 2021 48 MA THEMA TICS: KASNER A ND DE CICCO PROC. N. A. S.

entiable curves are tangent at a given point 0, then the ratio p of their departures from the common tangent line is a projective invariant. 2. An algebraic curve Cn, of degree n is defined in cartesian coordinates (x, y) by the equation n Cn: (p(x, y) = j (x, y) = 0, (1) k=O Pk where Pk(x, y) is a homogeneous polynomial in (x, y) of degree k, and P,,(x, y) is not identically zero. It is convenient to use non-homogeneous cartesian coordinates (x, y) since we are dealing with the metric character of the polar curves. / Upon taking the given point 0 at the origin, we find that the first polar C.-, of degree n - 1 of 0 with respect to C,, is2 n-i , = (2) Cn-1: k =O(n- k)Pk(x, y) 0. The rth polar Cn-r of degree n -r of the point 0 with respect to C,, may be defined by induction. It is the first polar of the same point 0 with respect to the (r - 1) polar C,,-+ 1 of degree n - r + 1. Thus the equation of the rth polar of the origin 0 is n -r Cn-r: , (n -k)(n- k -1). ... (n- k-r + 1) Pk(x, y) = O. (3) k =O The (n - 1) polar C1of the origin 0 is the polar line C,: nPo + Pi(x, y) = 0. (4) 3. Let the origin 0 be an ordinary point of the algebraic curve C,, so that Po = 0 and Pi(x, y) is tiot identically zero. The tangent line of C,, at 0 is given by the equation Pi(x, y) = 0, and hence is identical with the polar line (4). By equations (2) and (3), it is seen that all the polar curves of 0 with respect to C. pass through 0 and touch C,, at 0. THEOREM 1. The ratio pi of the curvature of the first polar C,,- to the curvature of Cn, constructed at an ordinary point 0 of C,, is n-2 Pi n- . (5) This can be deduced from equations (1) and (2). By considering the equations (1) and (3), we obtain the following generalization. This can be also obtained by iteration. THEOREM 2. The ratio Pr of the curvature of the rth polar curve C,,-r to the curvature of the algebraic curve C,, evaluated at an ordinary point 0 of C,,, is Downloaded by guest on September 30, 2021 VOL. 33; 1947 MA THEMA TICS: KASNER AND DE CICCO 49

Pr n- 1 * (6) In particular, consider a quartic curve C4. At an ordinary point 0 of C4, construct the polar cubic C3, the polar conic C2, and the polar line Cl. These all touch C4 at 0 and the corresponding ratios of the curvatures are 2/3, 1/3, 0. 4. Before continuing with our work, it is necessary to consider some definitions. At a point P of a curve C, construct the tangent line t. From a point Q on the curve C near P, draw a line perpendicular to t and inter- secting t in the point R. The order of contact y of the curve C with its tangent line t at P is the positive number y for which lim RQ/(PR)P+1, R--P is a finite non-zero number. It is remarked that y is any positive real number. Let two curves C1 and C2 be tangent at a point P. Through a point R on the common tangent line t near P, erect a perpendicular to t, which intersects the two curves C1 and C2 in the points Qi and, Q2, both of which are near P. The ratio p of the departure of C2 to that of C1 from the common tangent line t is defined by the expression: p = lim RQ2/RQ1. If ,c(r) denotes the rth derivative of the curvature K of a curve C with respect to its arc length, and if y, the order of contact of Ci or C2 with the common tangent line t at P, is- a positive integer, then it can be shown that p = K2(Y-)/KK1(yl). According to the theorem of Mehmke-Segre, this ratio p is a projective invariant. 5. By the equations (1), (2) and (3), it is found that if 0 is an ordinary point of the algebraic curve C,,, and if the order of contact of C. with its tangent line at 0 is y, then all the polar curves of 0 with respect to C. are tangent to Cn at 0, and the order of contact of each of these polars with the common tangent line is also y. THEOREM 3. The ratio Pr of the departure of the r th polar curve C,-7 to the departure of Cn, constructed at a point 0 of Cn in the case where the order of contact with the common tangent line at 0 is y which may be any positive integer, is _(n-7n-1)(n-7-2)...(n-r -r) Pr-= (n-1)(n-2). . .(n-r) * (7 In particular, the corresponding ratios P1 P2, PS for the first, second and third polars are n-7-1 (n- -1)(n- -2) P1 = n P2 (n 1) (n 2) (n- -1)(n-7-2)(n-7-3) (n-1)(n-2)(n-3) (8) Downloaded by guest on September 30, 2021 50 MA TITEMA TICS: KASNER A ND DE CICCO PROC. N. A. S.

6. Now we consider the case where the origin 0 is a cusp of lowest order of the algebraic curve C,,. Then in equation (1), we must have that Po = 0, Pi(x, y) is identically zero, and P2(x, y) is a constant non-zero multiple of the square of a linear homogeneous form in (x, y). From equations (2) and (3), it is deduced that all the polars (r = 1, 2, .... n - 2) have a cusp of the same qualitative nature at 0. THEOREM 4. The ratio Pr of the departure of the r th polar C,-T to the departure of C,, constructed at a simple cusp 0 of C. in the case where the order of contact is 1/2, is /n- r- /2 Pr = n-2 }(9) In particular, the (nr 2)/ In particular, the corresponding ratios P1, P2, P3for the first, second and third polars are P n 13) /2 P 1n)4/2n / 15 1/2 pi t ) p2= n2 P3 (n-2)

7. In this section, we study the ratio Pr constructed at a simple cusp O of the algebraic curve C,, in the case where the order of contact with the common tangent line is greater than 1/2. We have proved the following result. THEOREM 5. Let 0 be a simple cusp of the algebraic curve C. so that the order of contact y of C, with the tangent line at 0 is an integral multiple of 1/2. The r th polar C-r- of 0 with respect to C,, also has a simple cusp at O with the same tangent line and the same order of contact y at 0. If y - (2q - 1)12, the ratio Pr of the departure of C,,-r to the departure of C, from the common tangent line at 0 is [(n -2q -l1)(n-2q -2) ..(n-2q -r)]/ Pr = r)/2 (1

In particular, the corresponding ratios Pl, P2 P3 for the first, second and third polars are (n-2q - 1 1/2 [(n 2q- 1)(n-2q-- 2)11/2 P=\n-2 P L (n -2)(n-3) J [(n - 2q - 1)(n - 2q - 2)(n - 2q - 3) 1/2 "~~L (n - 2)(n - 3)(n - 4) 1 (2 On the other hand, if -y is an integer, then Pr depends on. the coefficients of the polynomial equation defining the algebraic curve C,. It is remarked that if 0 -is a simple cusp where the order of contact is a positive integer y, we meet for the first time a case where the ratio pr is Downloaded by guest on September 30, 2021 VOL. 33, 1947 MA THEMA TICS: W. F. EBERLEIN 51

not an arithmetic invariant but actually is a differential invariant, that is, pr depends on the coefficients of the original algebraic curve C,. 8. In our later work, we shall study this ratio PT for a singular point 0 of higher order of the algebraic curve C,. We shall generalize the results of this article to algebraic surfaces and to higher manifolds in later papers. Our theorems since they deal with curvature appear to belong to differential geometry, that is, geometry in the small. But actually the notion of polar curve involves the total algebraic curve and therefore our theorems belong to geometry in the large, that is, to global geometry. * Presented to the American Mathematical Society, Feb. 1947. 1 Salmon, Treatis,e on Higher Plane Curves (1852). 2 Kasner, "Determination of the Algebraic Curves Whose Polar Curves Are Parabolas," Amer. Jour. Math., 26, 164-168 (1903).

WEAK COMPACTNESS IN BANACHSPACES I BY W. F. EBERLEIN DEPARTMENT OF MATHEMATICS, UNIVERSITY OF MICHIGAN Communicated February 5, 1947 1. In recent years the "weak" topology of Banach spaces has aroused interest, principally because a notion of weak compactness is vital to the theory of infinite dimensional linear spaces. Advances in the compactness theory, however, have been retarded by apparent dichotomy in nomenclature-"compactness" versus "bicompactness." From the standpoint of existing applications the basic tool is the weakly convergent sub- sequence or "compactness"; from the standpoint of topological structure the concept of "bicompactness" is fundamental. We prove that for weakly closed subsets of any Banach space these two seemingly disparate notions are one. 2. Let E be any Banach space with elements x and unit sphere S, E* the conjugate space with elements f and unit sphere S*, and E** the second conjugate space with elements X. Denote by ET the linear topological space (weak topology of E) formed by the elements of E under the neighborhood system U(xo) ix Ifi(x -' xo) < 6, = 1, ..., n], where the fi are arbitrary elements of E*. A like topology can be introduced in E* by taking elements from E**, but a more functional topological space TE* (weak* topology of E*) arises from the generic neighborhoods U(fo)= [f If(x) - fo(xi) < 1E i = 1, , n], where the xi are arbitrary elements of E. A space TE** = T(E*)* (weak* topology of E**) is similarly generated; and the mapping J(x) = X defined by X(f) = f(x) imbeds E Downloaded by guest on September 30, 2021