Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP Posting-Version: version B 2.10.2 9/5/84; site sjuvax.UUCP Path: utzoo!watmath!clyde!burl!ulysses!allegra!princeton!astrovax!sjuvax!bhuber From: bhuber@sjuvax.UUCP (B. Huber) Newsgroups: net.math Subject: Re: Question came up at a party Message-ID: <2731@sjuvax.UUCP> Date: Mon, 27-Jan-86 17:20:56 EST Article-I.D.: sjuvax.2731 Posted: Mon Jan 27 17:20:56 1986 Date-Received: Thu, 30-Jan-86 05:16:35 EST References: <532@well.UUCP> <144@linus.UUCP> <2597@sdcrdcf.UUCP> Reply-To: bhuber@sjuvax.UUCP (B. Huber) Distribution: net Organization: St. Joseph's University, Phila. PA. Lines: 137 Keywords: special affine curvature, projective transformations Summary: This one's a genuinely different solution! >>> Is there a formula which describes *all* conic sections, which >>> will generate a particular class of same (e.g., circles, hyperbolas) >>> when certain coefficients are plugged into it? >> >> 2 2 >> F(x,y) = Ax + Cy + Dx + Ey + F >> >>F(x,y) = 0 is the general equation representing a conic section. The following >>conditions hold: > ... >This only produces thoses ellipses, parabolas and hyperbolas that have an >axis parallel to the X and Y axis. If you want rotated versions you have to >add in the Bxy term giving the equation: > > Ax**2 + Bxy + Cy**2 + Dx + Ey + F = 0 > (Thus the family of conic sections in the plane is five-dimensional, since one can always eleminate one of the real parameters A ... F by rescaling the equation.) This is essentially the answer everyone came up with. It's a good one, but suffers on two grounds: it does not generalize to higher dimensions very nicely, and it does not reveal the geometry which inspired the original intuition that fundamentally all the conic sections are "the same". I offer two ways to remedy this, one very well-known, the other fairly obscure. One could parametrize the ellipses, parabolae, and circles in the plane by intersecting the standard cone in R**3 (given by x**2 + y**2 = z**2) with various planes not passing through the origin, and then translating and rescaling the various curves which result. A better, but equivalent way, is to move the cone and keep the plane fixed. Briefly, one moves the cone via any nonsingular linear transformation of R**3. The lines through the origin are mapped to lines through the origin, so that the image of the cone becomes an 'elliptic' cone. The curve given by intersec- ting this cone with the plane z=1 is the image of the standard circle x**2 + y**2 = 1. Thus one could uniformly parametrize all of the conic sections by the interval [0,2*pi] beginning with the standard parametrization of the unit circle. The result can be written down immediately by anyone at all familiar with projective coordinates; the result is, t ----> (a3cos(t)+b3sin(t)+c3)**(-1) * (a1cos(t)+b1sin(t)+c1, a2cos(t)+b2sin(t)+c2) where (a1 b1 c1 a2 b2 c2 a3 b3 c3) is any nonsingular real matrix. There is a redundancy in this parametrization of the conic sections; four of these parameters are really unnecessary. The result is computationally messy, but is distinguished from the others in that the conic sections are given explicit parametrizations. (Incidentally, these parametrizations are singular in general; for a small, finite number of values of t, one or both of the coordinates become infinite. These are the 'points at infinity' on the conics.) The other method is nicer in many ways. A more detailed description can be found in M. Spivak's Differential Geometry, Vol. 2, pp 52 - 58. There he shows that the conic sections are precisely those parametrized curves c: R -----> R**2 whose special affine curvature k is constant. There is a one-parameter family of representatives: A) c(t) = ((1/k)sin(kt), -(1/k**2)cos(kt)), whose image is the curve (kx)**2 + (k**2y)**2 = 1, an ellipse (k>0); B) c(t) = ((1/k)sinh(kt), -(1/k**2)cosh(kt)), whose image is the curve (kx)**2 - (k**2y)**2 = 1 (k>0); C) c(t) = (t, .5t**2), whose image is the curve y=x**2/2. (NB. These equations differ from his. I have hastily tried to correct his solutions to a simple O.D.E. I might be wrong too.) k thus amounts to one real parameter, equal to the square of the special affine curvature. (The hyperbolae have the negative curvature.) The other five parameters are those of the special affine group. In short, every conic section is the image of one of these after applying a translation (two real parameters) and then an element of SL(2,R), which is the group of 2x2 real matrices of determinant one. The beautiful thing about this particular group of transformations is that it preserves the special affine curvature. (Each standard conic is preserved by a one-dimensional subgroup of this group.) The relationship between the solutions so far is that of subgroup: the largest group around is PGL(3,R), a real 8-dimensional group which acts transitively on the conic sections in R**2. The special affine group is a five-dimensional subgroup whose orbits on the conic sections are parametrized by the special affine curvature. The euclidean group for R**2 is the 3-dimensional group of translations and rotations of the plane. Its orbits on the conic sections are parametrized by the family of all ellipses, hyperbolae, and parabolae which are 'centered at the origin' and 'straight up and down' (it would be too pedantic to describe these more fully; the language is not meant to be technical). Each member of this family is sent into itself only by a finite subgroup of the euclidean group. The elements of this family are determined by an eccentricity and a 'size'. One way to understand all this is to look at invariants of each group. The smaller the group, the more invariants it has. Here's a table. Group Invariants in the family of conic sections ____ __________________________________________ PGL(3,R) None. All are equivalent. R^2 + SL(2,R) Special affine curvature. Similarity group S.A. curvature. Euclidean group SA curvature, eccentricity. Translation group SA curvature, eccentricity, orientation. Trivial group SA curvature, eccentricity, orientation, position (two parameters there). (The similarity group includes homogeneous rescalings, as well as translations and rotations.) The natural way, then, to generalize the study of conics, is to look for in- variants of families of hypersurfaces (hyperquadrics, say) in R**n with respect to subgroups of PGL(n+1,R). For one interesting very incomplete beginning, see Spivak vol. 3 pp 113-194. Incidentally, if one replaces 'hyperquadrics' above with 'submanifolds' then what I have written is one interpretation of Klein's Erlangerprogram (?), c. 1880. Bill Huber St Joseph's University 5600 City Ave Philadelphia, PA 19131