From: rusin@vesuvius.math.niu.edu (Dave Rusin) Newsgroups: sci.math Subject: Re: Quadrics: geometric parameterization? Date: 28 May 1997 17:20:51 GMT In article , Rainer Dunker wrote: >Quadrics in 3D-space are usually described in the polynomial form >a*x^2 + b*y^2 + c*z^2 + d*ab + e*ac + ... = 0, which is not very >suggestive. Also, it's hardly predictable how much the shape is >altered by a certain change of a coefficient value (or, is it?). > >Thus, for the sake of greater clarity: Given a coefficient vector for >the above equation, is it possible to retrieve a set of geometric >properties which fully describes the quadric (as e.g. a cylinder is >determined by its axis position, axis orientation and radius), and >which is valid for all quadric shape classes (sphere, cone, ...)? I'm not sure I understand the request. You have on the one hand a 10-dimensional real vector space, the set of quadratic polynomials in 3 variables. On the other hand you have the set of their solution sets (their zero loci). The Euclidean group, generated by all translations, rotations, and reflections, acts on both these sets. I'm guessing you are looking for descriptions of the equivalence classes (orbits) under the action of this group in both sets. Algebraically, it's fairly easy to see what's going on. Every quadratic may be written v A v^t + w v + g for a unique symmetric matrix v , a unique vector w, and a unique constant g. There is an orthogonal matrix M such that M A M^t is diagonal (and M is unique up to a permutation unless A has repeated eigenvalues). Replacing v by v M, we may assume A is diagonal, that is, the quadratic is a x^2 + b y^2 + c z^2 + d x + e y + f z + g If a<>0, there is a (unique) linear substitution under which the linear term vanishes, so we may assume d=0. Likewise if b<>0 we may assume e=0, and similarly either c=0 or f=0. Actually if more than one of the first three coefficients is zero, we can perform an additional rotation to ensure that at most one of the linear terms is zero. Moreover, if any quadratic coefficient is zero but the corresponding linear term is not, a further translation allows us to assume the constant term is zero. Thus every quartic is equivalent under this group action to precisly one of the following types: a x^2 + b y^2 + c z^2 + g (the "generic" case) a x^2 + b y^2 + f z a x^2 + b y^2 + g a x^2 + f z a x^2 + g f z g where a, b, c, and f, if present, are nonzero. You can then use scalings along the three axes to assume a, b, and c are equal to +-1 (or zero), but your use of the word "geometric" suggests you are not interested in equivalence classes under this larger group of all invertible substitutions. The effect of these transformations on the zero loci is to translate to the origin and rotate along the principal axes. Note that although there are 4-parameter families of equivalence classes of quadratic polynomials, there are only 3-parameter families of quadric surfaces since two polynomials which differ only by a scalar multiple determine the same surface. Each algebraic substitution corresponds to a translation or rotation of the corresponding quadric surface (we don't use reflections since all these surfaces have a plane of symmetry already). In the generic case as you may know, the surface is an ellipsoid, if a, b, c have one sign and g the opposite sign empty, if a, b, c have one sign and g the same sign a hyperboloid of two sheets, if three of a,b,c,g have one sign a hyperboloid of one sheet, if not. You can usually interpret the three parameters geometrically. For example, if a, b, c, d are all positive, they can easily be used to compute the lengths of the principal axes of this ellipsoid. I didn't know from your question how interested you were in the finer points of this analysis -- the special cases are boring or interesting depending on your perspective. For example, the distinct quadratics with a, b, c all positive but g<0 cannot be distinguished by geometric conditions because their real loci are empty! Likewise those with g=0 have a singular point at the origin, and I suppose are not usually considered quadric surfaces at all. Here's how the other cases look: a x^2 + b y^2 + c z^2 (i.e., g=0): this is an elliptic cone, unless a, b, and c have the same sign (in which case it's a single point). a x^2 + b y^2 + f z : a paraboloid, if a and b have the same sign, and a saddle surface otherwise. a x^2 + b y^2 + g : if a and b have the same sign, this is a cylinder, if g has the opposite sign; empty, if g has the same sign; a line, if g=0. If a and b have opposite signs this is a hyperbolic sheet if g <>0; two crossed sheets if g=0. a x^2 + f z: a parabolic sheet a x^2 + g : If g has the same sign as a, this is empty; if g has the oppsite sign, it's two planes; if g=0, it's one plane. f z: a plane g : the zero locus is either nothing or everything, according to whether g<>0 or g=0! dave