From: Cassiano Durand Newsgroups: sci.math,sci.math.num-analysis Subject: Lines tangent to 2 spheres Date: Mon, 20 Apr 1998 13:39:18 -0500 Hi Everybody, I'd like to know how to parametrize the family of lines tangent to 2 spheres. I know that the common tangents to two spheres are the generators of a one-parameter family (call the parameter Alpha) of circular-section hyperboloids of one sheet, with a common axis containing the centres of the two spheres. (This family includes two limiting elements corresponding to the extremes of the valid range of Alpha: if the two spheres do not intersect then there are two tangent circular cones; if they do intersect then there is one tangent cone and there is also the set of tangents to their circle of intersection.). However I could not manage to write the parametric equations. Any help is greatly appreciated. Thanks Cassiano Durand ============================================================================== From: rusin@vesuvius.math.niu.edu (Dave Rusin) Newsgroups: sci.math,sci.math.num-analysis Subject: Re: Lines tangent to 2 spheres Date: 28 Apr 1998 04:55:27 GMT In article <353B9656.197476CE@alcor.cs.purdue.edu>, Cassiano Durand wrote: >I'd like to know how to parametrize the family of lines >tangent to 2 spheres. > >I know that... [description of surface in language I could only understand after the fact] >...However I could not >manage to write the parametric equations. That's because you seem to have the misfortune of only describing sets whose parameterization would involve parameterizing elliptic curves. With suitably drawn coordinates we may assume the spheres are centered at (1,0,0) and (-1,0,0) with radii r and s respectively. The lines may be characterized by their points of tangency (x1,y1,z1) and (x2,y2,z2) with these spheres. You then have two vectors joining these points to their centers. This gives two equations in the 6 variables to specifiy the lengths of these vectors, and two more to state that they are perpendicular to the vector between the two points. These four equations describe a surface in R^6. Obviously the group of rotations around the x-axis acts transitively on this variety (and without fixed points). We may form a subvariety C consisting of those lines in R^3 for which the midpoint of the segment of interest is in the x-y plane; then it is clear that every point in the variety is the translate of a unique element of C under the action of the circle group; that is, the variety is precisely the direct product C x (circle-group). Well, parameterizing the rotations is easy; what about C? It's a curve in R^6 specified by the previous four equations and z1+z2 = 0. Well, with a little algebra, one may play off the four equations to discover x1+x2 = (s^2-r^2)/2; with a little more algebra and z2=-z1, one can eliminate first z1^2 then x1 and x2, and get the equation 4(y1+y2) (4(y1-y2) + (r^2-s^2)(y1+y2)) =(r^2-s^2)(2+r+s)(2+r-s)(2-r+s) which is the image of the hyperbola XY=c under a linear transformation. So we can use the parameterization X=t, Y=c/t to get y1,y2,x1,x2 as rational functions of t. Then we find we must have (-64t z1)^2 = (A-t^2)(B-t^2)(C-t^2) for some expressions A, B, C in terms of r and s. So if you could parameterize the points in the curve C then (t^2, -64t z1) would be a parameterization of an elliptic curve. Now you're down to the standard question here: what kind of parameterization did you have in mind. Rational functions? No can do. Elementary functions? Sure, just extract some square roots and remember to take both branches. Natural functions? Sure -- Weierstrass pe-function. Take your pick. dave ============================================================================== From: rusin@vesuvius.math.niu.edu (Dave Rusin) Newsgroups: sci.math,sci.math.num-analysis Subject: Re: Lines tangent to 2 spheres Date: 28 Apr 1998 22:42:55 GMT I need to retract a claim that a certain parameterization cannot exist. Cassiano Durand wrote: > >I'd like to know how to parametrize the family of lines >tangent to 2 spheres. I responded with an analysis of the variety in question, including the remark >Obviously the group of rotations around the x-axis acts transitively on >this variety (and without fixed points). We may form a subvariety C >consisting of those lines in R^3 for which the midpoint of the segment of >interest is in the x-y plane; then it is clear that every point in the >variety is the translate of a unique element of C under the action of >the circle group; that is, the variety is precisely the direct product >C x (circle-group). The assertion of a product structure is incorrect in the algebraic category (there is no splitting map from the surface to the circle group). It is still true that the variety is birationally equivalent to one given by X^2 + Y^2 + (Z^2+A)(Z^2+B)(Z^2+C) where the parameters A, B, C depend on the radii of the spheres (relative to the distance between the spheres); I think the relation is 16A=(A+B)(A+C). There is as I remarked no rational parameterization of the curve which results from setting X=0. However, it _may_ indeed be true that there is a rational parameterization of this surface. Indeed, if only the "+" before the product were a "-", one could obtain a parameterization of the surface just by taking X and Y to be appropriate cubics in Z. (There is a one-parameter family of such pairs of cubics.) In particular, the corresponding complex surface admits a rational parameterization. I don't believe that answers the question one way or the other about the existence of a rational parameterization define over the reals (say), though. Someone with a better understanding of algebraic surfaces than I have might be able to address this point more carefully. Sorry to have led anyone astray. dave (I have no idea why this thread started in sci.math.num-analysis !)