Common Tangents to Four Unit Balls in R 3

Abstract

Abstract. We answer a question of David Larman, by proving the following result. Any four unit balls in three-dimensional space, whose centers are not collinear, have at most twelve common tangent lines. This bound is tight. IntroductionThe screen of a computer monitor consists of small pixels. Suppose that we are given a three-dimensional scene consisting of several objects and a viewpoint. Generating an image of this scene ("rendering" the scene) is a basic task in computer graphics and in computational geometry, which amounts to determining the visible object(s) at each pixel. The methods developed for the solution of such problems have an extensive literature under the labels "ray tracing" and "hidden surface removal" (see, e.g., [7] and [20]). This field has served as a rich source of problems on geometric, combinatorial, and algebraic properties of systems of lines in their interaction with geometric objects.For instance, we can assume that all of our objects are unit balls in a region A, and we want to determine which balls are not visible from any viewpoint outside of A (see [29]

Cylinders Through Five Points: Complex and Real Enumerative Geometry

“…To understand why this cylinder problem is not generic for the mixed volume computation, note that the initials are identical up to a constant multiple; any random perturbation (say, add b 3 to the first polynomial) will give only one solution at infinity, and therefore yield 8 finite solutions. 15 V bad for which we obtain six real cylinders provided we count solutions by multiplicity.…”

“…This ansatz would lead us to expect twice as many solutions for this problem as we obtained for counting cylinders through five points. That there are in fact twelve (not necessarily real valued) cylinders of given radius through four generically placed points is a theorem in [15]. In the special case that the points are coplanar, that there are eight such cylinders is a result of [18].…”

“…If the tetrahedron edge length is 3 then the common cylinder radius is 9 10. It is related to a configuration from [15], wherein we have four points and a fixed radius for which there are twelve real cylinders through the points. For that we take vertices of one of the tetrahedra as our four points, and 9 10 as the common radius.…”

“…We can use the computational construction (0) to shed light on the problem of counting the number of cylinders of a given fixed radius through four points (which, as noted in [15], is equivalent to the problem of counting the number of lines simultaneously tangent to four given spheres of equal radius). As the radius is fixed (say, to 1), we are no longer free to rescale so we would use x 1 , 0, 0 for our second point.…”

“…To specify the former we use "real cylinders". Typically we will use the term "parameters" to refer to coordinates in the configuration space of the five points (which we may identify with 15 or even 15 ). The values that specify a cylinder, to wit, radius and axial line features, are generally referred to as "variables" (since they are what we solve for in finding cylinders) or as "cylinder parameters" to distinguish them from the coordinate parameters already mentioned.…”