Cylinders Through Five Points: Complex and Real Enumerative Geometry

Lichtblau, Daniel

doi:10.1007/978-3-540-77356-6_6

Cited by 5 publications

(12 citation statements)

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“…This shows that the cylinder through five points problem is nongeneric with respect to the theory presented in (Huber & Sturmfels, 1997). A hint as to why this is so may be gleaned from yet another computational proof of theorem 1 presented in (Lichtblau, 2007). This sort of nongeneric example is also noted in (Huber & Sturmfels, 1997).…”

Section: Remarkmentioning

confidence: 64%

“…We also work with cylinders through four points, of either extremal or fixed radius, using similar techniques. Section 6 poses some further questions regarding cylinders through five points, and proves a conjecture from (Lichtblau, 2007) about the case of an infinite solution set. Following that is a brief summary.…”

Section: Outline Of the Problem And Related Workmentioning

confidence: 97%

“…It is also clear from experiments (and from theory in Lichtblau, 2007) that there are other configurations that give no real cylinders. We now use symbolic computation to derive a particular family of such configurations.…”

Section: More Configurations That Give No Real Cylindersmentioning

confidence: 98%

“…Thus, for the no-real-cylinder examples given a uniform distribution of points in a cube we surmise that almost three out of four cases do not arise in this way. A partly proven conjecture in (Lichtblau, 2007) states, in effect, that the remaining configurations with no real solutions may be regarded as perturbations of the ones wherein one point is enclosed by the other four.…”

Section: Enumerating Real Cylindersmentioning

confidence: 99%

“…Other proofs of varying levels of complexity may be found in (Bottema & Veldkamp, 1977;Chaperon & Goulette, 2003;Devillers et al, 2003;Lichtblau, 2007). An algorithm that effectively automates finding the cardinality of generic solution sets to geometric configuration problems will be the subject of future research.…”

Section: Remarkmentioning

confidence: 99%

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Cylinders Through Five Points: Computational Algebra and Geometry

Lichtblau¹

2012

JMR

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We address the following question: Given five points in R 3 , determine a right circular cylinder containing those points. We obtain algebraic equations for the axial line and radius parameters and show that these give six solutions in the generic case. An even number (0, 2, 4, or 6) will be real valued and hence correspond to actual cylinders in R 3 . We will investigate computational and theoretical matters related to this problem. In particular we will show how exact and numeric Gröbner bases, equation solving, and related symbolic-numeric methods may be used to advantage. We will also discuss some applications.Keywords: computational geometry, enumerative geometry, Gröbner bases, nonlinear systems, symbolic-numeric computation Outline of the Problem and Related WorkGiven five points in R 3 , we are to determine all right circular cylinders containing those points. We do this by solving equations for the axial line and radius parameters. We will show that generically one obtains six solutions to these equations. Of these an even number are real valued, as the complex valued ones appear in conjugate pairs (an immediate consequence is that there is no "unique" real cylinder through five given points unless it a solution with multiplicity). Moreover there are open regions in the real configuration space that give each of these possibilities so none are disallowed.The basic problem of determining cylinders from five points may be recast in a computational geometry setting: Given five points in R 3 , find the smallest positive r and orientation parameters such that the cylinder of radius 2r with those parameters encloses tangentially the balls of radius r centered at the points.Here are some questions we will consider. The first three are classical; we address them here to illustrate the utility of symbolic computation in such investigations. The last ones are related to more recent work in computational and integral geometry.(1) Given the points and corresponding cylinder parameters, how might we display them graphically?(2) Given the cylinder parameters, how may we obtain its implicit equation as a hypersurface in R 3 ?(3) Reversing this, how can one obtain parameters from the implicit form?(4) Given five points chosen with random uniform distribution in a cube, what is the expected probability that one lies inside the convex hull of the other four (this is related to the "no real cylinder" case).(5) How might we rigorously provide, via straightforward computation, the generic number of solutions to the algebraic equations that describe cylinders through five indeterminate points.(6) Given six or more points, how do we find the coordinates of a (generically unique) cylinder in R 3 that "best" fits those points? (7) Given six or more points, how do we find the cylinder(s) of smallest radius enclosing them?The problem of finding cylinders through five points may be recast in a computational geometry setting: Given five points in R 3 , find the smallest positive r, and corresponding orientation parameters. ...

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Section: Remarkmentioning

confidence: 64%

Section: Outline Of the Problem And Related Workmentioning

confidence: 97%

Section: More Configurations That Give No Real Cylindersmentioning

confidence: 98%

Section: Enumerating Real Cylindersmentioning

confidence: 99%