Thomas Cecil

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Let M be an isoparametric hypersurface in the sphere Sn with four distinct principal curvatures. Münzner showed that the four principal curvatures can have at most two distinct multiplicities m1,m2, and Stolz showed that the pair (m1,m2) must either be (2, 2), (4, 5), or be equal to the multiplicities of an isoparametric hypersurface of FKM-type,(More)
We present a low-cost stereo vision implementation suitable for use in autonomous vehicle applications and designed with agricultural applications in mind. This implementation utilizes the Census transform algorithm to calculate depth maps from a stereo pair of automotive-grade CMOS cameras. The final prototype utilizes commodity hardware, including a(More)
In this paper we extend the two dimensional methods set forth in [4], proposing a variational approach to a path planning problem in three dimensions using a level set framework. After defining an energy integral over the path, we use gradient flow on the defined energy and evolve the entire path until a locally optimal steady state is reached. We follow(More)
This is a survey of the closely related fields of taut submanifolds and Dupin submanifolds of Euclidean space. The emphasis is on stating results in their proper context and noting areas for future research; relatively few proofs are given. The important class of isoparametric submanifolds is surveyed in detail, as is the relationship between the two(More)
If M is an isoparametric hypersurface in a sphere Sn with four distrinct principal curvatures, then the principal curvatures κ1, . . . , κ4 can be ordered so that their multiplicities satisfy m1 = m2 and m3 = m4, and the cross-ratio r of the principal curvatures (the Lie curvature) equals −1. In this paper, we prove that if M is an irreducible connected(More)
In this paper we propose a variational approach to a path planning problem in 2 dimensions using a level set framework. After defining an energy integral over the path, we use gradient flow on the defined energy and evolve the entire path until a locally optimal steady state is reached. Unlike typical level set implementations where the interface being(More)
A hypersurface Mn−1 in a real space-form R, S or H is isoparametric if it has constant principal curvatures. For R and H, the classification of isoparametric hypersurfaces is complete and relatively simple, but as Élie Cartan showed in a series of four papers in 1938–1940, the subject is much deeper and more complex for hypersurfaces in the sphere S. A(More)
We prove that any connected proper Dupin hypersurface in Rn is analytic algebraic and is an open subset of a connected component of an irreducible algebraic set. We prove the same result for any connected Dupin hypersurface in Rn that satisfies a finiteness condition. Hence any taut submanifold of Rn, whose unit normal bundle satisfies the finiteness(More)
We prove that any connected proper Dupin hypersurface in R is analytic algebraic and is an open subset of a connected component of an irreducible algebraic set. We prove the same result for any connected non-proper Dupin hypersurface in R that satisfies a certain finiteness condition. Hence any taut submanifold M in R, whose tube M satisfies this finiteness(More)