Journal ArticleDOI
Ridges, crests and sub-parabolic lines of evolving surfaces
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This paper study in detail the structure of the ridge lines, crest lines and sub-parabolic lines on a generic surface, and on a surface which is evolving in a generic (one-parameter) family.Abstract:
The ridge lines on a surface can be defined either via contact of the surface with spheres, or via extrema of principal curvatures along lines of curvature. Certain subsets of ridge lines called crest lines have been singled out by some authors for medical imaging applications. There is a related concept of sub-parabolic line on a surface, also defined via extrema of principal curvatures. In this paper we study in detail the structure of the ridge lines, crest lines and sub-parabolic lines on a generic surface, and on a surface which is evolving in a generic (one-parameter) family. The mathematical details of this study are in Bruce et al. (1994c).read more
Citations
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Journal ArticleDOI
Curves and singularities, by J. W. Bruce and P. J. Giblin. Pp 222. 1984. ISBN 0-521-24945-7 (Hardback) £25, 27091-X (Paperback) £8·95 (Cambridge University Press)
Journal ArticleDOI
A survey of methods for recovering quadrics in triangle meshes
TL;DR: This work is interested in the class of quadric surfaces, that is, algebraic surfaces of degree 2, instances of which are the sphere, the cylinder and the cone.
Journal ArticleDOI
Multiscale Medial Loci and Their Properties
TL;DR: Three major multiscale medial axes have been proposed: iteratively pruned trees of Voronoi edges, shock loci of reaction-diffusion equations, and height ridges of medialness (cores) and each has different mathematical properties of generic branching and ending properties, singular transitions, and geometry of implied boundary.
Journal ArticleDOI
A formal classification of 3D medial axis points and their local geometry
Peter Giblin,Benjamin B. Kimia +1 more
TL;DR: This paper proposes a novel hypergraph skeletal representation for 3D shape based on a formal derivation of the generic structure of its medial axis, and derives a pointwise reconstruction formula to reconstruct a surface from this medial axis hypergraph together with the radius function.
Journal ArticleDOI
Determining the Geometry of Boundaries of Objects from Medial Data
TL;DR: This work considers a region Ω in R2 or R3 with generic smooth boundary B and Blum medial axis M, on which is defined a multivalued “radial vector field” U from points x on M to the points of tangency of the sphere at x with B, and defines a “geometric medial map” on M which corresponds to the differential geometric properties of B.
References
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Book
Elementary differential geometry
TL;DR: In this paper, the authors introduce the geometry of curves and surfaces and provide an introduction to the theory of differential geometrical geometry of surfaces and curves, as well as their properties.
MonographDOI
Curves and singularities : a geometrical introduction to singularity theory
J. W. Bruce,Peter Giblin +1 more
TL;DR: In this paper, the authors introduce a gravitational catastrophe machine and a set of generic properties of curves, such as transversality, normal values, regular values and smooth manifolds.