Analysis and applications of pipe surfaces

Maekawa, Takashi; Patrikalakis, Nicholas M.; Sakkalis, Takis; Yu, Guoxin

doi:10.1016/s0167-8396(97)00042-3

Cited by 88 publications

(67 citation statements)

References 10 publications

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“…In response to the example of Figure 9, a theorem was published for ambient isotopic PL approximations of 1-manifolds [119], with an illustrative outcome shown in Figure 10. The proof utilizes 'pipe surfaces' from classical differential geometry [128].…”

Section: Correctly Embedded Approximations For Graphics and Applicationsmentioning

confidence: 99%

Computational topology

Blackmore¹,

Peters²

2007

Open Problems in Topology II

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Section: Correctly Embedded Approximations For Graphics and Applicationsmentioning

confidence: 99%

Computational topology

Blackmore¹,

Peters²

2007

Open Problems in Topology II

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“…As approximation is unavoidable in solid modeling, the question of whether an approximation is 'good enough' to preserve the essential features of the object is of central importance. Previous publications by the first two authors have invoked the concept of ambient isotopy, a topological notion of equivalence for admissibility of approximations to curves [21] and surfaces [7].…”

Section: Related Workmentioning

confidence: 99%

Isotopic approximations and interval solids

Sakkalis

Peters

Bisceglio

2004

Computer-Aided Design

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Given a nonsingular compact two-manifold F without boundary, we present methods for establishing a family of surfaces which can approximate F so that each approximant is ambient isotopic to F: The methods presented here offer broad theoretical guidance for a rich class of ambient isotopic approximations, for applications in graphics, animation and surface reconstruction. They are also used to establish sufficient conditions for an interval solid to be ambient isotopic to the solid it is approximating. Furthermore, the normals of the approximant are compared to the normals of the original surface, as these approximating normals play prominent roles in many graphics algorithms.The methods are based on global theoretical considerations and are compared to existing local methods. Practical implications of these methods are also presented. For the global case, a differential surface analysis is performed to find a positive number r so that the offsets F o ð^rÞ of F at distances^r are nonsingular. In doing so, a normal tubular neighborhood, FðrÞ; of F is constructed. Then, each approximant of F lies inside FðrÞ: Comparisons between these global and local constraints are given. q

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“…Applying the implicit function theorem, the level sets of the distance function are also C 2 in ν δ . Similarly, we see that the gradient field of the distance function in B δ is the unit C 1 field generated by the radial rays emanating from the boundary of M. This is an easy consequence of our choice of δ [10,15]. One can, in fact, show that the map…”

Section: Definition 52 An Embedded Manifold (M F ) Is Cmentioning

confidence: 75%

“…where c i , 1 ≤ i ≤ n is a boundary curve of M and δ(c i ) is the maximal radius of the regularly embedded pipe surfaces E r (c i ) [15]. Let r 0 be a sufficiently small positive number so that E r 0 (M ) is well defined and C 1 except for along the curves S 1,r 0 's where it is G 1 .…”

Section: The Definition Of the Envelopementioning

confidence: 99%

“…The construction of Ω proceeds locally upon each triangle in K(r) and on each line segment used to approximate subsets of Ψ(∂M ). Then these local isotopies are pasted together via Lemma 6.2, to achieve the final desired ambient isotopic PL approximation to M , where this pasting is similar to other ambient isotopic approximation techniques in the literature [11,15,19].…”

Section: Isotopy Of the Manifold With Boundarymentioning

confidence: 99%

See 1 more Smart Citation

Computational topology for reconstruction of surfaces with boundary: integrating experiments and theory

Abe

Bisceglio

Peters

et al.

International Conference on Shape Modeling and Applications 2005 (SMI' 05)

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Abstract. This paper presents new mathematical foundations for topologically correct surface reconstruction techniques that are applicable to 2-manifolds with boundary, where provable techniques previously had been limited to surfaces without boundary. This is done by an intermediate construction of the envelope (as defined herein) of the original surface. For any compact C 2 manifold M it is shown that its envelope is C 1,1 and this envelope can be reconstructed with topological guarantees. Then it is shown that there exists a piecewise linear (PL) subset of the reconstruction of the envelope that is ambient isotopic to M , whenever M is orientable. The emphasis of this paper is upon the mathematical proofs needed for these extensions, where more practical applications and examples are presented in a companion paper.

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Analysis and applications of pipe surfaces

Cited by 88 publications

References 10 publications

Computational topology

Computational topology

Isotopic approximations and interval solids

Computational topology for reconstruction of surfaces with boundary: integrating experiments and theory

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