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.

this report, the reconstruction and modeling of cerebral vascular structures are addressed. The main contributions brought are the following:

We present an efficient and robust algorithm to compute the intersection curve of two ringed surfaces, each being the sweep ∪uC u generated by a moving circle. Given two ringed surfaces ∪uC u 1 and ∪vC v 2, we formulate the condition C u 1 ∩ Cv 2 �=∅(i.e., that the intersection of the two circles C u 1 and Cv 2 is nonempty) as a bivariate equation λ(u,v) = 0 of relatively low degree. Except for redundant solutions and degenerate cases, there is a rational map from each solution ofλ(u,v) = 0 to the intersection point C u 1 ∩ Cv 2. Thus it is trivial to construct the intersection curve once we have computed the zero-set of λ(u,v) = 0. We also analyze exceptional cases and consider how to construct the corresponding intersection curves. A similar approach produces an efficient algorithm for the intersection of a ringed surface and a ruled surface, which can play an important role in accelerating the ray-tracing of ringed surfaces. Surfaces of linear extrusion and surfaces of revolution reduce their respective intersection algorithms to simpler forms than those for ringed surfaces and ruled surfaces. In particular, the bivariate equation λ(u,v) = 0 is reduced to a decomposable form, f (u) = g(v)or�f(u) − g(v)�=|r(u)|, which can be solved more efficiently than the

Given a nonsingular compact 2-manifold ¦ without boundary, we present methods for establishing a family of surfaces which can approximate ¦ so that each approximant is ambient isotopic to ¦. 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 § so that the offsets ¦©¨�����§� � of ¦ at distances �� § are nonsingular. In doing so, a normal tubular neighborhood, ¦���§� � , of ¦ is constructed. Then, each approximant of ¦ lies inside ¦���§� �. Comparisons between these global and local constraints are given. Key words: Ambient isotopy; computational topology; surface reconstruction; interval solids; offsets and deformations; reverse engineering Preprint submitted to Elsevier Science 30 July 2003 1

Abstract. New computational topology techniques are presented for surface reconstruction of 2-manifolds with boundary, while rigorous proofs have previously 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 embedded in R 3, it is shown that its envelope is C 1,1. 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 formal mathematical proofs needed for these extensions. (Practical application examples have already been published in a companion paper.) Possible extensions to non-orientable manifolds are also discussed. The mathematical exposition relies heavily on known techniques from differential geometry and topology, but the specific new proofs are intended to be sufficiently specialized to prompt further algorithmic discoveries.

This paper proposes a new representation scheme of the cerebral blood vessels. This model provides information on the semantics of the vascular structure: the topological relationships between vessels and the labeling of vascular accidents such as aneurysms and stenoses. In addition, the model keeps information of the inner surface geometry as well as of the vascular map volume properties, i.e. the tissue density, the blood flow velocity and the vessel wall elasticity.

We present a method to reconstruct a pipe or a canal surface from a point cloud (a set of unorganized points). A pipe surface is defined by a spine curve and a constant radius of a swept sphere, while a variable radius may be used to define a canal surface. In this paper, by using the shrinking and moving least-squares methods, we reduce a point cloud to a thin curve-like point set which will be approximated to the spine curve of a pipe or canal surface. The distance between a point in the thin point cloud and a corresponding point in the original point set represents the radius of the pipe or canal surface. 1.

The unresolved subtleties of floating point computations in geometric modeling become considerably more difficult in animations and scientific visualizations. Some emerging solutions based upon topological considerations will be presented. A novel geometric seeding algorithm for Newton’s method was used in experiments to determine feasible support for these visualization applications. 1 Computing the pipe surface radius Parametric curves have been shown to have a particular neighborhood whose boundary is non-self-intersecting [5]. It has also been shown that specified movements of the curve within this neighborhood preserve the topology of the curve [9, 8], as is desired in visualization. This neighborhood is defined by a single value, which is the radius of a pipe surface, where that radius depends on curvature and the minimum length over those line segments which are normal to the curve at both endpoints of the line segment [5]. Since computation of curvature is a well-treated problem, the focus of this paper is efficient and accurate floating point techniques to compute the other dependeancy for that radius.

We present a method to reconstruct a pipe or a canal surface from a set of unorganized points. A pipe surface is de ned by a spine curve and a constant radius of a swept sphere, while a variable radius can be used to de ne a canal surface. In this paper, by using the shrinking and moving least-squares methods, a point cloud is reduced to a thin curve-like point set that can be easily approximated with the spine curve of a pipe or canal surface. The distance between a point in the thin point cloud and a corresponding point in the original point set represents the radius of a pipe or canal surface. Keywords: pipe surface, canal surface, surface reconstruction Abbreviated article title: Reconstruction of pipe and canal surfaces Introduction A pipe surface is de ned as the envelope of a set of spheres with a constant radius r and with centers on a spine curve C(t). A canal surface is a generalization of a pipe surface, where a variable radius function r(t) can be used instead of r. Pipe a...

Abstract. The art of animation relies uopn modeling objects that change over time. A sequence of static images is displayed to produce an illusion of motion, which is frequently trusted to be topologically meaningful. A careful analysis exposes that formal topological guarantees are often lacking. This lack of formal justification can lead to subtle, but significant, flaws regarding topological integrity. A modified approach is proposed that integrates topological rigor with a continuous model of time. Examples will be given for splines widely used in many applications, with particular emphasis upon scientific visualization for molecular modeling. Moreover, the approach of choosing a family of functions and studying their topological properties over time should be broadly applicable to other domains. Prototype animations are available for viewing over the web.