Oleg G. Okunev

Function Representation of Solids Reconstructed from Scattered Surface Points and Contours

This paper presents a novel approach to the reconstruction of geometric models and surfaces from given sets of points using volume splines. It results in the representation of a solid by the inequality f(x,y,z) ≥ 0. The volume spline is based on use of the Greens function for interpolation of scalar function values of a chosen “carrier” solid. Our algorithm is capable of generating highly concave and branching objects automatically. The particular case where the surface is reconstructed from cross‐sections is discussed too. Potential applications of this algorithm are in tomography, image processing, animation and CAD for bodies with complex surfaces.

Fast collision detection between complex solids using rasterizing graphics hardware

We present an efficient method for detecting collisions between complex solid objects. The method features a stable processing time and low sensitivity to the complexity of contact between objects. The algorithm handles both concave and convex objects; however, the best performance is achieved when at least one object is convex in the proximity of the collision zone (our techniques check the required convexity property as a byproduct of the calculations). The method achieves real-time performance when calculations are supported by the standard functionality of graphics hardware available on high-end workstations.

Hierarchic shape description via singularity and multiscaling

We introduce a new concept of ridges, ravines and related structures (skeletons) associated with surfaces in three-dimensional space that generalizes the medial axis transformation approach. The concept is based on singularity theory and involves both local and global geometric properties of the surface; it is invariant with respect to translations and rotations of the surface. It leads to a method of hierarchic description of surfaces that yields new approaches to shape coding, rendering and design. The extraction of the features is based on differential geometry of surfaces with consequent segregation via multiscale analysis. Terrain feature recognition, dental shape reconstruction and medical imagery are a partial list of applications.<<ETX>>

RIDGES AND RAVINES: A SINGULARITY APPROACH

We propose a new approach for recognition, description and extraction of ridges and ravines based on singularity theory; the approach may be used for shape coding and animation.

A method for constructing examples of M-equivalent spaces

Abstract Two retractions r1 and r2 of a space X are called parallel if r1 ∘ r2 = r1 and r2 ∘ r1 = r2. Images of the space X under parallel retractions are called parallel retracts of X. We show that if K1 and K2 are parallel retracts of a space X, then the R-quotient spaces X K 1 and X K 2 are M-equivalent (i.e., their free topological groups in the sense of Markov are topologically isomorphic). This assertion yields a number of examples of M-equivalent spaces.

Articulation simulation for an intelligent dental care system

Abstract CAD/CAM techniques are used increasingly in dentistry for design and fabrication of teeth restorations. An important issue is preserving occlusal contacts of teeth after restoration. Traditional techniques based on the use of casts with mechanical articulators require manual adjustment of the occlusal surface, which becomes impractical when hard restoration materials like porcelain are used; they are also time- and labour-consuming. Most existing computer systems ignore completely such an articulation check, or perform the check at the level of a tooth and its immediate neighbours. We present a new mathematical model and a related user interface for global articulation simulation, developed for the Intelligent Dental Care System project. The aim of the simulation is elimination of the use of mechanical articulators and manual adjustment in the process of designing dental restorations and articulation diagnostics. The mathematical model is based upon differential topological modelling of the jaws considered as a mechanical system. The user interface exploits metaphors that are familiar to dentist from everyday practice. A new input device designed specifically for use with articulation simulation is proposed.

On Lindelof sets of continuous functions

Abstract We prove that, under MA+¬CH, if X is a compact, separable space, then every subspace Y of C p ( X ) with Y n Lindelof for all n ϵ N has countable network.

Lindelöf powers and products of function spaces

We give criteria for finite and countable powers of a space similar to the Michael line being Lindelöf. As applications, we give examples related to Lindelöf property in products of spaces of Michael line type and in products of spaces of continuous functions on separable σ-compact spaces. All spaces considered below are assumed to be Tychonoff (= completely regular Hausdorff). We denote by Cp(X) the space of all continuous real-valued functions endowed with the topology of pointwise convergence on X ; this topology can be obtained as the restriction of the Tychonoff product topology on the set R of all real-valued functions onX to its subset C(X) (see [Arh1]). Cp(X, 2) is the subspace of Cp(X) consisting of all functions to 2 = {0, 1}. The symbols ω, R, I and C stand for the set of naturals, the real line, the segment [0, 1], and the Cantor cube 2. If P and Q are sets, then P denotes the set of all functions from Q to P ; if κ is a cardinal, then X is the κth power of X (with the Tychonoff product topology); the projection of X to its ith factor is denoted by πi. For j ∈ 2 and σ ∈ 2, denote σaj = σ ∪ {〈i, j〉} ∈ 2. The symbol c denotes the cardinality of the continuum. Polish spaces are separable completely metrizable spaces.

Density properties and points of uncountable order for families of open sets in function spaces

Abstract We characterize the given extent in finite powers of X in terms of the topology of C p ( X ). It is shown that many properties of C p ( X ) are determined by dense subsets of C p ( X ). We introduce the density tightness and establish that for compact spaces of π -weight ⩽ ω 1 countable density tightness implies countable π -character.

Evaluation of human jaw articulation [computer animation]

Computer-aided diagnosis of occlusal disorders and design of dental restorations requires an automated evaluation of jaw occlusion and chewing ability. This requires simulation of the motion of the jaws and characterization of contacts between the surfaces of teeth. We propose approaches to evaluation of the load on teeth and of the grinding process. These characteristics are derived in interactive time, and are based on distance maps and topological structure of the contact zones. The proposed approaches are general and usable in applications where modeling of contact between objects with complex geometry is required.<<ETX>>