오프셋 곡선 및 곡면의 자가 교차 검출 및 제거

학위논문(박사)--서울대학교 대학원 :공과대학 컴퓨터공학부,2020. 2. 김명수.Offset curves and surfaces have many applications in computer-aided design and manufacturing, but the self-intersections and redundancies must be trimmed away for their practical use. We present a new method for offset curve and surface trimming that detects the self-intersections and eliminates the redundant parts of an offset curve and surface that are closer than the offset distance to the original curve and surface. We first propose an offset trimming method based on constructing geometric constraint equations. We formulate the constraint equations of the self-intersections of an offset curve and surface in the parameter domain of the original curve and surface. Numerical computations based on the regularity and intrinsic properties of the given input curve and surface is carried out to compute the solution of the constraint equations. The method deals with numerical instability around near-singular regions of an offset surface by using osculating tori that can be constructed in a highly stable way, i.e., by offsetting the osculating torii of the given input regular surface. We reveal the branching structure and the terminal points from the complete self-intersection curves of the offset surface. From the observation that the trimming method based on the multivariate equation solving is computationally expensive, we also propose an acceleration technique to trim an offset curve and surface. The alternative method constructs a bounding volume hierarchy specially designed to enclose the offset curve and surface and detects the self-collision of the bounding volumes instead. In the case of an offset surface, the thickness of the bounding volumes is indirectly determined based on the maximum deviations of the positions and the normals between the given input surface patches and their osculating tori. For further acceleration, the bounding volumes are pruned as much as possible during self-collision detection using various geometric constraints imposed on the offset surface. We demonstrate the effectiveness of the new trimming method using several non-trivial test examples of offset trimming. Lastly, we investigate the problem of computing the Voronoi diagram of a freeform surface using the offset trimming technique for surfaces. By trimming the offset surface with a gradually changing offset radius, we compute the boundary of the Voronoi cells that appear in the concave side of the given input surface. In particular, we interpret the singular and branching points of the self-intersection curves of the trimmed offset surfaces in terms of the boundary elements of the Voronoi diagram.오프셋 곡선 및 곡면은 computer-aided design (CAD)와 computer-aided manufacturing (CAM)에서 널리 이용되는 연산들 중 하나이다. 하지만 실용적인 활용을 위해서는 오프셋 곡선 및 곡면에서 생기는 자가 교차를 찾고 이를 기준으로 오프셋 곡선 및 곡면에서 원래의 곡선 및 곡면에 가까운 불필요한 영역을 제거하여야한다. 본 논문에서는 오프셋 곡선 및 곡면에서 생기는 자가 교차를 계산하고, 오프셋 곡선 및 곡면에서 생기는 불필요한 영역을 제거하는 알고리즘을 제안한다. 본 논문은 우선 오프셋 곡선 및 곡면의 자가 교차점들과 그 교차점들이 기인한 원래 곡선 및 곡면의 점들이 이루는 평면 이등변 삼각형 관계로부터 오프셋 곡선 및 곡면의 자가 교차점의 제약 조건을 만족시키는 방정식들을 세운다. 이 제약식들은 원래 곡선 및 곡면의 변수 공간에서 표현되며, 이 방정식들의 해는 다변수 방정식의 해를 구하는 solver를 이용하여 구한다. 오프셋 곡면의 경우, 원래 곡면의 주곡률 중 하나가 오프셋 반지름의 역수와 같을 때 오프셋 곡면의 법선이 정의가 되지 않는 특이점이 생기는데, 오프셋 곡면의 자가 교차 곡선이 이 부근을 지날 때는 자가 교차 곡선의 계산이 불안정해진다. 따라서 자가 교차 곡선이 오프셋 곡면의 특이점 부근을 지날 때는 오프셋 곡면을 접촉 토러스로 치환하여 더 안정된 방법으로 자가 교차 곡선을 구한다. 계산된 오프셋 곡면의 자가 교차 곡선으로부터 교차 곡선의 $xyz$-공간에서의 말단 점, 가지 구조 등을 밝힌다. 본 논문은 또한 바운딩 볼륨 기반의 오프셋 곡선 및 곡면의 자가 교차 곡선 검출을 가속화하는 방법을 제시한다. 바운딩 볼륨은 기저 곡선 및 곡면을 단순한 기하로 감싸고 기하 연산을 수행함으로써 가속화에 기여한다. 오프셋 곡면의 자가 교차 곡선을 구하기 위하여, 본 논문은 오프셋 곡면의 바운딩 볼륨 구조를 기저 곡면의 바운딩 볼륨과 기저 곡면의 법선 곡면의 바운딩 볼륨의 구조로부터 계산하며 이때 각 바운딩 볼륨의 두께를 계산한다. 또한, 바운딩 볼륨 중에서 실제 오프셋 곡선 및 곡면의 자가 교차에 기여하지 않는 부분을 깊은 재귀 전에 찾아서 제거하는 여러 조건들을 나열한다. 한편, 자가 교차가 제거된 오프셋 곡선 및 곡면은 기저 곡선 및 곡면의 보로노이 구조와 깊은 관련이 있는 것이 알려져 있다. 본 논문에서는 자유 곡면의 연속된 오프셋 곡면들로부터 자유 곡면의 보로노이 구조를 유추하는 방법을 제시한다. 특히, 오프셋 곡면의 자가 교차 곡선 상에서 나타나는 가지 점이나 말단 점과 같은 특이점들이 자유 곡면의 보로노이 구조에서 어떻게 해석되는지 제시한다.1. Introduction 1 1.1 Background and Motivation 1 1.2 Research Objectives and Approach 7 1.3 Contributions and Thesis Organization 11 2. Preliminaries 14 2.1 Curve and Surface Representation 14 2.1.1 Bezier Representation 14 2.1.2 B-spline Representation 17 2.2 Differential Geometry of Curves and Surfaces 19 2.2.1 Differential Geometry of Curves 19 2.2.2 Differential Geometry of Surfaces 21 3. Previous Work 23 3.1 Offset Curves 24 3.2 Offset Surfaces 27 3.3 Offset Curves on Surfaces 29 4. Trimming Offset Curve Self-intersections 32 4.1 Experimental Results 35 5. Trimming Offset Surface Self-intersections 38 5.1 Constraint Equations for Offset Self-Intersections 38 5.1.1 Coplanarity Constraint 39 5.1.2 Equi-angle Constraint 40 5.2 Removing Trivial Solutions 40 5.3 Removing Normal Flips 41 5.4 Multivariate Solver for Constraints 43 5.A Derivation of f(u,v) 46 5.B Relationship between f(u,v) and Curvatures 47 5.3 Trimming Offset Surfaces 50 5.4 Experimental Results 53 5.5 Summary 57 6. Acceleration of trimming offset curves and surfaces 62 6.1 Motivation 62 6.2 Basic Approach 67 6.3 Trimming an Offset Curve using the BVH 70 6.4 Trimming an Offset Surface using the BVH 75 6.4.1 Offset Surface BVH 75 6.4.2 Finding Self-intersections in Offset Surface Using BVH 87 6.4.3 Tracing Self-intersection Curves 98 6.5 Experimental Results 100 6.6 Summary 106 7. Application of Trimming Offset Surfaces: 3D Voronoi Diagram 107 7.1 Background 107 7.2 Approach 110 7.3 Experimental Results 112 7.4 Summary 114 8. Conclusion 119 Bibliography iDocto

Evaluating the boundary and covering degree of planar Minkowski sums and other geometrical convolutions

AbstractAlgorithms are developed, based on topological principles, to evaluate the boundary and “internal structure” of the Minkowski sum of two planar curves. A graph isotopic to the envelope curve is constructed by computing its characteristic points. The edges of this graph are in one-to-one correspondence with a set of monotone envelope segments. A simple formula allows a degree to be assigned to each face defined by the graph, indicating the number of times its points are covered by the Minkowski sum. The boundary can then be identified with the set of edges that separate faces of zero and non-zero degree, and the boundary segments corresponding to these edges can be approximated to any desired geometrical accuracy. For applications that require only the Minkowski sum boundary, the algorithm minimizes geometrical computations on the “internal” envelope edges, that do not contribute to the final boundary. In other applications, this internal structure is of interest, and the algorithm provides comprehensive information on the covering degree for different regions within the Minkowski sum. Extensions of the algorithm to the computation of Minkowski sums in R3, and other forms of geometrical convolution, are briefly discussed

Doctor of Philosophy

dissertationThe medial axis of an object is a shape descriptor that intuitively presents the morphology or structure of the object as well as intrinsic geometric properties of the object’s shape. These properties have made the medial axis a vital ingredient for shape analysis applications, and therefore the computation of which is a fundamental problem in computational geometry. This dissertation presents new methods for accurately computing the 2D medial axis of planar objects bounded by B-spline curves, and the 3D medial axis of objects bounded by B-spline surfaces. The proposed methods for the 3D case are the first techniques that automatically compute the complete medial axis along with its topological structure directly from smooth boundary representations. Our approach is based on the eikonal (grassfire) flow where the boundary is offset along the inward normal direction. As the boundary deforms, different regions start intersecting with each other to create the medial axis. In the generic situation, the (self-) intersection set is born at certain creation-type transition points, then grows and undergoes intermediate transitions at special isolated points, and finally ends at annihilation-type transition points. The intersection set evolves smoothly in between transition points. Our approach first computes and classifies all types of transition points. The medial axis is then computed as a time trace of the evolving intersection set of the boundary using theoretically derived evolution vector fields. This dynamic approach enables accurate tracking of elements of the medial axis as they evolve and thus also enables computation of topological structure of the solution. Accurate computation of geometry and topology of 3D medial axes enables a new graph-theoretic method for shape analysis of objects represented with B-spline surfaces. Structural components are computed via the cycle basis of the graph representing the 1-complex of a 3D medial axis. This enables medial axis based surface segmentation, and structure based surface region selection and modification. We also present a new approach for structural analysis of 3D objects based on scalar functions defined on their surfaces. This approach is enabled by accurate computation of geometry and structure of 2D medial axes of level sets of the scalar functions. Edge curves of the 3D medial axis correspond to a subset of ridges on the bounding surfaces. Ridges are extremal curves of principal curvatures on a surface indicating salient intrinsic features of its shape, and hence are of particular interest as tools for shape analysis. This dissertation presents a new algorithm for accurately extracting all ridges directly from B-spline surfaces. The proposed technique is also extended to accurately extract ridges from isosurfaces of volumetric data using smooth implicit B-spline representations. Accurate ridge curves enable new higher-order methods for surface analysis. We present a new definition of salient regions in order to capture geometrically significant surface regions in the neighborhood of ridges as well as to identify salient segments of ridges

Representing rational curve segments and surface patches using semi-algebraic sets

We provide a framework for representing segments of rational planar curves or patches of rational tensor product surfaces with no singularities using semi-algebraic sets. Given a rational planar curve segment or a rational tensor product surface patch with no singularities, we find the implicit equation of the corresponding unbounded curve or surface and then construct an algebraic box defined by some additional equations and inequalities associated to the implicit equation. This algebraic box is proved to include only the given curve segment or surface patch without any extraneous parts of the unbounded curve or surface. We also explain why it is difficult to construct such an algebraic box if the curve segment or surface patch includes some singular points such as self-intersections. In this case, we show how to isolate a neighborhood of these special points from the corresponding curve segment or surface patch and to represent these special points with small curve segments or surface patches. This framework allows us to dispense with expensive approximation methods such as voxels for representing surface patches.National Natural Science Foundation of ChinaMinisterio de Ciencia, Innovación y Universidade

Effective algorithms for the study of the degree of algebraic varieties in offsetting processes

El trabajo que se presenta en esta tesis pertenece al Área del Cálculo Simbólico, y en particular, al subárea de la Geometría Algebraica (Simbólica) Efectiva para curvas y superficies. Concretamente, en esta tesis se estudia la estructura de grados del polinomio multivariable que define el objeto geométrico que resulta al aplicar procesos de offsetting. Es decir, estudiamos su grado total y sus grados parciales con respecto a cada una de las variables, incluyendo la variable distancia. Para llevar a cabo este objetivo, la tesis se compone de cuatro capítulos y dos apéndices, cuya estructura se detalla a continuación: En el Capítulo 1, titulado Preliminaries and Statement of the Problem, se introducen las nociones de offset genérica y de polinomio de la offset genérica, junto con sus propiedades básicas. En este capítulo se sientan las bases teóricas de nuestro objeto de estudio. En particular, se prueba la propiedad fundamental del polinomio de la offset genérica,que afirma que dicho polinomio especializa bien; es decir, para casi todo valor que se asigne a la variable distancia la especialización del polinomio ,de la offset genérica, es el polinomio que define a la offset para ese valor concreto tomado como distancia. Una vez establecida dicha conexión con la teoría clásica, se define el problema central de esta tesis, que es el problema del grado de la offset genérica. Además se presenta la notación y terminología asociadas a ese problema. Se incluyen también en este capítulo algunos lemas técnicos, que tratan sobre la aplicación de la resultante para el análisis de problemas de intersección de curvas. El Capítulo 2, titulado Total Degree Formulae for Plane Curves, trata del problema del grado total para la offset genérica de una curva plana. Nuestro estudio incluye el caso general en el que la curva viene dada por su ecuación implícita, y también, para curvas racionales, el caso de curvas dadas paramétricamente. En ambos casos obtenemos fórmulas eficientes para el grado total de la offset genérica. Además se presentan otras fórmulas que pueden utilizarse para el estudio teórico del grado total de la offset. En este capítulo se introducen las nociones de sistema offset-recta, curva auxiliar y puntos intrusos. Estas tres nociones juegan un papel esencial en nuestro tratamiento del problema del grado. Estas nociones se utilizan para establecer un marco común para el desarrollo de fórmulas para el grado basadas en resultantes. En el siguiente capítulo ese marco común se aplica para obtener diversas fórmulas de grado. El Capítulo 3, titulado Partial Degree Formulae for Plane Curves, es una continuación natural del capítulo precedente. Aplicando la estrategia, métodos y lenguaje del Capítulo 2, en este capítulo se completa el análisis de la estructura de grados de la offset genérica para curvas planas. En concreto, obtenemos fórmulas para calcular cualquier grado parcial de la offset genérica, y también el grado con respecto a la variable distancia. Estas fórmulas cubren tanto el caso implícito como el caso paramétrico. Además se muestran otras fórmulas que pueden utilizarse para el análisis teórico del problema del grado. El Capítulo 4, titulado Degree Formulae for Rational Surfaces, trata el problema del grado para superficies. La mayor parte del capítulo se dedica a la demostración de una fórmula de grado total para superficies racionales, dadas paramétricamente. Esta fórmula puede aplicarse siempre que la superficie generadora satisfaga cierta condición muy general. En concreto, tenemos que asumir que existe a lo sumo una cantidad finita de valores de la distancia para los que la offset de la superficie pasa por el origen. La fórmula requiere el cálculo de una resultante generalizada univariada, y del máximo común divisor de polinomios con coeficientes simbólicos. La sección final de este capítulo contiene un enfoque alternativo para el estudio de la estructura de grados de una superficie de revolución, independiente de los resultados previos de este capítulo. Con este enfoque se obtiene una solución completa y efectiva para el problema del grado en este caso. La tesis se completa con dos apéndices, que contienen, respectivamente, un resumen de las fórmulas de grado obtenidas en esta tesis y los resultados de algunos cálculos, correspondientes a demostraciones o ejemplos, que, por su longitud, resulta más conveniente incluir aquí

Computing the topology of a plane or space hyperelliptic curve

International audienceWe present algorithms to compute the topology of 2D and 3D hyperelliptic curves. The algorithms are based on the fact that 2D and 3D hyperelliptic curves can be seen as the image of a planar curve (the Weierstrass form of the curve), whose topology is easy to compute, under a birational mapping of the plane or the space. We report on a Maple implementation of these algorithms, and present several examples. Complexity and certification issues are also discussed

Effective algorithms for the study of the degree of algebraic varieties in offsetting processes

El trabajo que se presenta en esta tesis pertenece al Área del Cálculo Simbólico, y en particular, al subárea de la Geometría Algebraica (Simbólica) Efectiva para curvas y superficies. Concretamente, en esta tesis se estudia la estructura de grados del polinomio multivariable que define el objeto geométrico que resulta al aplicar procesos de offsetting. Es decir, estudiamos su grado total y sus grados parciales con respecto a cada una de las variables, incluyendo la variable distancia. Para llevar a cabo este objetivo, la tesis se compone de cuatro capítulos y dos apéndices, cuya estructura se detalla a continuación: En el Capítulo 1, titulado Preliminaries and Statement of the Problem, se introducen las nociones de offset genérica y de polinomio de la offset genérica, junto con sus propiedades básicas. En este capítulo se sientan las bases teóricas de nuestro objeto de estudio. En particular, se prueba la propiedad fundamental del polinomio de la offset genérica,que afirma que dicho polinomio especializa bien; es decir, para casi todo valor que se asigne a la variable distancia la especialización del polinomio ,de la offset genérica, es el polinomio que define a la offset para ese valor concreto tomado como distancia. Una vez establecida dicha conexión con la teoría clásica, se define el problema central de esta tesis, que es el problema del grado de la offset genérica. Además se presenta la notación y terminología asociadas a ese problema. Se incluyen también en este capítulo algunos lemas técnicos, que tratan sobre la aplicación de la resultante para el análisis de problemas de intersección de curvas. El Capítulo 2, titulado Total Degree Formulae for Plane Curves, trata del problema del grado total para la offset genérica de una curva plana. Nuestro estudio incluye el caso general en el que la curva viene dada por su ecuación implícita, y también, para curvas racionales, el caso de curvas dadas paramétricamente. En ambos casos obtenemos fórmulas eficientes para el grado total de la offset genérica. Además se presentan otras fórmulas que pueden utilizarse para el estudio teórico del grado total de la offset. En este capítulo se introducen las nociones de sistema offset-recta, curva auxiliar y puntos intrusos. Estas tres nociones juegan un papel esencial en nuestro tratamiento del problema del grado. Estas nociones se utilizan para establecer un marco común para el desarrollo de fórmulas para el grado basadas en resultantes. En el siguiente capítulo ese marco común se aplica para obtener diversas fórmulas de grado. El Capítulo 3, titulado Partial Degree Formulae for Plane Curves, es una continuación natural del capítulo precedente. Aplicando la estrategia, métodos y lenguaje del Capítulo 2, en este capítulo se completa el análisis de la estructura de grados de la offset genérica para curvas planas. En concreto, obtenemos fórmulas para calcular cualquier grado parcial de la offset genérica, y también el grado con respecto a la variable distancia. Estas fórmulas cubren tanto el caso implícito como el caso paramétrico. Además se muestran otras fórmulas que pueden utilizarse para el análisis teórico del problema del grado. El Capítulo 4, titulado Degree Formulae for Rational Surfaces, trata el problema del grado para superficies. La mayor parte del capítulo se dedica a la demostración de una fórmula de grado total para superficies racionales, dadas paramétricamente. Esta fórmula puede aplicarse siempre que la superficie generadora satisfaga cierta condición muy general. En concreto, tenemos que asumir que existe a lo sumo una cantidad finita de valores de la distancia para los que la offset de la superficie pasa por el origen. La fórmula requiere el cálculo de una resultante generalizada univariada, y del máximo común divisor de polinomios con coeficientes simbólicos. La sección final de este capítulo contiene un enfoque alternativo para el estudio de la estructura de grados de una superficie de revolución, independiente de los resultados previos de este capítulo. Con este enfoque se obtiene una solución completa y efectiva para el problema del grado en este caso. La tesis se completa con dos apéndices, que contienen, respectivamente, un resumen de las fórmulas de grado obtenidas en esta tesis y los resultados de algunos cálculos, correspondientes a demostraciones o ejemplos, que, por su longitud, resulta más conveniente incluir aquí