Efficient merging of multiple segments of B\'ezier curves

This paper deals with the merging problem of segments of a composite B\'ezier curve, with the endpoints continuity constraints. We present a novel method which is based on the idea of using constrained dual Bernstein polynomial basis (P. Wo\'zny, S. Lewanowicz, Comput. Aided Geom. Design 26 (2009), 566--579) to compute the control points of the merged curve. Thanks to using fast schemes of evaluation of certain connections involving Bernstein and dual Bernstein polynomials, the complexity of our algorithm is significantly less than complexity of other merging methods

www.elsevier.com/locate/cagd A local fitting algorithm for converting planar curves to B-splines

In this paper we present a local fitting algorithm for converting smooth planar curves to B-splines. For a smooth planar curve a set of points together with their tangent vectors are first sampled from the curve such that the connected polygon approximates the curve with high accuracy and inflexions are detected by the sampled data efficiently. Then, a G1 continuous Bézier spline curve is obtained by fitting the sampled data with shape preservation as well as within a prescribed accuracy. Finally, the Bézier spline is merged into a C2 continuous B-spline curve by subdivision and control points adjustment. The merging is guaranteed to be within another error bound and with no more inflexions than the Bézier spline. In addition to shape preserving and error control, this conversion algorithm also benefits that the knots are selected automatically and adaptively according to local shape and error bound. A few experimental results are included to demonstrate the validity and efficiency of the algorithm

АППРОКСИМАЦИЯ СОПРЯЖЕНИЯ КРИВЫХ БЕЗЬЕ С СОХРАНЕНИЕМ ПОРЯДКА ГЛАДКОСТИ И ДОПОЛНИТЕЛЬНЫМИ ОГРАНИЧЕНИЯМИ

Bezier curves are a mandatory component of the geometric core of modern computer-aided design (CAD). The article proposes a mathematical approach that makes it possible to approximate the conjugation (connection) of Bezier curves of arbitrary degree, so that at the conjugation point the conditions of smoothness (continuity) are satisfied up to the order of an equal degree of the given Bezier curves. This approach helps to represent conjugate curves of a single Bezier curve with the degree equal to the degrees of the given curves. Additional restrictions can be imposed on the conjugate curves and the approximating curve in the form of a complete coincidence with one of the given curves, or the passage of the approximating curve through a given point and the equality of the derivatives to the given values at this point. To solve these problems, we introduced two difference metrics between the given curves and the approximating curve, and formulated optimization problems with constraints in the form of equalities. We applied the method of Lagrange multipliers which solves the corresponding system of linear algebraic equations. To represent Bezier curves, it is proposed to use the basic functions of B-splines, which allows you to use the software functions included in the geometric core of modern CAD systems. This greatly simplifies the derivation of all degree derivatives for curves and, without significant changes in the future, will make it possible to extend the results to conjugation problems of B-splines. The paper provides examples of approximations using various metrics and their limitations.Кривые Безье являются обязательной составляющей геометрического ядра современных систем автоматизированного проектирования (CAD – Computer-Aided Design). В статье предлагается математический подход, позволяющий выполнить аппроксимацию сопряжения (соединения) кривых Безье произвольной степени таким образом, чтобы в точке сопряжения выполнялись условия гладкости (непрерывности) до порядка, равного степени заданных Безье кривых. Данный подход позволяет представить сопряженные кривые одной кривой Безье со степенью, равной степеням заданных кривых. На сопряженные кривые и аппроксимирующую кривую могут быть наложены дополнительные ограничения в виде полного совпадения с одной из заданных кривых или прохождения аппроксимирующей кривой через заданную точку и равенства производных заданным значениям в этой точке. Для решения указанных задач вводятся две различные метрики разности между заданными кривыми и аппроксимирующей кривой, формулируются оптимизационные задачи с ограничениями в виде равенств, для решения которых применяется метод множителей Лагранжа, который сводится к решению соответствующей системы линейных алгебраических уравнений. Для представления кривых Безье предлагается использовать базисные функции В-сплайнов, что позволяет пользоваться программными функциями входящих в геометрическое ядро современных CAD-систем. Это существенно упрощает получение производных всех степеней для кривых и без существенных изменений в последующем позволит распространить результаты на задачи сопряжения B-сплайнов. Приводятся примеры аппроксимаций с использованием различных метрик и с учетом ограничений

Method of trimming PDE surfaces

A method for trimming surfaces generated as solutions to Partial Differential Equations (PDEs) is presented. The work we present here utilises the 2D parameter space on which the trim curves are defined whose projection on the parametrically represented PDE surface is then trimmed out. To do this we define the trim curves to be a set of boundary conditions which enable us to solve a low order elliptic PDE on the parameter space. The chosen elliptic PDE is solved analytically, even in the case of a very general complex trim, allowing the design process to be carried out interactively in real time. To demonstrate the capability for this technique we discuss a series of examples where trimmed PDE surfaces may be applicable

Achieving high data reduction with integral cubic B-splines

During geometry processing, tangent directions at the data points are frequently readily available from the computation process that generates the points. It is desirable to utilize this information to improve the accuracy of curve fitting and to improve data reduction. This paper presents a curve fitting method which utilizes both position and tangent direction data. This method produces G(exp 1) non-rational B-spline curves. From the examples, the method demonstrates very good data reduction rates while maintaining high accuracy in both position and tangent direction

Adaptive isocurves based rendering for freeform surfaces

technical reportFreeform surface rendering is traditionally performed by approximating the surface with polygons and then rendering the polygons This approach is extremely common because of the complexity in accurately rendering the surfaces directly Recently?? several papers presented methods to render surfaces as sequences of isocurves Unfortunately?? these methods start by assuming that an appropriate collection of isocurves has already been derived The algorithms themselves neither automatically create an optimal or almost optimal set of isocurves so the whole surface would be correctly rendered without having pixels redundantly visited nor automatically compute the parameter spacing required between isocurves to guarantee such coverage In this paper?? a new algorithm is developed to ll these needs An algorithm is introduced that automat ically computes a set of almost optimal isocurves covering the entire surface area This algorithm can be combined with a fast curve rendering method?? to make surface rendering without polygonal approximation practica

Adaptive isocurves based rendering for freeform surfaces

Journal ArticleFreeform surface rendering is traditionally performed by approximating the surface with polygons and then rendering the polygons. This approach is extremely common because of the complexity in accurately rendering the surfaces directly. Recently, several papers presented methods to render surfaces as sequences of isocurves. Unfortunately, these methods start by assuming that an appropriate collection of isocurves has already been derived. The algorithms themselves neither automatically create an optimal or almost optimal set of isocurves so t h e whole surface would be correctly rendered without having pixels redundantly visited nor automatically compute the parameter spacing required between isocurves to guarantee such coverage. In this paper, a new algorithm is developed to fill these needs. An algorithm is introduced that automatically computes a set of almost optimal isocurves covering the entire surface area. This algorithm can be combined with a fast curve rendering method, to make surface rendering without polygonal approximation practical

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

A Graphical Method of Computing Offset Curves

Computation of offset curves is an operation critical to many computer-aided design and manufacturing (CAD/CAM) applications. Though simple on the surface, differences between the straightforward mathematical definition and the demands of CAD/CAM environment in the formulation and expression of an offset curve create a problem for which only complicated, approximate solutions are presently available. This thesis explores one of the newest methods of offset curve computation, using graphics hardware to directly compute the offset curve for arbitrary input geometry. Linear segments of the input curve are represented as meshes in 3D space, and the rendering process is used to create a field of depth values from which the offset curve is extracted as an isoline. This results in significant performance enhancements over previous, similar methods. Combined with a quantification of the errors involved in a graphical approach, these advances bring the technique closer to industrial readiness. Algorithm performance is shown to be linear with respect to geometric complexity of the input curve