쿤스 패치를 이용한 자유곡면 모델링과 베지에 곡면으로의 근사

학위논문 (석사)-- 서울대학교 대학원 : 자연과학대학 수리과학부, 2018. 2. 강명주.우리는 자유곡면의 형태를 띈 비균일 유리 B-스플라인 곡면을 쌍삼차로 혼합된 쿤스 패치를 사용하여 근사하였다. 이러한 쿤스 곡면의 설계에 있어서, 모든 부분 패치들은 네 모서리로부터 정의된 쌍삼차 곡면을 사용하여 매우 효율적으로 유계시킬 수 있다. 많은 기하학적 알고리즘 등에서 사용되는 용적 제한 체계를 쓰기 위해서는, 제어점의 명확한 값을 아는것이 필수적이다. 또한, 근사를 하는데 있어서 원래 곡면을 정의역에서 분할할 때에 분할된 정의역의 조각의 크기와 자유곡면 근사에서의 근사율의 크기(또는 근사속도)가 서로 상관관계가 있다. 그래서 우리는 쌍삼차로 혼합된 쿤스 패치의 베지에 변환으로 이것의 제어점을 구하는 방법을 제시하였고, 분할 전과 분할 후 쌍삼차 쿤스 패치로 근사한 것들의 근사율의 크기차이(또는 근사 속도)에 대하여 이전보다 구체화된 가설을 제시하였다.We approximate a freeform NURBS surface using the bicubically blended Coons patches. From the Coons surface construction, each subpatch of a NURBS object can be bounded very efficiently using the bicubic surface determined by the four boundary curves. To use the bounding volume hierarchy for a freeform surface modeling in many geometric algorithms, the explicit value of control points are essential and required. Also, there is a interrelation between the size of the domain of a surface and the approximation rate (or the approximation speed) from a subdivision in the domain. We present the control points of the bicubically blended Coons patches for a B\'{e}zier surfaces. Also we proposed a hypothesis about the decay rate in a freeform surface approximation, how to find approximation rate of errors for the bicubically blended Coons patch between the previous step of subdivision state and the subsequent step of subdivision state from the error analysis.1 Introduction 1 1.1 Introduction 1 1.2 Related works 4 2 Coons patch with Hermite Basis 6 2.1 Preliminaries 6 2.2 Bicubic Coons patch 10 2.3 Bounding Volume Hierarchy 12 3 Experimental Results 15 3.1 For an arbitrary Bezier surface 16 3.2 For the Utah teapot 18 3.3 For the Stanford Bunny 21 3.4 Error analysis and Property for the decay rate 24 4 Conclusion 27 The bibliography 28 국문초록 30Maste

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

Doctor of Philosophy

dissertationWhile boundary representations, such as nonuniform rational B-spline (NURBS) surfaces, have traditionally well served the needs of the modeling community, they have not seen widespread adoption among the wider engineering discipline. There is a common perception that NURBS are slow to evaluate and complex to implement. Whereas computer-aided design commonly deals with surfaces, the engineering community must deal with materials that have thickness. Traditional visualization techniques have avoided NURBS, and there has been little cross-talk between the rich spline approximation community and the larger engineering field. Recently there has been a strong desire to marry the modeling and analysis phases of the iterative design cycle, be it in car design, turbulent flow simulation around an airfoil, or lighting design. Research has demonstrated that employing a single representation throughout the cycle has key advantages. Furthermore, novel manufacturing techniques employing heterogeneous materials require the introduction of volumetric modeling representations. There is little question that fields such as scientific visualization and mechanical engineering could benefit from the powerful approximation properties of splines. In this dissertation, we remove several hurdles to the application of NURBS to problems in engineering and demonstrate how their unique properties can be leveraged to solve problems of interest

New geometric algorithms and data structures for collision detection of dynamically deforming objects

Any virtual environment that supports interactions between virtual objects and/or a user and objects, needs a collision detection system to handle all interactions in a physically correct or plausible way. A collision detection system is needed to determine if objects are in contact or interpenetrates. These interpenetrations are resolved by a collision handling system. Because of the fact, that in nearly all simulations objects can interact with each other, collision detection is a fundamental technology, that is needed in all these simulations, like physically based simulation, robotic path and motion planning, virtual prototyping, and many more. Most virtual environments aim to represent the real-world as realistic as possible and therefore, virtual environments getting more and more complex. Furthermore, all models in a virtual environment should interact like real objects do, if forces are applied to the objects. Nearly all real-world objects will deform or break down in its individual parts if forces are acted upon the objects. Thus deformable objects are becoming more and more common in virtual environments, which want to be as realistic as possible and thus, will present new challenges to the collision detection system. The necessary collision detection computations can be very complex and this has the effect, that the collision detection process is the performance bottleneck in most simulations. Most rigid body collision detection approaches use a BVH as acceleration data structure. This technique is perfectly suitable if the object does not change its shape. For a soft body an update step is necessary to ensure that the underlying acceleration data structure is still valid after performing a simulation step. This update step can be very time consuming, is often hard to implement and in most cases will produce a degenerated BVH after some simulation steps, if the objects generally deform. Therefore, the here presented collision detection approach works entirely without an acceleration data structure and supports rigid and soft bodies. Furthermore, we can compute inter-object and intraobject collisions of rigid and deformable objects consisting of many tens of thousands of triangles in a few milliseconds. To realize this, a subdivision of the scene into parts using a fuzzy clustering approach is applied. Based on that all further steps for each cluster can be performed in parallel and if desired, distributed to different GPUs. Tests have been performed to judge the performance of our approach against other state-of-the-art collision detection algorithms. Additionally, we integrated our approach into Bullet, a commonly used physics engine, to evaluate our algorithm. In order to make a fair comparison of different rigid body collision detection algorithms, we propose a new collision detection Benchmarking Suite. Our Benchmarking Suite can evaluate both the performance as well as the quality of the collision response. Therefore, the Benchmarking Suite is subdivided into a Performance Benchmark and a Quality Benchmark. This approach needs to be extended to support soft body collision detection algorithms in the future.Jede virtuelle Umgebung, welche eine Interaktion zwischen den virtuellen Objekten in der Szene zulässt und/oder zwischen einem Benutzer und den Objekten, benötigt für eine korrekte Behandlung der Interaktionen eine Kollisionsdetektion. Nur dank der Kollisionsdetektion können Berührungen zwischen Objekten erkannt und mittels der Kollisionsbehandlung aufgelöst werden. Dies ist der Grund für die weite Verbreitung der Kollisionsdetektion in die verschiedensten Fachbereiche, wie der physikalisch basierten Simulation, der Pfadplanung in der Robotik, dem virtuellen Prototyping und vielen weiteren. Auf Grund des Bestrebens, die reale Umgebung in der virtuellen Welt so realistisch wie möglich nachzubilden, steigt die Komplexität der Szenen stetig. Fortwährend steigen die Anforderungen an die Objekte, sich realistisch zu verhalten, sollten Kräfte auf die einzelnen Objekte ausgeübt werden. Die meisten Objekte, die uns in unserer realen Welt umgeben, ändern ihre Form oder zerbrechen in ihre Einzelteile, wenn Kräfte auf sie einwirken. Daher kommen in realitätsnahen, virtuellen Umgebungen immer häufiger deformierbare Objekte zum Einsatz, was neue Herausforderungen an die Kollisionsdetektion stellt. Die hierfür Notwendigen, teils komplexen Berechnungen, führen dazu, dass die Kollisionsdetektion häufig der Performance-Bottleneck in der jeweiligen Simulation darstellt. Die meisten Kollisionsdetektionen für starre Körper benutzen eine Hüllkörperhierarchie als Beschleunigungsdatenstruktur. Diese Technik ist hervorragend geeignet, solange sich die Form des Objektes nicht verändert. Im Fall von deformierbaren Objekten ist eine Aktualisierung der Datenstruktur nach jedem Schritt der Simulation notwendig, damit diese weiterhin gültig ist. Dieser Aktualisierungsschritt kann, je nach Hierarchie, sehr zeitaufwendig sein, ist in den meisten Fällen schwer zu implementieren und generiert nach vielen Schritten der Simulation häufig eine entartete Hüllkörperhierarchie, sollte sich das Objekt sehr stark verformen. Um dies zu vermeiden, verzichtet unsere Kollisionsdetektion vollständig auf eine Beschleunigungsdatenstruktur und unterstützt sowohl rigide, wie auch deformierbare Körper. Zugleich können wir Selbstkollisionen und Kollisionen zwischen starren und/oder deformierbaren Objekten, bestehend aus vielen Zehntausenden Dreiecken, innerhalb von wenigen Millisekunden berechnen. Um dies zu realisieren, unterteilen wir die gesamte Szene in einzelne Bereiche mittels eines Fuzzy Clustering-Verfahrens. Dies ermöglicht es, dass alle Cluster unabhängig bearbeitet werden und falls gewünscht, die Berechnungen für die einzelnen Cluster auf verschiedene Grafikkarten verteilt werden können. Um die Leistungsfähigkeit unseres Ansatzes vergleichen zu können, haben wir diesen gegen aktuelle Verfahren für die Kollisionsdetektion antreten lassen. Weiterhin haben wir unser Verfahren in die Physik-Engine Bullet integriert, um das Verhalten in dynamischen Situationen zu evaluieren. Um unterschiedliche Kollisionsdetektionsalgorithmen für starre Körper korrekt und objektiv miteinander vergleichen zu können, haben wir eine Benchmarking-Suite entwickelt. Unsere Benchmarking- Suite kann sowohl die Geschwindigkeit, für die Bestimmung, ob zwei Objekte sich durchdringen, wie auch die Qualität der berechneten Kräfte miteinander vergleichen. Hierfür ist die Benchmarking-Suite in den Performance Benchmark und den Quality Benchmark unterteilt worden. In der Zukunft wird diese Benchmarking-Suite dahingehend erweitert, dass auch Kollisionsdetektionsalgorithmen für deformierbare Objekte unterstützt werden

Collision Detection and Merging of Deformable B-Spline Surfaces in Virtual Reality Environment

This thesis presents a computational framework for representing, manipulating and merging rigid and deformable freeform objects in virtual reality (VR) environment. The core algorithms for collision detection, merging, and physics-based modeling used within this framework assume that all 3D deformable objects are B-spline surfaces. The interactive design tool can be represented as a B-spline surface, an implicit surface or a point, to allow the user a variety of rigid or deformable tools. The collision detection system utilizes the fact that the blending matrices used to discretize the B-spline surface are independent of the position of the control points and, therefore, can be pre-calculated. Complex B-spline surfaces can be generated by merging various B-spline surface patches using the B-spline surface patches merging algorithm presented in this thesis. Finally, the physics-based modeling system uses the mass-spring representation to determine the deformation and the reaction force values provided to the user. This helps to simulate realistic material behaviour of the model and assist the user in validating the design before performing extensive product detailing or finite element analysis using commercially available CAD software. The novelty of the proposed method stems from the pre-calculated blending matrices used to generate the points for graphical rendering, collision detection, merging of B-spline patches, and nodes for the mass spring system. This approach reduces computational time by avoiding the need to solve complex equations for blending functions of B-splines and perform the inversion of large matrices. This alternative approach to the mechanical concept design will also help to do away with the need to build prototypes for conceptualization and preliminary validation of the idea thereby reducing the time and cost of concept design phase and the wastage of resources

Distance based heterogeneous volume modelling.

Natural objects, such as bones and watermelons, often have a heterogeneous composition and complex internal structures. Material properties inside the object can change abruptly or gradually, and representing such changes digitally can be problematic. Attribute functions represent physical properties distribution in the volumetric object. Modelling complex attributes within a volume is a complex task. There are several approaches to modelling attributes, but distance functions have gained popularity for heterogeneous object modelling because, in addition to their usefulness, they lead to predictability and intuitiveness. In this thesis, we consider a unified framework for heterogeneous volume modelling, specifically using distance fields. In particular, we tackle various issues associated with them such as the interpolation of volumetric attributes through time for shape transformation and intuitive and predictable interpolation of attributes inside a shape. To achieve these results, we rely on smooth approximate distance fields and interior distances. This thesis deals with outstanding issues in heterogeneous object modelling, and more specifically in modelling functionally graded materials and structures using different types of distances and approximation thereof. We demonstrate the benefits of heterogeneous volume modelling using smooth approximate distance fields with various applications, such as adaptive microstructures, morphological shape generation, shape driven interpolation of material properties through time and shape conforming interpolation of properties. Distance based modelling of attributes allows us to have a better parametrization of the object volume and design gradient properties across an object. This becomes more important nowadays with the growing interest in rapid prototyping and digital fabrication of heterogeneous objects and can find practical applications in different industries

자유형상의 기하연산 가속화

We introduce compact hierarchical data structures for freeform geometric models, which are constructed in a preprocessing stage. Using these data structures, we develop efficient geometric algorithms which demonstrate interactive speed. The performance improvement is often in the range of 100 ∼ 1000 times speed up when compared with the conventional algorithms which are based on on-line recursive subdivisions of freeform curves and surfaces. For planar freeform curves, we represent the recursive subdivision structure for a G1-biarc approximation as a prebuilt hierarchical data structure. Circular arcs are considerably easier to deal with than the original curves due to the simple geometrical properties of circles. Using simple tests for circular arcs we can eliminate many redundant curve segments from further consideration in many geometric computations. For the remaining curve segments, we analyze the topological structure of the solution space and efficiently compute accurate solutions numerically or using a sequence of osculating circles to the given curves. We propose new data structures for geometric operations on freeform surfaces. For the distance related computations, we employ a bounding volume hierarchy(BVH) of NURBS surfaces using Coons patches. The BVH of freeform surfaces is represented as a hierarchy of Coons patches providing very tight approximation to the surfaces. The BVH for each Coons patch can be represented very compactly using the bilinear structure of the Coons Patch. Using the Coons BVH, we can accelerate distance-related computations (collision detection, minimum distance computation, and Hausdorff distance computation) on freeform surfaces to an interactive speed. For geometric computations related to surface normals, we build hierarchical data structures for both the unit normal vector field and a special distance function, called the support distance function, which describes the signed distance of each surface point from the origin along the surface normal direction at the point. Using the special hierarchical data structure we can trim out the majority of redundant surface patches by considering the upper envelope of the support distance function defined on the Gaussian sphere. In the final stage, we compute the precise exact 1D or 2D manifold by numerically tracing the solution space. We demonstrate the effectiveness of our compact hierarchical data structure by solving several non-trivial geometric problems such as collision detection for ten thousands NURBS models colliding each other, minimum distance computation between fairly complex models, Hausdorff distance computation for freeform surfaces in close proximity and convex hull computation for freeform surfaces. The experimental results show considerable performance improvement over the conventional algorithms in computing time as well as efficient memory usage._x0013_ 본 논문에서는 자유형상을 위한 저용량의 계층적 전처리 자료구조를 소개하고 이를 활용하여 자유형상의 여러 기한연산들을 가속화하는 알고리즘을 제시한다. 이러한 전처리 자료구조를 이용한 가속화 알고리즘은 기존의 재귀적 분할을 기반으로 하는 기하연산의 성능을 100 ∼ 1000배 가량 향상시킨다. 본 논문에서는 평면곡선의 전처리 자료구조로 평면곡선의 재귀적인 G1-biarc 근사를 사용한다. 원호들은 곡선을 잘 근사할 뿐만 아니라 곡선에 비해 훨씬 간단한 기하학적인 성질을 가지고 있다. 본 논문에서는 이러한 원호의 기하학적 성질을 활용하여 기하연산의 결과에 영향을 미치지 않는 많은 부분곡선들을 효율적으로 제거하는 방법을 제시한다. 그리고 불필요한 부분을 제거한 뒤 남은 작은 곡선 구간들에 대해서 이들이 가지는 해공간의 위상적 구조를 분석하여 해의 유일성을 확인하고 연속적인 접촉원을 이용하여 정확한 해를 수치적으로 구하는 알고리즘을 제시한다. 자유곡면의 기하연산 가속화를 위해서 본 논문은 새로운 자료구조를 소개한다. 거리와 관련된 기하연산의 가속화를 위해 본 논문은 Coons 곡면을 이용한 NURBS 곡면의 BVH(Bounding Volume Hierarchy)를 도입한다. 자유곡면의 BVH는 곡면을 잘 근사하는 계층적인 Coons 곡면들로 이루어져 있으며 각각의 Coons 곡면의 BVH는 Coons 곡면의bilinear structure를 이용하여 효율적으로 표현이 가능하다. 본 논문에서는 이러한 Coons BVH를 이용하여 거리와 관련된 자유곡면의 기하연산들 (충돌감지, 최단거리 계산, Hausdorff 거리 계산) 을 가속화 시킨다. 법선벡터와 관련된 기하연산들을 가속화하기 위해서 본 논문은 단위 법선 벡터장과 곡면상의 각점과 원점사이의 법선 방향에 대한 거리를 나타내는 support distance function이라고 하는 특수한 거리 함수에 대한 BVH들을 사용한다. 법선벡터와 관련된 기하연산들은 공통적으로 가우스사상(Gaussian Map)에서 support distance function의 상위막(Upper Envlope)을 구한다. 법선벡터장의 BVH와 support distance function의 BVH를 활용하여 상위막에 기여하지 않는 부분곡면들을 효율적으로 제거하고 남은 부분곡면들로부터 정확한 1차원 또는 2차원 다양체를 수치적 추적(Numerical Tracing)으로 구한다. 본 논문에서는 제안한 저용량의 계층적 자료구조의 효율성을 보이기 위해서 기존의 방법으로는 수행시간이 너무 오래걸려 해결하기 어렵다고 여겨지는 여러 가지 문제들(만개의 자유 낙하하는 자유곡면들 사이의 충돌감지, 복잡한 자유 곡면 사이의 최단거리 계산, 매우 가까이 있는 자유곡면 사이의 Hausdorff 거리 계산, 자유곡면의 Convex Hull 계산)을 해결하는 알고리즘을 제시한다. 기존의 알고리즘과 비교실험 결과 제안된 알고리즘이 기존의 알고리즘에 비해 상당한 수행성능향상을 보였으며 훨씬 효율적인 메모리 사용을 하였음을 확인할 수 있었다.Docto