Developable Surfaces from Arbitrary Sketched Boundaries

International audienceDevelopable surfaces are surfaces that can be unfolded into the plane with no distortion. Although ubiquitous in our everyday surroundings, modeling them using existing tools requires significant geometric expertise and time. Our paper simplifies the modeling process by introducing an intuitive sketch-based approach for modeling developables. We develop an algorithm that given an arbitrary, user specified 3D polyline boundary, constructed using a sketching interface, generates a smooth discrete developable surface that interpolates this boundary. Our method utilizes the connection between developable surfaces and the convex hulls of their boundaries. The method explores the space of possible interpolating surfaces searching for a developable surface with desirable shape characteristics such as fairness and predictability. The algorithm is not restricted to any particular subset of developable surfaces. We demonstrate the effectiveness of our method through a series of examples, from architectural design to garments

Smooth quasi-developable surfaces bounded by smooth curves

Computing a quasi-developable strip surface bounded by design curves finds wide industrial applications. Existing methods compute discrete surfaces composed of developable lines connecting sampling points on input curves which are not adequate for generating smooth quasi-developable surfaces. We propose the first method which is capable of exploring the full solution space of continuous input curves to compute a smooth quasi-developable ruled surface with as large developability as possible. The resulting surface is exactly bounded by the input smooth curves and is guaranteed to have no self-intersections. The main contribution is a variational approach to compute a continuous mapping of parameters of input curves by minimizing a function evaluating surface developability. Moreover, we also present an algorithm to represent a resulting surface as a B-spline surface when input curves are B-spline curves.Comment: 18 page

Surface discretisation with rectifying strips on Geodesics

The use of geodesic curves of surfaces has enormous potential in architecture due to their multiple properties and easy geometric control using digital graphic tools. Among their numerous properties, the geodesic curves of a surface are the paths along which straight strips can be placed tangentially to the surface. On this basis, a graphical method is proposed to discretize surfaces using straight strips, which optimizes material consumption since rectangular straight strips take advantage of 100% of the material in the cutting process. The contribution of the article consists of presenting the geometric constraints that characterize this type of panelling from the idea of “rectifying surface”, considering the material inextensible. Experimental prototypes that have been part of the research are also described and the final theoretical results are presented

Surface Patches for 3D Curve Network Based on Its Design Intent

학위논문 (석사)-- 서울대학교 대학원 : 기계항공공학부, 2014. 2. 이건우.최근 스케치 기반 인터페이스를 이용한 자연스러운 기하모델링 방법들이 제시됨에 따라, 스케치를 통해 얻어진 3차원 곡선 네트워크로부터 모델을 생성하는 방법에 대한 다양한 연구가 진행되어왔다. 하지만 3차원 곡선들로부터 곡면 패치를 생성하는 것은 기하학적 정보의 부재로 인해 최적의 해가 존재하기 보다는 다양한 방법으로 해석할 수 있는 모호한 문제이다. 이 논문에서는 3차원 곡선 네트워크로부터 곡면 패치를 생성하는 알고리듬을 제안하며, 문제의 모호성을 해결하기 위해 스케치 특성에 따른 기하 구속조건을 추가적으로 부가한다. 각각의 기하 구속조건들은 곡면의 법선 벡터에 관하여 수식화되며, 최적 해를 찾는 방법으로 디자인 의도를 고려한 곡면의 법선 벡터를 추정한다. 이렇게 추정된 법선 벡터를 이용하여 곡선 네트워크가 이루는 곡면 모델을 생성하게 된다. 제안한 알고리듬은 다양한 곡선 네트워크에 대한 모델 생성을 통해 검증하였으며, 디자인 의도를 충분히 반영하기 위해 곡선에 기하 구속조건을 정의하여 새로운 기하정보를 추정했다는 점에서 기존의 논문들에 비해 시각적으로 합리적인 결과를 얻었다. 이 연구의 3차원 곡선의 기하정보를 추정하는 기술은 스케치를 기반으로 하는 모델링 방식에서 새로운 정보를 추출하기 위한 방법으로 사용될 수 있다. 더 나아가, 곡선 네트워크의 각각의 곡면에서 구한 곡선의 법선 벡터들은 인접 곡면의 법선 벡터 정보와 더해져 사용자 의도에 더욱 가까운 곡면 모델을 얻을 수 있을 것이다. 또한 곡선 네트워크 생성기술과의 결합은 3차원 모델링의 새로운 프레임워크를 제시할 수 있을 것으로 기대된다.초록 목차 그림 목차 표 목차 제 1 장 서론 1 1.1 곡선 네트워크 2 1.2 3차원 곡면 모델 생성 3 1.3 디자인 의도를 고려한 모델 생성 4 제 2 장 관련 연구 6 2.1 스케치 기반 모델링 인터페이스 6 2.2 곡선 네트워크로부터 곡면 패치/모델 생성 8 제 3 장 전체 과정 개괄 12 3.1 곡면 법선 벡터의 추정 12 3.2 곡면 모델 생성 13 제 4 장 기하정보 추정 15 4.1 기하정보 추정을 위한 기준 17 4.1.1 스케치의 연속성 17 4.1.2 곡선의 접선 벡터 조건 17 4.1.3 닫힌곡선 스케치 18 4.1.4 곡면 굽힘 최소화 18 4.2 법선 벡터 추정 20 4.2.1 회전 최소화 프레임 21 4.2.2 회전 최소화 프레임의 경계조건 23 4.2.3 곡면 굽힘 에너지 최소화 24 제 5 장 곡면 모델 생성 27 5.1 초기 메시 생성 27 5.2 메시 Refine 28 5.3 추정 법선 벡터 적용 31 제 6 장 결과 36 제 7 장 결론 42 참고문헌 43 Abstract 49Maste

Delaunay-restricted Optimal Triangulation of 3D Polygons

Triangulation of 3D polygons is a well studied topic of research. Existing methods for finding triangulations that minimize given metrics (e.g., sum of triangle areas or dihedral angles) run in a costly O(n4) time [BS95,BDE96], while the triangulations are not guaranteed to be free of intersections. To address these limitations, we restrict our search to the space of triangles in the Delaunay tetrahedralization of the polygon. The restriction allows us to reduce the running time down to O(n2) in practice (O(n3) worst case) while guaranteeing that the solutions are intersection free. We demonstrate experimentally that the reduced search space is not overly restricted. In particular, triangulations restricted to this space usually exist for practical inputs, and the optimal triangulation in this space approximates well the optimal triangulation of the polygon. This makes our algorithms a practical solution when working with real world data

Nested Explorative Maps: A new 3D canvas for conceptual design in architecture

International audienceIn this digital age, architects still need to alternate between paper sketches and 3D modeling software for their designs. Indeed, while 3D models enable to explore different views, creating them at very early stages might reduce creativity since they do not allow to superpose several tentative designs nor to refine them progressively, as sketches do. To enable exploratory design in 3D, we introduce Nested Explorative Maps, a new system dedicated to interactive design in architecture. Our model enables coarse to fine sketching of nested architectural structures, enabling to progressively sketch a 3D building from floor plan to interior design, thanks to a series of nested maps able to spread in 3D. Each map allows the visual representation of uncertainty as well as the interactive exploration of the alternative, tentative options. We validate the model through a user study conducted with professional architects, enabling us to highlight the potential of Nested Explorative Maps for conceptual design in architecture.En cette ère du numérique, les architectes doivent encore alterner entre le croquis papier et logiciels de modélisation 3D afin de réaliser leurs conceptions. En effet, les modèles 3D permettent d’explorer différentes vues mais leur création à un stade très précoce peut impliquer une perte de la créativité car ils ne permettent pas de superposer plusieurs plans provisoires ni de les affiner progressivement, comme le font les esquisses. Pour permettre la conception exploratoire dans l'espace 3D, nous présentons Nested Explorative Maps, un nouveau système dédié à la conception interactive en architecture. Notre modèle permet de dessiner du grossier aux détails des structures architecturales imbriquées, afin de dessiner progressivement un bâtiment en 3D, du plan à la décoration intérieure, grâce à une série de cartes imbriquées capables de se répandre en 3D. Chaque carte permet de représenter visuellement l’incertitude et d’explorer de manière interactive les différentes options possibles. Une étude utilisateur réalisée auprès d'architectes professionnels nous a permis de valider notre modèle et de mettre en évidence le potentiel des cartes exploratoires imbriquées pour la conception conceptuelle en architecture

Memoization method for storing of minimum-weight triangulation of a convex polygon

This study presents a practical view of dynamic programming, specifically in the context of the application of finding the optimal solutions for the polygon triangulation problem. The problem of the optimal triangulation of polygon is considered to be as a recursive substructure. The basic idea of the constructed method lies in finding to an adequate way for a rapid generation of optimal triangulations and storing - them in as small as possible memory space. The upgraded method is based on a memoization technique, and its emphasis is in storing the results of the calculated values and returning the cached result when the same values again occur. The significance of the method is in the generation of the optimal triangulation for a large number of n. All the calculated weights in the triangulation process are stored and performed in the same table. Results processing and implementation of the method was carried out in the Java environment and the experimental results were compared with the square matrix and Hurtado-Noy method

Non-smooth developable geometry for interactively animating paper crumpling

International audienceWe present the first method to animate sheets of paper at interactive rates, while automatically generating a plausible set of sharp features when the sheet is crumpled. The key idea is to interleave standard physically-based simulation steps with procedural generation of a piecewise continuous developable surface. The resulting hybrid surface model captures new singular points dynamically appearing during the crumpling process, mimicking the effect of paper fiber fracture. Although the model evolves over time to take these irreversible damages into account, the mesh used for simulation is kept coarse throughout the animation, leading to efficient computations. Meanwhile, the geometric layer ensures that the surface stays almost isometric to its original 2D pattern. We validate our model through measurements and visual comparison with real paper manipulation, and show results on a variety of crumpled paper configurations