Quasi-Bezier curves integrating localised information

Bezier curves (BC) have become fundamental tools in many challenging and varied applications, ranging from computer-aided geometric design to generic object shape descriptors. A major limitation of the classical Bezier curve, however, is that only global information about its control points (CP) is considered, so there can often be a large gap between the curve and its control polygon, leading to large distortion in shape representation. While strategies such as degree elevation, composite BC, refinement and subdivision reduce this gap, they also increase the number of CP and hence bit-rate, and computational complexity. This paper presents novel contributions to BC theory, with the introduction of quasi-Bezier curves (QBC), which seamlessly integrate localised CP information into the inherent global Bezier framework, with no increase in either the number of CP or order of computational complexity. QBC crucially retains the core properties of the classical BC, such as geometric continuity and affine invariance, and can be embedded into the vertex-based shape coding and shape descriptor framework to enhance rate-distortion performance. The performance of QBC has been empirically tested upon a number of natural and synthetically shaped objects, with both qualitative and quantitative results confirming its consistently superior approximation performance in comparison with both the classical BC and other established BC-based shape descriptor methods

Dynamic Bezier curves for variable rate-distortion

Bezier curves (BC) are important tools in a wide range of diverse and challenging applications, from computer-aided design to generic object shape descriptors. A major constraint of the classical BC is that only global information concerning control points (CP) is considered, consequently there may be a sizeable gap between the BC and its control polygon (CtrlPoly), leading to a large distortion in shape representation. While BC variants like degree elevation, composite BC and refinement and subdivision narrow this gap, they increase the number of CP and thereby both the required bit-rate and computational complexity. In addition, while quasi-Bezier curves (QBC) close the gap without increasing the number of CP, they reduce the underlying distortion by only a fixed amount. This paper presents a novel contribution to BC theory, with the introduction of a dynamic Bezier curve (DBC) model, which embeds variable localised CP information into the inherently global Bezier framework, by strategically moving BC points towards the CtrlPoly. A shifting parameter (SP) is defined that enables curves lying within the region between the BC and CtrlPoly to be generated, with no commensurate increase in CP. DBC provides a flexible rate-distortion (RD) criterion for shape coding applications, with a theoretical model for determining the optimal SP value for any admissible distortion being formulated. Crucially DBC retains core properties of the classical BC, including the convex hull and affine invariance, and can be seamlessly integrated into both the vertex-based shape coding and shape descriptor frameworks to improve their RD performance. DBC has been empirically tested upon a number of natural and synthetically shaped objects, with qualitative and quantitative results confirming its consistently superior shape approximation performance, compared with the classical BC, QBC and other established BC-based shape descriptor techniques

CAD interface and framework for curve optimisation applications

Computer Aided Design is currently expanding its boundaries to include more design features in its processes. Design is identified as an iterative process converging to solutions satisfying a set of constraints. Its close relation with optimisation indicate that there is strong potential for the integration of optimisation and CAD. The problem addressed in this thesis lies in interfacing the geometric representation of design with other non-geometric aspects. The example of free-form curve modelling is taken to investigate such relationships. Assumptions are made that Optimisation is powered by Evolutionary Computing algorithms like Genetic Algorithms (GA). The geometric definition of curves is commonly supported by NURBS, whose construction constraints are defined locally at the data points. Here the NURBS formulation is used with GA in an attempt to provide complementary handles on the curves shape other than the usual data point coordinates and control points weights. Differential properties are used for optimising NURBS, Hermite interpolation allows for the definition of higher order constraints (tangent, normal, bi-normal) at data points. The assignment of parameter values at the data points, known as parameterisation also provides control of the curve’s shape. Curve optimisation is also performed at the geometric modelling level. Old mathematical theorems established by Frénet and further developed by other mathematicians provide means of defining a curve’s shape with it’s intrinsic equations. Such representation is possible by using Function Representation (F-rep) algebra available in the ACIS software. Frep allows more generic and exact means of interfacing with the curve’s geometry and new functionality for curve inspection and optimisation are proposed in this thesis. The integration of optimisation findings and CAD are documented in the definition of a framework. The framework architecture proposed reconstructs a new CAD environment from separate elements bolted together in a generic Application Programming Interface (API) named “Oli interface”. Functionality created to interface optimisation and CAD makes a requirement list of the work that both sides should undertake to achieve design optimisation in the CAD environment.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

Coupled Experimentally-Driven Constraint Functions and Topology Optimization utilized in Design for Additive Manufacturing

Topology optimization (TO) is a structural optimization technique that searches for the proper material distribution inside a design space such that an objective function is maximized/ minimized. Rapid prototyping technologies such as additive manufacturing (AM) have allowed results from TO to be manufacturable. However, despite advancements in their ability to manufacture complex geometries, AM technologies still face certain constraints such as printing features at overhangs (unsupported features oriented at a certain angle from the axis normal to the build plate) and small feature sizes, amongst others. In the field of design for additive manufacturing (DfAM), it is common to only restrict one constraint to control the quality of the final parts. However, several studies have found that the final quality of a feature is heavily affected by at least two coupled constraints: the overhanging angle and the feature’s thickness. Modifying a structure’s layout while restricting only one constraint can uselessly increase the weight of a structure. To tackle this problem, the work done in this thesis considers the interplay between two geometrical constraints. The proposed research reviews some of the essential manufacturing constraints in topology optimization and emphasizes the need for coupling existing constraints. It first develops experiments to obtain a qualitative and a quantitative relationship between the design features’ surface qualities, orientation, and thickness. The relation between those parameters is used to update the layout of topologically optimized structures. The layout is changed by obtaining the medial axis of topologically optimized structures and then using implicit functions to conditionally thickening it. Throughout the analysis, it was observed that both the inclination and the thickness affect the surface quality. Furthermore, the effect of the parameters is more pronounced for low thicknesses and higher overhanging angles. The overhanging angle impacts the surface quality more than the thickness, which can be seen through ANOVA

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

Certificates of positivity in the simplicial Bernstein basis.

We study in the paper the positivity of real multivariate polynomials over a non-degenerate simplex V. We aim at obtaining certificates of positivity, {\it i.e.} algebraic identities certifying the positivity of a given polynomial on V, thus generalizing the work in \cite{BCR}. In order to do so, we use the Bernstein polynomials, which are more suitable than the usual monomial basis

Computational topology for approximations of knots

[EN] The preservation of ambient isotopic equivalence under piecewise linear (PL) approximation for smooth knots are prominent in molecular modeling and simulation. Sufficient conditions are given regarding:Hausdorff distance, anda sum of total curvature and derivative.High degree Bézier curves are often used as smooth representations, where computational efficiency is a practical concern. Subdivision can produce PL approximations for a given B\'ezier curve, fulfilling the above two conditions. The primary contributions are:       (i) a priori bounds on the number of subdivision iterations sufficient to achieve a PL approximation that is ambient isotopic to the original B\'ezier curve, and       (ii) improved iteration bounds over those previously established. The first, two authors acknowledge, with appreciation, partial support from NSF Grants 1053077 and 0923158 and also from IBM. The findings presented are the responsibility of these authors, not of the funding programs.Li, J.; Peters, TJ.; Jordan, KE. (2014). Computational topology for approximations of knots. Applied General Topology. 15(2):203-220. doi:http://dx.doi.org/10.4995/agt.2014.2281.SWORD203220152Amenta, N., Peters, T. J., & Russell, A. C. (2003). Computational topology: ambient isotopic approximation of 2-manifolds. Theoretical Computer Science, 305(1-3), 3-15. doi:10.1016/s0304-3975(02)00691-6L. E. Andersson, S. M. Dorney, T. J. Peters and N. F. Stewart, Polyhedral perturbations that preserve topological form, CAGD 12, no. 8 (1995), 785-799.Burr, M., Choi, S. W., Galehouse, B., & Yap, C. K. (2012). Complete subdivision algorithms, II: Isotopic meshing of singular algebraic curves. Journal of Symbolic Computation, 47(2), 131-152. doi:10.1016/j.jsc.2011.08.021Chazal, F., & Cohen-Steiner, D. (2005). A condition for isotopic approximation. Graphical Models, 67(5), 390-404. doi:10.1016/j.gmod.2005.01.005W. Cho, T. Maekawa and N. M. Patrikalakis, Topologically reliable approximation in terms of homeomorphism of composite Bézier curves, CAGD 13 (1996), 497-520.Denne, E., & Sullivan, J. M. (2008). Convergence and Isotopy Type for Graphs of Finite Total Curvature. Discrete Differential Geometry, 163-174. doi:10.1007/978-3-7643-8621-4_8G. E. Farin, Curves and surfaces for computer-aided geometric design: A practical code, Academic Press, Inc., 1996.Hirsch, M. W. (1976). Differential Topology. Graduate Texts in Mathematics. doi:10.1007/978-1-4684-9449-5J. Li, T. J. Peters, D. Marsh and K. E. Jordan, Computational topology counterexamples with 3D visualization of Bézier curves, Applied General Topology 13, no. 2 (2012), 115-134.Lin, L., & Yap, C. (2011). Adaptive Isotopic Approximation of Nonsingular Curves: the Parameterizability and Nonlocal Isotopy Approach. Discrete & Computational Geometry, 45(4), 760-795. doi:10.1007/s00454-011-9345-9T. Maekawa, N. M. Patrikalakis, T. Sakkalis and G. Yu, Analysis and applications of pipe surfaces, CAGD 15, no. 5 (1998), 437-458.Milnor, J. W. (1950). On the Total Curvature of Knots. The Annals of Mathematics, 52(2), 248. doi:10.2307/1969467G. Monge, Application de l'analyse à la géométrie, Bachelier, Paris, 1850.Moore, E. L. F., Peters, T. J., & Roulier, J. A. (2007). Preserving computational topology by subdivision of quadratic and cubic Bézier curves. Computing, 79(2-4), 317-323. doi:10.1007/s00607-006-0208-9G. Morin and R. Goldman, On the smooth convergence of subdivision and degree elevation for Bézier curves, CAGD 18 (2001), 657-666.J. Munkres, Topology, Prentice Hall, 2nd edition, 1999.D. Nairn, J. Peters and D. Lutterkort, Sharp, quantitative bounds on the distance between a polynomial piece and its Bézier control polygon, CAGD 16 (1999), 613-63.Reid, M., & Szendroi, B. (2005). Geometry and Topology. doi:10.1017/cbo978051180751

IST Austria Thesis

Fabrication of curved shells plays an important role in modern design, industry, and science. Among their remarkable properties are, for example, aesthetics of organic shapes, ability to evenly distribute loads, or efficient flow separation. They find applications across vast length scales ranging from sky-scraper architecture to microscopic devices. But, at the same time, the design of curved shells and their manufacturing process pose a variety of challenges. In this thesis, they are addressed from several perspectives. In particular, this thesis presents approaches based on the transformation of initially flat sheets into the target curved surfaces. This involves problems of interactive design of shells with nontrivial mechanical constraints, inverse design of complex structural materials, and data-driven modeling of delicate and time-dependent physical properties. At the same time, two newly-developed self-morphing mechanisms targeting flat-to-curved transformation are presented. In architecture, doubly curved surfaces can be realized as cold bent glass panelizations. Originally flat glass panels are bent into frames and remain stressed. This is a cost-efficient fabrication approach compared to hot bending, when glass panels are shaped plastically. However such constructions are prone to breaking during bending, and it is highly nontrivial to navigate the design space, keeping the panels fabricable and aesthetically pleasing at the same time. We introduce an interactive design system for cold bent glass façades, while previously even offline optimization for such scenarios has not been sufficiently developed. Our method is based on a deep learning approach providing quick and high precision estimation of glass panel shape and stress while handling the shape multimodality. Fabrication of smaller objects of scales below 1 m, can also greatly benefit from shaping originally flat sheets. In this respect, we designed new self-morphing shell mechanisms transforming from an initial flat state to a doubly curved state with high precision and detail. Our so-called CurveUps demonstrate the encodement of the geometric information into the shell. Furthermore, we explored the frontiers of programmable materials and showed how temporal information can additionally be encoded into a flat shell. This allows prescribing deformation sequences for doubly curved surfaces and, thus, facilitates self-collision avoidance enabling complex shapes and functionalities otherwise impossible. Both of these methods include inverse design tools keeping the user in the design loop