Computer Aided Geometric Design最新文献

This paper introduces an algorithm for approximating a set of data points with $G^1$ continuous arcs, leveraging covariance data associated with the points. Prior approaches to arc spline approximation typically assumed equal contribution from all data points, resulting in potential algorithmic instability when outliers are present. To address this challenge, we propose a robust method for arc spline approximation, taking into account the 2D covariance of each data point. Beginning with the definition of models and parameters for single-arc approximation, we extend the framework to support multiple-arc approximation for broader applicability. Finally, we validate the proposed algorithm using both synthetic noisy data and real-world data collected through vehicle experiments conducted in Sejong City, South Korea.

Triple orthogonal coordinate systems having coordinate lines as circles or straight lines are considered. Technically, they are represented by trilinear rational quaternionic maps and are called Dupin cyclidic cubes, naturally generalizing the bilinear rational quaternionic parametrizations of principal patches of Dupin cyclides. Dupin cyclidic cubes and their singularities are studied and classified up to Möbius equivalency in Euclidean space.

Euler spirals have linear varying curvature with respect to arc length and can be applied in fields such as aesthetics pleasing shape design, curve completion or highway design, etc. However, evaluation and interpolation of Euler spirals to prescribed boundary data is not convenient since Euler spirals are represented by Fresnel integrals but with no closed-form expression of the integrals. We investigate a class of Bézier or B-spline curves called Euler Bézier spirals or Euler B-spline spirals which have specially defined control polygons and approximate linearly varying curvature. This type of spirals can be designed conveniently and evaluated exactly. Simple but efficient algorithms are also given to interpolate $G^1$ boundary data by Euler Bézier spirals or cubic Euler B-spline spirals.

Voss nets are surface parametrizations whose parameter lines follow a conjugate network of geodesics. Their discrete counterparts, so called V-hedra, are flexible quadrilateral meshes with planar faces such that opposite angles at every vertex are equal; by replacing this equality condition with the closely related constraint that opposite angles are supplementary, we get so-called anti-V-hedra. In this paper, we study the problem of constructing and manipulating (anti-)V-hedra. First, we present a V-hedra generator that constructs a modifiable (anti-)V-hedron in a geometrically exact way, from a set of simple conditions already proposed by Sauer in 1970; our generator can compute and visualize the flexion of the (anti-)V-hedron in real time. Second, we present an algorithm for the design and interactive exploration of V-hedra using a handle-based deformation approach; this tool is capable of simulating the one-parametric isometric deformation of an imperfect V-hedron via a quad soup approach. Moreover, we evaluate the performance and accuracy of our tools by applying the V-hedra generator to constraints obtained by numerical optimization. In particular, we use example surfaces that originate from one-dimensional families of smooth Voss surfaces – each spanned by two isothermal conjugate nets – for which an explicit parametrization is given. This allows us to compare the isometric deformation of the smooth target surface with the rigid folding of the optimized (imperfect) V-hedron.

The use of subdivision schemes in applied and real-world contexts requires the development of conceptually simple algorithms that can be converted into fast and efficient implementation procedures. In the domain of interpolatory subdivision schemes, there is a demand for developing an algorithm capable of (i) reproducing all types of conic sections whenever the input data (in our case point-normal pairs) are arbitrarily sampled from them, (ii) generating a visually pleasing limit curve without creating unwanted oscillations, and (iii) having the potential to be naturally and easily extended to the bivariate case. In this paper we focus on the construction of an interpolatory subdivision scheme that meets all these conditions simultaneously. At the centre of our construction lies a conic fitting algorithm that requires as few as four point-normal pairs for finding new edge points (and associated normals) in a subdivision step. Several numerical results are included to showcase the validity of our algorithm.

Splines in the manifold setting have been defined as extensions from the standard Euclidean setting, but they are far more complicated. Alternative approaches, which are equivalent in the Euclidean case, lead to different results in the manifold case; the existence conditions are often quite restrictive; and the necessary computations are rather involved. All difficulties stem from the peculiar nature of the geodesic distance: in general, shortest geodesics may be not unique and the dependence on their endpoints may not be smooth; and distances cannot be computed in closed form. The former issue may impose strong limitations on the placement of control points. While the latter may greatly complicate the computations. Nevertheless, some recent results suggest that splines on surfaces may have practical impact on CAGD applications. We review the literature on this topic, accounting for both theoretical results and practical implementations.

Avoiding singularities is a crucial task in robotics and path planning. This paper proposes a novel algorithm for detecting the closest singularity to a given pose for nine interpretations of the 3-RPR manipulator. The algorithm utilizes intrinsic metrics based on the framework's total elastic strain energy density, employing the physical concept of Green-Lagrange strain. The constrained optimization problem for detecting the closest singular configuration with respect to these metrics is solved globally using tools from numerical algebraic geometry implemented in the software package Bertini. The effectiveness of the proposed algorithm is demonstrated on a 3-RPR manipulator executing a one-parametric motion. Additionally, the obtained intrinsic singularity distances are compared with extrinsic metrics. Finally, the paper illustrates the advantage of employing a well-defined metric for identifying the closest singularities in comparison with the existing methods in the literature, highlighting its application in design optimization.

Various interpolation and approximation methods arising in several practical applications in geometric modeling deal, at a particular step, with the problem of computing suitable rational patches (of low degree) on the unit sphere. Therefore, we are concerned with the construction of a system of spherical triangular patches with prescribed vertices that globally meet along common boundaries. In particular, we investigate various possibilities for tiling a given spherical triangular patch into quadratically parametrizable subpatches. We revisit the condition that the existence of a quadratic parameterization of a spherical triangle is equivalent to the sum of the interior angles of the triangle being π, and then circumvent this limitation by studying alternative scenarios and present constructions of spherical macro-elements of the lowest possible degree. Applications of our method include algorithms relying on the construction of (interpolation) surfaces from prescribed rational normal vector fields.

We investigate the construction of $C^2$ cubic spline quasi-interpolants on a given arbitrary triangulation $T$ to approximate a sufficiently smooth function f. The proposed quasi-interpolants are locally represented in terms of a simplex spline basis defined on the cubic Wang–Shi refinement of the triangulation. This basis behaves like a B-spline basis within each triangle of $T$ and like a Bernstein basis for imposing smoothness across the edges of $T$. Any element of the cubic Wang–Shi spline space can be uniquely identified by considering a local Hermite interpolation problem on every triangle of $T$. Different $C^2$ cubic spline quasi-interpolants are then obtained by feeding different sets of Hermite data to this Hermite interpolation problem, possibly reconstructed via local polynomial approximation. All the proposed quasi-interpolants reproduce cubic polynomials and their performance is illustrated with various numerical examples.

Reverse engineering Computer-Aided Design (CAD) models based on the original geometry is a valuable and challenging research problem that has numerous applications across various tasks. However, previous approaches have often relied on excessive manual interaction, leading to limitations in reconstruction speed. To mitigate this issue, in this study, we approach the reconstruction of a CAD model by sequentially constructing geometric primitives (such as vertices, edges, loops, and faces) and performing Boolean operations on the generated CAD modules. We address the complex reconstruction problem in four main steps. Firstly, we use a plane to cut the input mesh model and attain a loop cutting line, ensuring accurate normals. Secondly, the cutting line is automatically fitted to edges using primitive information and connected to form a primitive loop. This eliminates the need for time-consuming manual selection of each endpoint and significantly accelerates the reconstruction process. Subsequently, we construct the loop of primitives as a chunked CAD model through a series of CAD procedural operations, including extruding, lofting, revolving, and sweeping. Our approach incorporates an automatic height detection mechanism to minimize errors that may arise from manual designation of the extrusion height. Finally, by merging Boolean operations, these CAD models are assembled together to closely approximate the target geometry. We conduct a comprehensive evaluation of our algorithm using a diverse range of CAD models from both the Thingi10K dataset and real-world scans. The results validate that our method consistently delivers accurate, efficient, and robust reconstruction outcomes while minimizing the need for manual interactions. Furthermore, our approach demonstrates superior performance compared to competing methods, especially when applied to intricate geometries.