Computer Aided Geometric Design最新文献

The Pythagorean-Hodograph curve (PH curve) is a valuable curve type extensively utilized in computer-aided geometric design and manufacturing. This paper presents an approach to approximate a planar algebraic curve within a bounding box by employing piecewise quintic PH spline curves, while maintaining $G^2$ smoothness of the approximating curve and preserving second-order geometric details at singularities. The bounding box encompasses all x-coordinates of key topological points, ensuring accurate representation. The paper explores the analysis of the $G^2$ interpolation problem for quintic PH curves with invariant convexity, transforming the quest for interpolation solutions into identifying positive roots within a set of algebraic equations. Through infinitesimal order analysis, it is established that a solution necessarily exists following adequate subdivision, laying the groundwork for practical application. Finally, the paper introduces a novel algorithm that integrates prior research to construct the approximating curve while maintaining control over the desired error levels.

It is known that a camera can be calibrated using three pictures of either squares, or spheres, or surfaces of revolution. We give a new method to calibrate a camera with the picture of a single torus.

A smooth T-surface can be thought of as a generalization of a surface of revolution in such a way that the axis of rotation is not fixed at one point but rather traces a smooth path on the base plane. Furthermore, the action, by which the aforementioned surface is obtained does not need to be merely rotation but any “suitable” planar equiform transformation applied to the points of a certain smooth profile curve. In analogy to the smooth setting, if the axis footpoints sweep a polyline on the base plane and if the profile curve is discretely chosen then a T-hedra (discrete T-surface) with trapezoidal faces is obtained.

The goal of this article is to reconstruct a T-hedron from an already given unorganized point cloud of a T-surface. In doing so, a kinematic approach is taken into account, where the algorithm at first tries to find the aforementioned axis direction associated with the point cloud. Then the algorithm finds a polygonal path through which the axis footpoint moves. Finally, by properly cutting the point cloud with the planes passing through the axis and its footpoints, it reconstructs the surface. The presented method is demonstrated using examples.

From an applied point of view, the straightforwardness of the generation of these surfaces predestines them for building and design processes. In fact, one can find many built objects belonging to the sub-classes of T-surfaces such as surfaces of revolution and moulding surfaces. Furthermore, the planarity of the faces of the discrete version paves the way for steel/glass construction in industry. Finally, these surfaces are also suitable for transformable designs as they allow an isometric deformation within their class.

Geometric surface models are extensively utilized in brain imaging to analyze and compare three-dimensional anatomical shapes. Due to the intricate nature of the brain surface, rather than examining the entire cortical surface, we are introducing a new set of signatures focused on characteristics of the hippocampal region, which is linked to aspects of Alzheimer's disease. Our approach focuses on Ricci flow as a conformal parameterization method, permitting us to calculate the conformal factor and mean curvature as conformal surface representations to identify distinct regions within a three-dimensional mesh. For the first time for such settings, we propose a simple while elegant formulation by employing the well-established concept of Shannon entropy on these well-known features. This compact while rich feature formulation turns out to lead to an efficient local surface encoding. We are validating its effectiveness through a series of preliminary experiments on 3D MRI data from the Alzheimer's Disease Neuroimaging Initiative (ADNI), with the aim of diagnosing Alzheimer's disease. The feature vectors generated and inputted into the XGBoost classifier demonstrate a remarkable level of accuracy, further emphasizing their potential as a valuable additional measure for surface-based cortical morphometry in Alzheimer's disease research.

Whilst Paul de Casteljau is now famous for his fundamental algorithm of curve and surface approximation, little is known about his other findings. This article offers an insight into his results in geometry, algebra and number theory.

Related to geometry, his classical algorithm is reviewed as an index reduction of a polar form. This idea is used to show de Casteljau's algebraic way of smoothing, which long went unnoticed. We will also see an analytic polar form and its use in finding the intersection of two curves. The article summarises unpublished material on metric geometry. It includes theoretical advances, e.g., the 14-point strophoid or a way to link Apollonian circles with confocal conics, and also practical applications such as a recurrence for conjugate mirrors in geometric optics. A view on regular polygons leads to an approximation of their diagonals by golden matrices, a generalisation of the golden ratio.

Relevant algebraic findings include matrix quaternions (and anti-quaternions) and their link with Lorentz' equations. De Casteljau generalised the Euclidean algorithm and developed an automated method for approximating the roots of a class of polynomial equations. His contributions to number theory not only include aspects on the sum of four squares as in quaternions, but also a view on a particular sum of three cubes. After a review of a complete quadrilateral in a heptagon and its angles, the paper concludes with a summary of de Casteljau's key achievements.

The article contains a comprehensive bibliography of de Casteljau's works, including previously unpublished material.

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.