2D Typical curves (Mineur et al., 1998) are a class of special Bézier curves with monotone curvature, which play a key role in designing aesthetically pleasing surfaces for the automotive industry. To deal with 3D typical curves, Farin (2006) introduces the more general concept of Class A Bézier curves. These curves are defined by so-called Class A matrix that oughts to satisfy some appropriate conditions

Shape morphing is a continuous deformation in time between two shapes (curves, surfaces,..). For planar curves, most efficient methods for blending between two closed curves are based on the construction of the morph curve involving its signed curvature function. The latter is obtained by linear interpolation of the signed curvature functions of the source and target curves (Sederberg et al. (1993)

In this paper, we introduce a feature preserving surface reconstruction algorithm to produce a high fidelity triangulated mesh from an input point set. The concept of local Delaunay triangulation is applied to speed up the reconstruction procedure and to preserve features. The proposed algorithm has running time complexity of O(nlogn), where n is the number of points. Additionally, the local Delaunay

Blending and filleting are well established operations in solid modeling and computer-aided geometric design. The creation of a transition surface which smoothly connects the boundary surfaces of two (or more) objects has been extensively investigated. In this work, we introduce several algorithms for the construction of, possibly heterogeneous, trivariate fillets, that support smooth filleting operations

This foreword provides an overview of the contributions presented at the fifteenth International Conference on Geometric Modeling and Processing (GMP2021 - originally scheduled for May 10-13, 2021 in Pilsen, Czech Republic).

Action involves rich geometric and physical properties hidden in the spatial structure and temporal dynamics. However, there is a lack of synergy in investigating these properties and their joint embedding in the existing literature. In this paper, we propose a multi-derivative physical and geometric embedding network (PGEN) for action recognition from skeleton data. We model the skeleton joint and

We present a novel method to compute bijective domain parameterizations with low distortion for isogeometric analysis. Our key insight is that instead of solving the difficult mapping optimization problem, we fit the spline function to a piecewise linear map between computational and parametric domains while ensuring bijection. The basis of our key insight is that the bijection of the piecewise linear

We present an efficient algorithm for computing the precise Hausdorff Distance (HD) between two freeform surfaces. The algorithm is based on a hybrid Bounding Volume Hierarchy (BVH), where osculating toroidal patches (stored in the leaf nodes) provide geometric properties essential for the HD computation in high precision. Intrinsic features from the osculating geometry resolve computational issues

We present an efficient and robust algorithm for computing the self-intersection of a freeform surface, based on a special representation of miter points, using sufficiently small quadrangles in the parameter domain. A self-intersecting surface changes its normal direction quite dramatically around miter points, located at the open endpoints of the self-intersection curve. This undesirable behavior

Surface representations play a major role in a variety of applications throughout a diverse collection of fields, such as biology, chemistry, physics, or architecture. From a simulation point of view, it is important to simulate the surface as good as possible, including the usage of a wide range of different approximating elements. However, when it comes to manufacturing, it is desirable to have as

In this paper, a volume parametric model is computed from a piecewise smooth skeleton. Generating a volume model from a stick figure S defined in 3D is an intuitive process: given S whose topology is a pseudo-graph and whose edges are embedded as Bézier curves in R3, we propose a method for creating a thick volume parametric model “around” S. The volume model we generate is based on semi-simploidal

The theoretical basis of Floater's parameterization technique for triangulated surfaces is simultaneously a generalization (to non-barycentric weights) and a specialization (to a plane near-triangulation, which is an embedding of a planar graph with the property that all bounded faces are – possibly curved – triangles) of Tutte's Spring Embedding Theorem. Extensions of this technique cover surfaces

Human action is a spatio-temporal motion sequence where strong inter-dependencies between the spatial geometry and temporal dynamics of motion exist. However, in existing literature for human action recognition from a video, there is a lack of synergy in investigating spatial geometry and temporal dynamics in a joint representation and embedding space. In this paper, we propose a dilated Silhouette

We propose a novel approach to image segmentation based on combining implicit spline representations with deep convolutional neural networks. This is done by predicting the control points of a bivariate spline function whose zero-set represents the segmentation boundary. We adapt several existing neural network architectures and design novel loss functions that are tailored towards providing implicit

Exponential polynomials are essential in subdivision for the reconstruction of specific families of curves and surfaces, such as conic sections and quadric surfaces. It is well known that if a linear subdivision scheme is able to reproduce a certain space of exponential polynomials, then it must be level-dependent, with rules depending on the frequencies (and eventual multiplicities) defining the considered

We present a method to construct quadratic birational planar maps, which is based on the observation that any rational planar map with complex rational representation possesses two special syzygies. After establishing the relation between degree one complex rational Bézier curves and quadratic rational Bézier curves, we derive conditions to determine when a quadratic rational planar map has a complex

Polar parameterizations of star-shaped domains are based on the line segments that connect a suitably chosen center point with the points on the domain's boundary. Valid (i.e., regular everywhere except at the center point) polar parameterizations are obtained when choosing a center from the kernel of the domain. Recently, the flexibility of these polar parameterizations has been enhanced by considering

Finding the optimal parameterization for fitting a given sequence of data points with a parametric curve is a challenging problem that is equivalent to solving a highly non-linear system of equations. In this work, we propose the use of a residual neural network to approximate the function that assigns to a sequence of data points a suitable parameterization for fitting a polynomial curve of a fixed

The aim of this paper is to derive the curvatures and Frenet vectors of the intersection curve of (n−1) transversally intersecting hypersurfaces represented either in implicit or in parametric form in Euclidean n-space. Since the intersection of (n−1) implicit hypersurfaces defines an implicit curve in n-dimensions, the method derived for the implicit case can be considered as a partial answer to the

Kinematic interpretation of quaternionic–Bézier curves and surfaces is revealed. This combination of kinematic and geometric modeling methods is used to characterize orbits of quadratic surfaces in the Study quadric under the kinematic map as certain Darboux cyclides. In particular, earlier unknown rational parametrizations of one-oval Darboux cyclides are found.

Motivated by applications in CNC machining, we provide a characterization of surfaces which are enveloped by a one-parametric family of congruent rotational cones. As limit cases, we also address ruled surfaces and their offsets. The characterizations are higher order nonlinear PDEs generalizing the ones by Gauss and Monge for developable surfaces and ruled surfaces, respectively. The derivation includes

Additive manufacturing has recently enabled the creation of heterogeneous objects in which different materials can be specified at different locations. This creates new exciting perspectives in additive crafting but also presents several challenges, including the problem of specifying rules for varying materials at particular locations as well as encoding user specified properties (such as stress tensors)

We are witnessing a proliferation of textured 3D models captured from the real world with automatic photo-reconstruction tools by people and professionals without a proper technical background in computer graphics. Digital 3D models of this class come with a unique set of characteristics and defects – especially concerning their parametrization – setting them starkly apart from 3D models originating

A mesh is locally quad-dominant (lqd) if all non-4-sided facets are surrounded by quadrilaterals. Lqd meshes allow for irregular nodes where n≠4 quads meet and for multi-sided facets, called T-gons, that end quad-strips and so adjust mesh density. This paper introduces a new class of bi-cubic (bi-3) Geometric T-joint (GT) splines whose control nets are τ-nets, i.e. T-gons surrounded by quads. These

A well–known feature of the Pythagorean–hodograph (PH) curves is the multiplicity of solutions arising from their construction through the interpolation of Hermite data. In general, there are four distinct planar quintic PH curves that match first–order Hermite data, and a two–parameter family of spatial quintic PH curves compatible with such data. Under certain special circumstances, however, the

Existing definitions for inflection point on 3D curves lack the direct relation to local shape-characteristics of the 3D curve that the corresponding definition for planar curves has. This paper presents a new generalization of the definition of planar-curve inflection to the case of a 3D curve that is directly related to the local convexity of the 3D curve and to that of orthogonal projections of

In this paper, we present a progressive and iterative approximation method with memory for least square fitting (MLSPIA). It adjusts the control points and the weighted sums iteratively to construct a series of fitting curves (surfaces) with three weights. For any normalized totally positive basis, even when the collocation matrix is of deficient column rank, we obtain a condition to guarantee that

Dubuc pioneered a method of interpolation iterative scheme, which is called four-point interpolatory subdivision scheme. Since then, the method has been developed extensively and deeply. However some precise problems proposed by Dubuc are still unsolved after thirty years. In this paper, we will partially solve these problems.

We present an algorithm for curve skeleton extraction from unstructured points using visibility as a guide. We introduce visible cells inside the point samples. By locating a viewpoint in the space, visible points from the viewpoint are connected to indicate the visible region, which locally captures the interior structure of the points and is called a visible cell. We then analyze and clean the visible

This paper presents a self-supervised learning network called SP-Flow to generate keypoints in real-time for SLAM systems. Optical flows are employed to match the keypoints between two successive frames in the training process of SP-Flow. This approach enables the network to use datasets without manual annotations. To show the efficacy of our SP-Flow, we built an SP-Flow SLAM system by replacing ORB

For any (real) algebraic variety X in a Euclidean space V endowed with a nondegenerate quadratic form q, we introduce a polynomial EDpolyX,u(t2) which, for any u∈V, has among its roots the distance from u to X. The degree of EDpolyX,u is the Euclidean Distance degree of X. We prove a duality property when X is a projective variety, namely EDpolyX,u(t2)=EDpolyX∨,u(q(u)−t2) where X∨ is the dual variety

Extending a Ball B-spline Curve (BBSC) is a useful function in the shape modelling of freeform tubular objects. In this paper, we aim to obtain a cubic BBSC B‾(t‾) that can smoothly and fairly extend a given cubic BBSC B(t) to a target ball R. BBSCs with one endpoint satisfying G2 continuity with B(t) and the other endpoint passing through R are optional extending results. We choose the fairest of

Point cloud based shape completion has great significant application values and refers to reconstructing a complete point cloud from a partial input. In this paper, we propose a multi-stage point completion network (MSPCN) with critical set supervision. In our network, a cascade of upsampling units is used to progressively recover the high-resolution results with several stages. Different from the

Kim and Moon (2017) have recently proposed rectifying control polygons as an alternative to Bézier control polygons and a way of controlling planar PH curves by the rectifying control polygons. While a Bézier control polygon determines a unique polynomial curve, a rectifying control polygon gives a multitude of PH curves. This multiplicity of PH curves naturally raises the selection problem of the

In this paper, we present an efficient approach to compute the integral of monomials and polynomials over polyhedra and regions defined by parametric curved boundary surfaces. We use Euler's theorem for homogeneous functions in combination with Stokes's theorem to reduce the integration of a monomial over a three-dimensional solid to its boundary. If the solid is a polytope, through a recursive application

In this paper, we propose a novel framework for dynamic spline bas-relief modeling based on isogeometric collocation method. In the proposed method, the bas-relief model is separated into two parts: the base surface and the relief surface; and both of them are represented in spline form. To simplify the construction, the spline bases for representing the base surface and relief surface are in the same

This paper presents a geometric construction for fitting planar motions to three control poses within a particular geometric algebra. The immediate impact of the geometric construction along with the control pose representation is the provision of simple, usable tools for the design and manipulation of motions in a similar way to the highly successful B-spline approach for curves and surfaces in CAD

Grasp saliency map is an important analysis tool to explore human grasping skills and has many potential applications in visual and robotic fields. Currently, few works concentrate on the 3D grasp saliency detection for novel challenging instances, since the calculation of the grasp saliency map depends on the insufficient human grasping data and geometry of the 3D shapes. To address the above problem

Feature analysis plays a crucial role in various applications in both computer vision and computer graphics. The semantic gap between 2D images and 3D graphical models is the major obstacle to improve the universality of existing valuable technologies. To bridge the gap, we propose an effective and robust representation of 3D models, named multi-scale Graphical Image (GI), which is constructed by introducing

We investigate the problem of recognizing a generalization of surfaces of revolution appearing in the field of affine differential geometry, namely affine rotation surfaces. By using some notions from affine differential geometry, we determine how to detect whether or not a given implicit algebraic surface is an affine rotation surface. These results generalize some previous results of the authors

We consider an adaptive algorithm for finite element methods for the isogeometric analysis (IGAFEM) of elliptic (possibly non-symmetric) second-order partial differential equations. We employ analysis-suitable T-splines of arbitrary odd degree on T-meshes generated by the refinement strategy of Morgenstern and Peterseim (2015) in 2D and Morgenstern (2016) in 3D. Adaptivity is driven by some weighted

Geographical visualization systems, such as online maps, provide interactive operations of continuous zooming and panning. With consistent dynamic map labeling, users can navigate continuously in the map areas such that labels are not allowed to exhibit abrupt change in terms of their positions or sizes, and labels should not suddenly disappear or reappear when zooming in or pop up when zooming out

A polar configuration is a node surrounded by m triangles. Polar configurations are common to cap off cylinders and spheres. When the triangles, interpreted as quadrilaterals with one edge collapsed, are surrounded by a quad-strip then the extended polar configuration qualifies as part of a locally quad-dominant (lqd) mesh. Recent constructions, referred to as semi-structured splines, can use lqd meshes

This paper deals with the homology computation of a subdivided object during its construction. In this paper, we focus on the construction operation consisting of merging cells. For each step of the construction, a homological equivalence is maintained. This algebraic structure connects the chain complex associated with the object to a smaller object (i.e. containing less cells) having the same homology

A circular biarc can be defined by using two points K and L with their (oriented) tangents gK and gL as input. It is well-known that one can determine a one parametric set of circular arc pairs k,ℓ such that k starts at K with tangent gK, ℓ ends at L with tangent gL and k and ℓ meet with a common tangent in an intermediate point P. In this paper we investigate a similar construction where we replace

Weighted φ-transformed systems include many systems of functions useful in C.A.G.D. It is proved that these systems inherit some geometric properties of a given initial system, such as shape preservation or optimal shape preservation. A general class of important rational bases can be obtained as a particular example of weighted φ-transformed systems. For these bases, evaluation and subdivision algorithms

Given points P1,P2,…,Pn in Rd (d≥2), we consider the problem of constructing a fair interpolating curve. For d=2, we proposed and analyzed, in Johnson and Johnson (2016), a method which first generates a family of G1 interpolating curves, where each piece is a parametric cubic. An energy functional, that loosely approximates bending energy, is defined on this family and then one seeks a curve with

The aim of this paper is a construction of quartic parametric polynomial interpolants of a circular arc, where two boundary points of a circular arc are interpolated. For every unit circular arc of an inner angle not greater than 2π we find the best interpolant, where the optimality is measured by the simplified radial error.

We present a robust framework to conduct material modeling suitable for continuously varying materials with a local material composition control. The geometry and attribute representations are defined using two trivariate T-splines based on the same parametric domain. The optimization process decreases the mismatch of geometry and attribute between the input tetrahedral mesh and the trivariate T-splines

In applications like computer aided design, geometric models are often represented numerically as polynomial splines or NURBS, even when they originate from primitive geometry. For purposes such as redesign and isogeometric analysis, it is of interest to extract information about the underlying geometry through reverse engineering. In this work we develop a novel method to determine these primitive