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

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

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

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

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

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

In this paper we study the dimension of splines of mixed smoothness on axis-aligned T-meshes. This is the setting when different orders of smoothness are required across the edges of the mesh. Given a spline space whose dimension is independent of its T-mesh's geometric embedding, we present constructive and sufficient conditions that ensure that the smoothness across a subset of the mesh edges can

We define general geometric subdivision schemes generating curves on the 2-dimensional unit sphere by using geodesic polygons and the spherical distance. We show that a spherical interpolatory geometric subdivision scheme is convergent if the sequence of maximum edge lengths is summable and the limit curve is G1-continuous if in addition the sequence of maximum angular defects is summable. In particular

Isogeometric shape optimization has been receiving great attention due to its advantages of exact geometry representation and smooth representation of the boundaries. In the present study, the optimum shape of a horn speaker was found to minimize the back reflection and improve impedance matching. The acoustic field was estimated by isogeometric analysis (IGA). Horn reflections are sensitive to the

Given a set of points S∈R2, reconstruction is a process of identifying the boundary edges that best approximates the set of points. In general, the set of points can either be derived from only the boundaries of the curves (called as boundary sample) or can be derived from both boundary and interior of the curves (called as dot pattern). Most of the existing algorithms focus towards reconstruction

C1 splines over box-complexes generalize C1 degree 3 (cubic) tensor-product splines. A box-complex is a collection of 3-dimensional boxes forming an unstructured hexahedral mesh that can include irregular points and irregular edges where the layout deviates from the tensor-product grid layout. For example, an edge shared and enclosed by five boxes is irregular. Where the mesh is locally regular, the

Following and extending the recent results of Albrecht et al. (2017) for planar Pythagorean hodograph (PH) B-spline curves to the Minkowski 3-space, we introduce a class of Minkowski Pythagorean hodograph (MPH) B-spline curves. The distinguished property of these curves is that the Minkowski norm of their hodograph is a B-spline function. We focus mainly on the clamped case and using Clifford algebra

The medial axis transform (MAT) of a 3D shape includes the set of centers and radii of the maximally inscribed spheres, and is a complete shape descriptor that can be used to reconstruct the original shape. It is a compact representation that jointly describes geometry, topology, and symmetry properties of a given shape. In this work, we present P2MAT-NET, a neural network which learns the pattern

Mechanisms and robots often share the following fundamental property: the instantaneous twist space generated by the end-effector at a generic pose is a rigidly-displaced copy of the one generated at the home configuration, i.e., the tangent spaces at all points of its motion manifold (a manifold of the Lie group of rigid displacements SE(3)) are mutually congruent. A manifold of this kind, hereafter

How to reconstruct the complete 3D mesh model from a single natural image is now still considered as a challenging problem. Most existing methods describe 3D shapes in the form of voxel or point cloud, and it is not always trivial to convert them into meshes with high quality. In this paper, we present a novel method to effectively address this problem by using a specially-designed GAN model to map

Nonlinear behaviors are commonplace in many engineering applications, e.g., metal forming and vehicle crash tests. Different from linear systems, nonlinear problems cannot be solved by using a system of linear equations and there is no guarantee that a unique solution can be found. In this work, we develop a unified framework, NLIGA (Non-Linear Isogeometric Analysis), for mainly solving two and three-dimensional

This article deals with the spatial counterpart of the recently introduced class of planar Pythagorean-Hodograph (PH) B–Spline curves. Spatial Pythagorean-Hodograph B–Spline curves are odd-degree, non-uniform, parametric spatial B–Spline curves whose arc length is a B–Spline function of the curve parameter and can thus be computed explicitly without numerical quadrature. After giving a general definition

We propose a new isogeometric method using Toric surface patches for trimmed CAD planar surfaces. This method converts each trimmed spline element into a Toric surface patch with conforming boundary representation and converts each non-trimmed spline element into a Bézier element. Because the Toric surface patches are a multi-sided generalization of classical Bézier surface patches, all trimmed and

Volumetric modeling is an important topic for material modeling and isogeometric simulation. In this paper, two kinds of interpolatory Catmull-Clark volumetric subdivision approaches over unstructured hexahedral meshes are proposed based on the limit point formula of Catmull-Clark subdivision volume. The basic idea of the first method is to construct a new control lattice, whose limit volume by the

The problem of constructing spatial Pythagorean hodograph (PH) curves with a given Gauss–Legendre polygon is addressed. For planar/spatial PH curves of degree 2n+1, the Gauss–Legendre polygon, which consists of n+1 edges, obtained by evaluating the hodograph at the nodes of the Gauss–Legendre quadrature, is the rectifying polygon, which has the same length as the PH curve. On the other hand, if a planar

This paper is devoted to techniques for adaptive spline projection via quasi-interpolation, enabling the efficient approximation of given functions. We employ local least-squares fitting in restricted hierarchical spline spaces to establish novel projection operators for hierarchical splines of degree p. This leads to efficient spline projectors that require O(pd) floating point operations and O(1)

We classify planar polynomial Pythagorean-hodograph curves of any degree with respect to Euclidean similarities. We also analyze possible global shapes of the Pythagorean-hodograph curves of degree four and five and describe all possible configurations of their singular points. This description results in a classification of Pythagorean-hodograph quartics and quintics with respect to the homeomorphisms

The computation of offset and bisector curves/surfaces has always been considered a challenging problem in geometric modeling and processing. In this work, we investigate a related problem of approximating offsets of curves on surfaces (OCS) and bisectors of curves on surfaces (BCS). While at times the precise geodesic distance over the surface between the curve and its offset might be desired, herein

Traditional portrait bas-relief modeling methods are limited by the shape of the input 3D model. In this paper, we introduce a novel approach to generate portrait bas-relief with various poses and expressions. In the alignment stage, we segment the input model into several meaningful regions so that the facial features of each region can be separately aligned. To guaranty the quality of expression

Curve skeletons of 3D objects are central to many geometry analysis tasks in the field of computer graphics. A desirable skeleton has to meet at least four requirements: (1) topologically homotopic to the primitive shape, (2) truly well-centred, (3) feature preserving and (4) has a reasonable degree of smoothness. There are at least a couple of difficulties with skeletonization. On the one hand, finding

Voronoi diagram based partitioning of a 2-manifold surface in R3 is a fundamental operation in the field of geometry processing. However, when the input object is a thin-plate model or contains thin branches, the traditional restricted Voronoi diagrams (RVD) cannot induce a manifold structure that is conformal to the original surface. Yan et al. (2014) are the first who proposed a localized RVD (LRVD)

Triangular mesh is one of the most popular 3D modalities, which usage spans a wide variety of applications in computer vision, computer graphics and multimedia. Raw mesh surfaces generated by 3D scanning devices often suffer from mesh irregularity and also lack the implicit ordered structure that characterizes 2D images. Therefore, they are not suitable to be processed as such, especially for applications

The paper investigates a quasi-interpolation framework for the construction of operators that provide approximations for bivariate functions from the space of Argyris macro-element splines on a general triangulation. An operator of such type is determined by local approximation operators that produce quintic polynomials and are naturally associated with vertices and triangles of the triangulation.

Given two spatial C1 PH spline curves, aim of this paper is to study the construction of a tensor–product spline surface which has the two curves as assigned boundaries and which in addition incorporates a single family of isoparametric PH spline curves. Such a construction is carried over in two steps. In the first step a bi–patch is determined in a ‘Coons–like’ way having as boundaries two quintic

Provided that they are in appropriate configurations (tight data), given planar G1 Hermite data generate a unique cubic Pythagorean hodograph (PH) spline curve interpolant. On a given associated knot-vector, the corresponding spline function cannot be C1, save for exceptional cases. By contrast, we show that replacing cubic spaces by cubic-like sparse spaces makes it possible to produce infinitely

In materials science and engineering, auxetic behavior refers to deformations of flexible structures where stretching in some direction involves lateral widening, rather than lateral shrinking. We address the problem of detecting auxetic behavior for flexible periodic bar-and-joint frameworks. Currently, the only known algorithmic solution is based on the rather heavy machinery of fixed-dimension semi-definite

We construct and analyze piecewise approximations of functional data on arbitrary 2D bounded domains using generalized barycentric finite elements, and particularly quadratic serendipity elements for planar polygons. We compare approximation qualities (precision/convergence) of these partition-of-unity finite elements through numerical experiments, using Wachspress coordinates, natural neighbor coordinates

Computational fluid dynamics (CFD) simulations of flow over complex objects have been performed traditionally using fluid-domain meshes that conform to the shape of the object. However, creating shape conforming meshes for complicated geometries like automobiles require extensive geometry preprocessing. This process is usually tedious and requires modifying the geometry, including specialized operations

Cubic Hermite hexahedral finite element meshes have some well-known advantages over linear tetrahedral finite element meshes in biomechanical and anatomic modeling using isogeometric analysis. These include faster convergence rates as well as the ability to easily model rule-based anatomic features such as cardiac fiber directions. However, it is not possible to create closed complex objects with only

Building on a result of U. Reif on removable singularities, we construct C1 bi-3 splines that may include irregular points where less or more than four tensor-product patches meet. The resulting space complements PHT splines, is refinable and the refined spaces are nested, preserving for example surfaces constructed from the splines. As in the regular case, each quadrilateral has four degrees of freedom

Gk (geometrically continuous surface) constructions were developed to create surfaces that are smooth also at irregular points where, in a quad-mesh, three or more than four elements come together. Isogeometric elements were developed to unify the representation of geometry and of engineering analysis. We show how matched Gk constructions for geometry and analysis automatically yield Ck isogeometric

Motivated by applications in freeform architecture, we study surfaces which are composed of smoothly joined bilinear patches. These surfaces turn out to be discrete versions of negatively curved affine minimal surfaces and share many properties with their classical smooth counterparts. We present computational design approaches and study special cases which should be interesting for the architectural

Despite its great success in improving the quality of a tetrahedral mesh, the original optimal Delaunay triangulation (ODT) is designed to move only inner vertices and thus cannot handle input meshes containing "bad" triangles on boundaries. In the current work, we present an integrated approach called boundary-optimized Delaunay triangulation (B-ODT) to smooth (improve) a tetrahedral mesh. In our

In this paper, we present efficient algorithms for generating hierarchical molecular skin meshes with decreasing size and guaranteed quality. Our algorithms generate a sequence of coarse meshes for both the surfaces and the bounded volumes. Each coarser surface mesh is adaptive to the surface curvature and maintains the topology of the skin surface with guaranteed mesh quality. The corresponding tetrahedral

By a d-dimensional B-spline object (denoted as ), we mean a B-spline curve (d = 1), a B-spline surface (d = 2) or a B-spline volume (d = 3). By regularization of a B-spline object we mean the process of relocating the control points of such that they approximate an isometric map of its definition domain in certain directions and is shape preserving. In this paper we develop an efficient regularization

This paper describes a comprehensive approach to construct quality meshes for implicit solvation models of biomolecular structures starting from atomic resolution data in the Protein Data Bank (PDB). First, a smooth volumetric electron density map is constructed from atomic data using weighted Gaussian isotropic kernel functions and a two-level clustering technique. This enables the selection of a

We use various nonlinear partial differential equations to efficiently solve several surface modelling problems, including surface blending, N-sided hole filling and free-form surface fitting. The nonlinear equations used include two second order flows, two fourth order flows and two sixth order flows. These nonlinear equations are discretized based on discrete differential geometry operators. The