scholarly journals Geodesic Hermite Spline Curve on Triangular Meshes

Symmetry ◽  
2021 ◽  
Vol 13 (10) ◽  
pp. 1936
Author(s):  
Yujin Ha ◽  
Jung-Ho Park ◽  
Seung-Hyun Yoon
Keyword(s):  
Geometric Modeling ◽  
Hermite Interpolation ◽  
Interpolation Method ◽  
Triangular Mesh ◽  
Triangular Meshes ◽  
Polygonal Mesh ◽  
Spline Curve ◽  
Geodesic Vector ◽  
Hermite Spline ◽  
The Given

Curves on a polygonal mesh are quite useful for geometric modeling and processing such as mesh-cutting and segmentation. In this paper, an effective method for constructing C1 piecewise cubic curves on a triangular mesh M while interpolating the given mesh points is presented. The conventional Hermite interpolation method is extended such that the generated curve lies on M. For this, a geodesic vector is defined as a straightest geodesic with symmetric property on edge intersections and mesh vertices, and the related geodesic operations between points and vectors on M are defined. By combining cubic Hermite interpolation and newly devised geodesic operations, a geodesic Hermite spline curve is constructed on a triangular mesh. The method follows the basic steps of the conventional Hermite interpolation process, except that the operations between the points and vectors are replaced with the geodesic. The effectiveness of the method is demonstrated by designing several sophisticated curves on triangular meshes and applying them to various applications, such as mesh-cutting, segmentation, and simulation.

scholarly journals Parametric Blending of Hole Patches Based on Shape Difference

Symmetry ◽  
2020 ◽  
Vol 12 (11) ◽  
pp. 1759
Author(s):  
Jung-Ho Park ◽  
Sanghun Park ◽  
Seung-Hyun Yoon
Keyword(s):  
Geometric Modeling ◽  
Positive Definite Matrix ◽  
3D Models ◽  
Triangular Mesh ◽  
Principal Curvatures ◽  
Third Order ◽  
Shape Difference ◽  
Curvature Variation ◽  
Laplacian Smoothing ◽  
The Given

A triangular mesh obtained by scanning 3D models typically contains holes. We present an effective technique for filling a hole in a triangular mesh in geometric modeling. Simple triangulation of a hole is refined and remeshed iteratively to generate an initial patch. The generated patch is then enhanced to become a target patch by minimizing the variation of principal curvatures. In discrete approximation, this produces a third-order Laplacian system of sparse symmetric positive definite matrix, and the symmetry can efficiently be used to find the robust solutions to the given Laplacian system. Laplacian smoothing of the target patch is defined as a source patch. The shape difference between two corresponding vertices of the source and the target patches is measured in terms of Euclidean distance and curvature variation. On the basis of the shape difference and a user-specified control parameter, different blending weights are determined for each vertex, and the final patch is generated by blending two patches. We demonstrate the effectiveness of our technique by discussing several examples. The experimental results show that our technique can effectively restore salient geometric features of the original shape.

scholarly journals GEOMETRIC MODELING OF A SHAPE OF PARALLELOGRAM PLATES IN A PROBLEM OF FREE VIBRATIONS USING CONFORMAL RADII

2021 ◽  
pp. 143-159
Author(s):  
A.A. Chernyaev ◽  
Keyword(s):  
Cross Sections ◽  
Geometric Modeling ◽  
Interpolation Method ◽  
Geometric Characteristic ◽  
Free Vibrations ◽  
Theory Of Elasticity ◽  
Constant Thickness ◽  
Basic Frequency ◽  
Plate Vibrations ◽  
The Subject

The paper considers a method of geometric modeling applied when solving basic twodimensional problems of the theory of elasticity and structural mechanics, in particular the applied problems of engineering. The subject of this study is vibrations of thin elastic parallelogram plates of constant thickness. To determine a basic frequency of vibrations, the interpolation method based on the geometric characteristic of the shape of plates (membrane, cross sections of a rod) is proposed. This characteristic represents a ratio of interior and exterior conformal radii of the plate. As is known from the theory of conformal mappings, conformal radii are those obtained by mapping of a plate onto the interior and exterior of a unit disk. The paper presents basic terms, tables, and formulas related to the considered geometric method with a comparative analysis of the curve diagrams obtained using various interpolation formulas. The original computer program is also developed. The main advantage of the proposed method of determining the basic frequency of plate vibrations is a graphic representation of results that allows one to accurately determine the required solution on the graph among the other solutions corresponding to the considered case of parallelogram plates. Although there are many known approximate approaches, which are used to solve the considered problems, only geometric modeling technique based on the conformal radii ratio gives such an opportunity.

An Algorithm for Geometric Modeling and Intersection in NC Milling Simulation Based on Triangular Mesh Model

2011 ◽  
Vol 48-49 ◽  
pp. 541-546 ◽  
Author(s):  
Dian Zhu Sun ◽  
Xin Cai Kang ◽  
Yan Rui Li ◽  
Yong Wei Sun
Keyword(s):  
Geometric Modeling ◽  
Tool Path ◽  
Triangular Mesh ◽  
Simulation Process ◽  
Mesh Model ◽  
Milling Simulation ◽  
Nc Simulation ◽  
Simulation Based ◽  
Triangular Mesh Model ◽  
Nc Milling

To achieve the accurate and efficient NC milling simulation based on the discrete triangular mesh model, we proposed an algorithm for geometric modeling and intersection. We construct the R*-tree index for upper-surface nodes of mesh model, based on which the nodes within cutting region can be obtained. We compute tool path segments within cutting projection region of node, and calculate the minimum adjustment height of node according to tool path segments within cutting projection region and then change the z-value of node. Thus, we complete the intersection calculation in simulation process. It has been proved by examples that the algorithm for geometric modeling and intersection in NC milling simulation has strong adaptation to tool path segment type and that it can accurately and efficiently reflect the effect of NC simulation process based on the discrete triangular mesh model of rough.

Simulation of Water Droplet Merging under Shock Wave Using Real Ghost Fluid Method

2013 ◽  
Vol 444-445 ◽  
pp. 628-632
Author(s):  
Ru Chao Shi ◽  
Sheng Li Xu ◽  
Ya Jun Zhang
Keyword(s):  
Shock Wave ◽  
Interpolation Method ◽  
Time Discretization ◽  
Problem Solver ◽  
Ghost Fluid Method ◽  
Numerical Accuracy ◽  
Air Water Interface ◽  
Fifth Order ◽  
The Given ◽  
Flow States

This paper presents a 3D numerical simulation of water droplets merging under a given shock wave. We couple interpolation method to RGFM (Real Ghost Fluid Method) to improve the numerical accuracy of RGFM. The flow states of air-water interface are calculated by ARPS (approximate Riemann problem solver). Flow field is solved by Euler equation with fifth-order WENO spatial discretization and fourth-order R-K (Runge-Kutta) time discretization. We also employ fifth-order HJ-WENO to discretize level set equation to keep track of gas-liquid interface. Numerical results demonstrate that droplets shape has little change before merging and the merged droplet gradually becomes umbrella-shaped under the given shock wave. We verify that combination of RGFM with interpolation method has the property of reducing numerical error by comparing to the results without employment of interpolation method.

scholarly journals A Geometric Modeling Method Based on TH-Type Uniform B-Splines

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Jin Xie
Keyword(s):  
Geometric Modeling ◽  
Modeling Method ◽  
Control Points ◽  
Hyperbolic Functions ◽  
Hyperbolic Polynomial ◽  
B Splines ◽  
Control Polygon ◽  
The Given

A geometric modeling method based on TH-type uniform B-splines which are composed of trigonometric and hyperbolic polynomial with parameters is introduced in this paper. The new splines possess many important properties of quadratic and cubic B-splines. Taking different values of the parameters, one can not only locally adjust the shape of the curves, but also change the type of some segments of a curve between trigonometric and hyperbolic functions as well. The given curves can also interpolate directly control polygon locally by selecting special parameters. Moreover, the introduced splines can represent some quadratic curves and transcendental curves with selecting proper control points and parameters.

scholarly journals FAST MESH INTERPOLATION AND MESH DECOMPOSITION WITH APPLICATIONS

2012 ◽  
Vol 12 (01) ◽  
pp. 1250006
Author(s):  
SHUHUA LAI ◽  
FUHUA (FRANK) CHENG
Keyword(s):  
Iterative Process ◽  
Interpolation Method ◽  
Low Frequency ◽  
Texture Mapping ◽  
Construction Process ◽  
Mesh Decomposition ◽  
Decomposition Scheme ◽  
New Approaches ◽  
Control Mesh ◽  
The Given

A new approach for constructing a smooth subdivision surface to interpolate the vertices of an arbitrary mesh is presented. The construction process does require setting up neither any linear systems, nor any matrix computation, but is simply done by iteratively moving vertices of the given mesh locally until control mesh of the required interpolating surface is reached. The new interpolation method has the simplicity of a local method in effectively dealing with meshes of a large number of vertices. It also has the capability of a global method in faithfully resembling the shape of a given mesh. Furthermore, the new method is fast and does not require a fairing step in the construction process because the iterative process converges to a unique solution at an exponential rate. Another important result of this work is, with the new iterative process, each mesh (surface) can be decomposed into a sum of simpler meshes (surfaces) which carry high-and low-frequency information of the given model. This mesh decomposition scheme provides us with new approaches to some classic applications in computer graphics such as texture mapping, denoising/smoothing/sharpening, and morphing. These new approaches are demonstrated in this paper and test results are included.

scholarly journals Approximation of Vehicle Trajectory with B-Spline Curve

2016 ◽  
Vol 19 (1) ◽  
pp. 1-5
Author(s):  
Dušan Páleš ◽  
Veronika Váliková ◽  
Ján Antl ◽  
František Tóth
Keyword(s):  
Basis Function ◽  
Extreme Case ◽  
Bézier Curve ◽  
Control Points ◽  
B Spline ◽  
Spline Curve ◽  
Planar Curve ◽  
Vehicle Trajectory ◽  
The Given ◽  
Real Vehicle

In this contribution, we present the description of a B-spline curve. We deal with creation of its basis function as well as with creation of the curve itself from entered control points. Following the literature, we formed an algorithm for B-spline modelling and we used it for the planar and spatial curve. The planar curve is made of chosen points. The spatial curve approximates the trajectory of a real vehicle, which trajectory was obtained by the set of measured points. The modelled curve very exactly describes the polygon created from the linked control points. With the lowering degree of the curve, this one is more clamping to the given polygon and for the extreme case it is transformed to the polygon itself. The advantage of the B-spline curve use is, for example in comparison with a Bézier curve, high adaptability, which is expressed in its parameters - besides entered control points, these are knots generated on the curve and degree of the curve.

scholarly journals ON ONE APPROXIMATION ESTIMATE

2017 ◽  
Vol 17 (5) ◽  
pp. 53-59
Author(s):  
M.B. Medegey
Keyword(s):  
Interpolation Method ◽  
Linear Operators ◽  
Approximation Estimate ◽  
The Given

Linear operators satisfying some conditions are considered. The given operators are a particular kind of class operators (by P.P. Korovkin). For the estimate derivation the interpolation method is used, described in the works by Yu.G. Abakumov and O.N. Shestakova

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