From NURBS to C-NURBS: I — C-NURBS Curves and Their Properties

1999 ◽  
Author(s):  
Manhong Wen ◽  
Kwun-Lon Ting
Keyword(s):  
Shape Control ◽  
Free Form ◽  
Control Points ◽  
Degree Elevation ◽  
Nurbs Curve ◽  
3D Space ◽  
Cad Cam ◽  
Form Model ◽  
And Control ◽  
The Given

Abstract This paper develops a new free-form model, called c-NURBS, which is a general model of NURBS. A c-NURBS curve or surface is the projection of a 6D B-spline curve from a 6D homogeneous space H6 into a 3D space R3. The construction procedure of a c-NURBS curve or surface is that using cubic curves or bicubic patches repeatedly and piecewisely interpolates the given control points. The distinct properties of c-NURBS include independent weight modification, super-convexity, strong c-convex hull, and hidden degrees and control points. These properties greatly enhance the shape control and modification capability. All techniques developed for NURBS, such as the de Boor-Cox algorithm, knot insertion, and degree elevation and reduction, can be applied to c-NURBS. The implementation of c-NURBS requires little improvement on the CAD/CAM systems based on NURBS.

From NURBS to C-NURBS: II — C-NURBS Surfaces and C-Bezier Triangles

1999 ◽  
Author(s):  
Manhong Wen ◽  
Kwun-Lon Ting
Keyword(s):  
Convex Hull ◽  
Shape Control ◽  
Control Points ◽  
Nurbs Surface ◽  
De Casteljau Algorithm ◽  
Homogenous Space ◽  
Nurbs Surfaces ◽  
And Control ◽  
The Given ◽  
Bézier Triangles

Abstract This paper develops c-NURBS surfaces and c-Bezier triangles. The projection from 6D homogenous space to 3D vector space developed in previous papers [12, 13] is applied to surfaces. As a result, a c-NURBS surface can be constructed using bicubic patches to interpolate the given control points with the de Boor-Cox algorithm. Based on this, c-NURBS surfaces have the properties of independent weight modification, super-convexity, strong c-convex hull, and hidden degrees and control points. A c-Bezier triangle can be constructed using cubic patches to interpolate the given control points with the de Casteljau algorithm. Based on this, the c-Bezier triangle has the properties of independent weight modification, super-convexity, and hidden degrees and control points. These properties provide great convenience for shape control and modification operations.

Insertion loss (IL) of finite sound barriers of different contours - An introduction to geometrical solutions in 3-D space

2021 ◽  
Vol 263 (6) ◽  
pp. 164-174
Author(s):  
Giora Rosenhouse
Keyword(s):  
Insertion Loss ◽  
Vector Analysis ◽  
Control Points ◽  
Noise Sources ◽  
3D Space ◽  
Sound Barriers ◽  
Environmental Acoustics ◽  
And Control ◽  
Rectangular Barrier

The design of finite sound barriers noise sources and control points requires calculations beyond those that are used when the Maekawa formula is applied, since the problem involves polygon sd barriers located in various possible orientations in 3D space. We present here some means that are linked to basic mathematical geometrical tools. Those means are relatively simple, as compared to the physical formulation of the relevant diffraction solutions for sound barriers (e.g. Rosenhouse, 2019, 2020). Such calculations can apply algebraic, trigonometric or vector analysis and their combinations to define the geometries of barrier IL. This approach includes the location of the sources and control points, which are essential as data for finding IL and other issues of environmental acoustics. We will show solutions including results of IL for a common rectangular barrier, as compared to IL of a barrier with a sloped top and side, among other possibilities.

α Extension of a Class of T-Bezier Curve

2014 ◽  
Vol 556-562 ◽  
pp. 3478-3482
Author(s):  
Cheng Wei Wang
Keyword(s):  
Shape Control ◽  
Bézier Curve ◽  
Bezier Curve ◽  
Nurbs Curve ◽  
Extension Curve ◽  
Cad Cam ◽  
Control Capability ◽  
Better Than

By introducing the concept of weights in NURBS curve into a blending technique,the paper extends the representation of the T-Bezier curve.The generalized T-Bezier curve is denoted as α Extension T-Bezier curve,whose shape-control capability is shown to be much better than that of T-Bezier curve.The representation and properties of the extension curve is studied.The curve is easy and intuitive to reshape by varying the parameters;so it is useful in some applications of CAD/CAM .

CLOSED FREE-FORM SURFACE GEOMETRICAL MODELING A NEW APPROACH WITH GLOBAL AND LOCAL CHARACTERIZATION

2004 ◽  
Vol 04 (02) ◽  
pp. 241-262 ◽  
Author(s):  
JEAN-LUC MARI ◽  
JEAN SEQUEIRA
Keyword(s):  
Closed Surface ◽  
Free Form ◽  
Control Points ◽  
New Approach ◽  
Geometrical Properties ◽  
Geometrical Modeling ◽  
Knowing That ◽  
Local Characterization ◽  
And Control ◽  
Global And Local

In this paper, we present a new approach to geometrical modeling which allows the user to easily characterize and control the shape defined to a closed surface. We will focus on dealing with the shape's topological, morphological and geometrical properties separately. To do this, we have based our work on the following observations concerning surfaces defined by control-points, and implicit surfaces with skeleton. They both provide complementary approaches to the surface's deformation, and both have specific advantages and limits. We thus attempted to conceive a model which integrates the local and geometrical characterization induced by the control points, as well as the representation of the morphology given by the skeleton. Knowing that the lattice of control points is close to the surface and that the skeleton is centered in the related shape, we thought of a 3-layer model. The transition layer separates the local geometrical considerations from those linked to the global morphology. We apply our model to shape design in order to modify an object in an interactive and ergonomic way, as well as to reconstruction which allows better shape understanding. To do so, we present the algorithms related to these processes.

scholarly journals Laplacian-based Design: Sketching 3D Shapes

2006 ◽  
Vol 5 (3) ◽  
pp. 59-65 ◽  
Author(s):  
Bin Sheng ◽  
Enhua Wu
Keyword(s):  
Free Form ◽  
3D Space ◽  
3D Profile ◽  
Free Form Surfaces ◽  
The Given ◽  
2D Sketch ◽  
Laplacian Equations ◽  
3D Shapes ◽  
Minimum Curvature ◽  
3D Surfaces

The sketch-based shape modeling is one of the most challenging and active problems in computer graphics. In this paper, we present an interactive modeling system for generating free-form surfaces using a 2D sketch interface. Since inferring 3D shape from 2D sketches is an one to many function with no unique solution, we propose to interpret the given 2D curve to be the projection of the 3D curve that has minimum curvature among all the candidates in 3D. In this way, firstly, we present an algorithm to efficiently find a close approximation of this minimum curvature 3D space curve. In the second step, our system could identify the 3D surfaces automatically, and then we apply Delaunay triangulation on these surfaces. Finally, the shape of the triangular surface mesh that follows the 3D profile curves is computed using harmonic interpolation by solving Laplacian equations. We present experimental results on various kinds of drawings by the interactive modeler

Sketching Free-Forms in Semi-Immersive Virtual Environments

2000 ◽  
Author(s):  
André Stork ◽  
Oliver Schimpke ◽  
Raffaele de Amicis
Keyword(s):  
Mechanical Design ◽  
Free Form ◽  
Control Points ◽  
3D Interaction ◽  
Immersive Virtual Environments ◽  
3D Space ◽  
Projection Display ◽  
Nurbs Surfaces ◽  
Immersive Environment ◽  
Free Form Surfaces

Abstract In spite of the widely used sophisticated software tools for mechanical design, serious difficulties are encountered in the styling phase of the design process when free-form surfaces (shortly free-forms) have to be modelled. In this paper we extend the ideas of Sachs19 pioneering 3-Draw which allowed curve sketching in 3D space in several ways: we support the creation of NURBS surfaces (not only curves), we use a semi-immersive environment centered around a table-like stereoscopic back-projection display (also know as Virtual Table or Responsive Workbench16) and we give the user an immediate preview while he sketches free-form surface. This combination of features makes our approach unique in comparison to similar systems. Some of them only work on polyhedral models and others are restricted to indirect 3D interaction with control points. Our approach follows the Walk-Up VR concept: the user can just step in front of a Virtual Table and work on his task intuitively.

Dynamics and Control of Electrostatic Structures

1996 ◽  
Vol 63 (2) ◽  
pp. 383-391 ◽  
Author(s):  
L. Silverberg ◽  
L. Weaver
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Shape Control ◽  
Symmetric System ◽  
Strain Component ◽  
Control Points ◽  
Circulatory Effects ◽  
Dynamics And Control ◽  
And Control ◽  
Control Surfaces

This paper formulates the equations governing the dynamics and control of electrostatic structures. Using a Lagrangian mechanics approach, a potential energy function composed of a strain component, an electric component, and a gravitational component is defined. The resulting system of nonlinear ordinary differential equations are linearized about the electrostatic equilibrium leading to a linear system of ordinary differential equations characterized by mass, stiffness, damping, gyroscopic, and circulatory effects. In the absence of feedback control, the damping, gyroscopic, and circulatory effects vanish resulting in a symmetric system that admits normal mode vibration. Voltages applied over the charged subsurfaces (control points) of the electrostatic structure can control its shape. In the presence of feedback controls, control gains can be tailored to produce desirable levels of stiffness and damping. Two different control approaches are studied, one using control points that are attached to the electrostatic structure and one where the control points are fixed in space. Example problems illustrate the dynamics and control; specifically, circumstances that lead to instabilities, shape control using attached control surfaces, shape control using fixed control surfaces, and electrostatic damping.

Change of the redox potential in the profile of light grey forest surface-gleyed soil under different anthropogenic

2018 ◽  
Vol 2 (95) ◽  
pp. 26-29
Author(s):  
O.S. Gavrishko ◽  
Yu.M. Olifir ◽  
T.V. Partyka
Keyword(s):  
Redox Potential ◽  
Mineral Fertilizer ◽  
Fertilizer Application ◽  
Soil Tillage ◽  
Mineral Fertilizers ◽  
Gleyed Soil ◽  
Agricultural Use ◽  
Fertilizer System ◽  
And Control ◽  
The Given

The results of studies of the change in redox potential in the profile of light gray forest surface-gleyed soil on variants with long-term agricultural use without applying fertilizers and mineral fertilizer system solely compared with the soil under the forest are presented. On the basis of the conducted analyzes it was established, that soil tillage without fertilizer application and with mineral fertilizer solely has a different effect on ROP in the profile. In the soil without fertilization (control) as compared to the forest a moderate oxidizing (514 mV) and slightly oxidizing (437 mV) processes are happening. Prolonged application of mineral fertilizers to the soil (N65R68K68) significantly reduced the redox potential of all genetic horizons compared with forest and control without fertilizers. For the given fertilizer system the highest values of ROP were obtained in arable HEgl and underarable HEgl layers: 426 mV and 416 mV respectively. Redox potential sharply decreases with the depth to 398-311 mV, which characterizes processes occurring in the soil profile, as weakly reducing and close to moderately reducing.

scholarly journals On the BIC for determining the number of control points in B-spline surface approximation in case of correlated observations

2020 ◽  
Vol 10 (1) ◽  
pp. 110-123
Author(s):  
Gaël Kermarrec ◽  
Hamza Alkhatib
Keyword(s):  
Shape Control ◽  
Simple Procedure ◽  
Spline Approximation ◽  
Autoregressive Process ◽  
Optimal Number ◽  
Information Criteria ◽  
General Effect ◽  
Control Points ◽  
B Spline ◽  
Correlated Observations

Abstract B-spline curves are a linear combination of control points (CP) and B-spline basis functions. They satisfy the strong convex hull property and have a fine and local shape control as changing one CP affects the curve locally, whereas the total number of CP has a more general effect on the control polygon of the spline. Information criteria (IC), such as Akaike IC (AIC) and Bayesian IC (BIC), provide a way to determine an optimal number of CP so that the B-spline approximation fits optimally in a least-squares (LS) sense with scattered and noisy observations. These criteria are based on the log-likelihood of the models and assume often that the error term is independent and identically distributed. This assumption is strong and accounts neither for heteroscedasticity nor for correlations. Thus, such effects have to be considered to avoid under-or overfitting of the observations in the LS adjustment, i.e. bad approximation or noise approximation, respectively. In this contribution, we introduce generalized versions of the BIC derived using the concept of quasi- likelihood estimator (QLE). Our own extensions of the generalized BIC criteria account (i) explicitly for model misspecifications and complexity (ii) and additionally for the correlations of the residuals. To that aim, the correlation model of the residuals is assumed to correspond to a first order autoregressive process AR(1). We apply our general derivations to the specific case of B-spline approximations of curves and surfaces, and couple the information given by the different IC together. Consecutively, a didactical yet simple procedure to interpret the results given by the IC is provided in order to identify an optimal number of parameters to estimate in case of correlated observations. A concrete case study using observations from a bridge scanned with a Terrestrial Laser Scanner (TLS) highlights the proposed procedure.

scholarly journals Two view NURBS reconstruction based on GACO model

2021 ◽  
Author(s):  
Deepika Saini ◽  
Sanoj Kumar ◽  
Manoj K. Singh ◽  
Musrrat Ali
Keyword(s):  
Optimization Algorithms ◽  
Three Dimensional ◽  
Optimization Procedure ◽  
Least Square ◽  
Ant Colony ◽  
Free Form ◽  
Reconstruction Process ◽  
Nurbs Curve ◽  
Data Points ◽  
Ant Colony Optimization Algorithms

AbstractThe key job here in the presented work is to investigate the performance of Generalized Ant Colony Optimizer (GACO) model in order to evolve the shape of three dimensional free-form Non Uniform Rational B-Spline (NURBS) curve using stereo (two) views. GACO model is a blend of two well known meta-heuristic optimization algorithms known as Simple Ant Colony and Global Ant Colony Optimization algorithms. Basically, the work talks about the solution of NURBS-fitting based reconstruction process. Therefore, GACO model is used to optimize the NURBS parameters (control points and weights) by minimizing the weighted least-square errors between the data points and the fitted NURBS curve. The algorithm is applied by first assuming some pre-fixed values of NURBS parameters. The experiments clearly show that the optimization procedure is a better option in a case where good initial locations of parameters are selected. A detailed experimental analysis is given in support of our algorithm. The implemented error analysis shows that the proposed methodology perform better as compared to the conventional methods.

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