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.

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.

Geometric Modeling of Compound NURBS Surfaces

2020 ◽  
Vol 38 (2A) ◽  
pp. 277-287
Author(s):  
Ali K. Alwan ◽  
Wisam K. Hamdan
Keyword(s):  
Geometric Modeling ◽  
Design Method ◽  
Control Points ◽  
Sculptured Surfaces ◽  
Nurbs Surface ◽  
Surface Design ◽  
Spline Basis ◽  
Automotive Parts ◽  
Nurbs Surfaces ◽  
Complex Features

The design of sculptured surfaces occupies an essential area in the field of modern industrial, aerospace, and medical applications. The challenge is to design products that have complex features efficiently with great flexibility of editing in certain regions without affecting other regions, which the designer has no intent to modify. In this paper, we propose a surface design method based on compound NURBS surface to model automotive parts with 400 control points. First, a Non-Uniform B-Spline basis function is derived with a cubic degree and 20 control points. This method is utilized to design car posterior door, car hood, and rear car door as case studies.

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.

A Conceptual Design of the Knee Rehabilitation Device

2012 ◽  
Vol 215-216 ◽  
pp. 225-228
Author(s):  
Liang Han ◽  
You Yang Li ◽  
Qian Zhang
Keyword(s):  
Pulse Width Modulation ◽  
Liquid Crystal Display ◽  
Main Body ◽  
Stepping Motor ◽  
Rehabilitation Process ◽  
Knee Rehabilitation ◽  
Body Design ◽  
And Control ◽  
Micro Controller Unit ◽  
The Given

This paper aims to design a knee rehabilitation device which can help the knee patients to perform movement in bending leg manner. And in that way the rehabilitation process will be improved greatly. The design of the device includes the mechanical main body design, driving motor, and control circuit which contains the keyboard setting, display unit and clock unit. Through the pulse width modulation (PWM) technology the stepping motor is driven and the automatic bent in leg is achieved. During the exercise the following information is known: the starting time, the lasting time, the angle of movement and the speed of movement. The micro controller unit (MCU) is responsible for the information processing from both the key and liquid crystal display (LCD). After the key inputs the given parameters the stepping motor can output the desired motions. Meanwhile, the LCD can display the input information. Now the project has accomplished the preliminary design, and the concrete scheme is shown in this paper.

Constrained Shape Scaling of Trimmed NURBS Surfaces

1999 ◽  
Author(s):  
Pifu Zhang ◽  
Caiming Zhang ◽  
Fuhua (Frank) Cheng
Keyword(s):  
Data Exchange ◽  
Surface Deformation ◽  
Current Data ◽  
Free Form ◽  
Test Results ◽  
Nurbs Surface ◽  
Curvature Distribution ◽  
The Given ◽  
Boundary Curves

Abstract A method to scale and deform a trimmed NURBS surface while holding the shape and size of specific features (trimming curves) unchanged is presented. The new surface is formed by scaling the given surface according to the scaling requirement first; and then attaching the (original) features to the scaled NURBS surface at appropriate locations. The attaching process requires several geometric operations and constrained free-form surface deformation. The resulting surface has the same features as the original surface and same boundary curves as the scaled surface while reflecting the shape and curvature distribution of the scaled surface. This is achieved by minimizing a shape-preserving objective function which covers all the factors in the deformation process such as bending, stretching and spring effects. The resulting surface maintains a NURBS representation and, hence, is compatible with most of the current data-exchange standards. Test results on several car parts with trimming curves are included. The, quality of the resulting surfaces is examined using the highlight line model.

Nonlinear Axial and Transverse Oscillation and Control of a PZT Laminated Distributed Structronic System

2000 ◽  
Author(s):  
H. S. Tzou ◽  
J. H. Ding ◽  
W. K. Chai
Keyword(s):  
Distributed Control ◽  
Shape Control ◽  
Geometrical Nonlinearity ◽  
Axial Displacement ◽  
Beam Model ◽  
Control Effect ◽  
Laminated Beam ◽  
Practical Applications ◽  
Axial Oscillation ◽  
And Control

Abstract Piezoelectric laminated distributed systems have broad applications in many new smart structures and structronic systems. As the shape control becomes an essential issue in practical applications, the nonlinear large deformation has to be considered, and thus, the geometrical nonlinearity has to be incorporated. Two electromechanical partial differential equations, one in the axial direction and the other in the transverse direction, are derived for the nonlinear PZT laminated beam model. The conventional approach is to neglect the axial oscillation and distributed sensing and control of the distributed laminated beam is evaluated, excluding the effect of axial oscillation. In this paper, influence of the axial displacement to the dynamics and distributed control effect is evaluated. Analysis results reveal that the axial displacement, indeed, has significant influence to the dynamic and distributed control responses of the nonlinear distributed PZT laminated beam structronics systems.

scholarly journals Fully Automatic Cell Segmentation with Fourier Descriptors

2021 ◽  
Author(s):  
Dominik Hirling ◽  
Peter Horvath
Keyword(s):  
Neural Networks ◽  
Convolutional Neural Networks ◽  
Fundamental Problem ◽  
Cell Segmentation ◽  
Fourier Descriptors ◽  
Control Points ◽  
Coefficient Vector ◽  
Fully Automatic ◽  
High Level ◽  
And Control

Cell segmentation is a fundamental problem in biology for which convolutional neural networks yield the best results nowadays. In this paper, we present HarmonicNet, a network, which is a modification of the popular StarDist and SplineDist architectures. While StarDist and SplineDist describe an object by the lengths of equiangular rays and control points respectively, our network utilizes Fourier descriptors, predicting a coefficient vector for every pixel on the image, which implicitly define the resulting segmentation. We evaluate our model on three different datasets, and show that Fourier descriptors can achieve a high level of accuracy with a small number of coefficients. HarmonicNet is also capable of accurately segmenting objects that are not star-shaped, a case where StarDist performs suboptimally according to our experiments.

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