ERROR BOUNDED VARIABLE DISTANCE OFFSET OPERATOR FOR FREE FORM CURVES AND SURFACES

1991 ◽  
Vol 01 (01) ◽  
pp. 67-78 ◽  
Author(s):  
GERSHON ELBER ◽  
ELAINE COHEN
Keyword(s):  
Approximation Algorithms ◽  
Error Bound ◽  
Free Form ◽  
Global Error Bound ◽  
New Methods ◽  
Curves And Surfaces ◽  
Polygon Mesh ◽  
Finite Set ◽  
Variable Distance ◽  
Improved Accuracy

Most offset approximation algorithms for freeform curves and surfaces may be classified into two main groups. The first approximates the curve using simple primitives such as piecewise arcs and lines and then calculates the (exact) offset operator to this approximation. The second offsets the control polygon/mesh and then attempts to estimate the error of the approximated offset over a region. Most of the current offset algorithms estimate the error using a finite set of samples taken from the region and therefore can not guarantee the offset approximation is within a given tolerance over the whole curve or surface. This paper presents new methods to globally bound the error of the approximated offset of freeform curves and surfaces and then automatically derive new approximations with improved accuracy. These tools can also be used to develop a global error bound for a variable distance offset operation and to detect and trim out loops in the offset.

Approximate Degree Reduction of NURBS Curves and Surfaces: An Integrated Approach

1994 ◽  
Author(s):  
Srinivasa P. Varanasi ◽  
Athamaram H. Soni
Keyword(s):  
Shape Control ◽  
Data Exchange ◽  
Data Transfer ◽  
Integrated Approach ◽  
Free Form ◽  
Aircraft Industry ◽  
Trimmed Surfaces ◽  
Curves And Surfaces ◽  
High Degree ◽  
Boundary Curves

Abstract Data exchange between different CAD systems usually requires conversion between different representations of free-form curves and surfaces. Also, trimmed surfaces give rise to high degree boundary curves. Accurate conversion of these forms becomes necessary for reliable data transfer. Also important is the issue of shape control, specially in the aircraft industry. The objective of this paper is to investigate conversion methods and effect of shape control on the design and choice of such methods.

New methods for projecting a 3D object onto a free-form surface

2020 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Jingyu Pei ◽  
Xiaoping Wang ◽  
Leen Zhang ◽  
Yu Zhou ◽  
Jinyuan Qian
Keyword(s):  
Approximate Solution ◽  
New Method ◽  
Free Form ◽  
Parallel Projection ◽  
Content Type ◽  
3D Object ◽  
New Methods ◽  
Free Form Surface ◽  
Newton Raphson ◽  
Raphson Iteration

Purpose This paper aims to provide a series of new methods for projecting a three-dimensional (3D) object onto a free-form surface. The projection algorithms presented can be divided into three types, namely, orthogonal, perspective and parallel projection. Design/methodology/approach For parametric surfaces, the computing strategy of the algorithm is to obtain an approximate solution by using a geometric algorithm, then improve the accuracy of the approximate solution using the Newton–Raphson iteration. For perspective projection and parallel projection on an implicit surface, the strategy replaces Newton–Raphson iteration by multi-segment tracing. The implementation takes two mesh objects as an example of calculating an image projected onto parametric and implicit surfaces. Moreover, a comparison is made for orthogonal projections with Hu’s and Liu’s methods. Findings The results show that the new method can solve the 3D objects projection problem in an effective manner. For orthogonal projection, the time taken by the new method is substantially less than that required for Hu’s method. The new method is also more accurate and faster than Liu’s approach, particularly when the 3D object has a large number of points. Originality/value The algorithms presented in this paper can be applied in many industrial applications such as computer aided design, computer graphics and computer vision.

New methods for fiber Bragg grating synthesis with improved accuracy

2007 ◽  
Author(s):  
O. V. Belai ◽  
L. L. Frumin ◽  
E. V. Podivilov ◽  
O. Ya. Schwarz ◽  
D. A. Shapiro
Keyword(s):  
Fiber Bragg Grating ◽  
Bragg Grating ◽  
New Methods ◽  
Improved Accuracy

Semi-automatic system for defining free-form curves and surfaces

1983 ◽  
Vol 15 (2) ◽  
pp. 65-72 ◽  
Author(s):  
Pierre E. Bézier ◽  
Salah Sioussiou
Keyword(s):  
Automatic System ◽  
Free Form ◽  
Curves And Surfaces

scholarly journals A COMPRESSED SENSING BASED APPROACH FOR SUBTYPING OF LEUKEMIA FROM GENE EXPRESSION DATA

2011 ◽  
Vol 09 (05) ◽  
pp. 631-645 ◽  
Author(s):  
WENLONG TANG ◽  
HONGBAO CAO ◽  
JUNBO DUAN ◽  
YU-PING WANG
Keyword(s):  
Gene Expression ◽  
Compressed Sensing ◽  
Gene Expression Analysis ◽  
Expression Data ◽  
Data Set ◽  
New Methods ◽  
Genome Wide ◽  
Different Types ◽  
Genome Wide Data ◽  
Improved Accuracy

With the development of genomic techniques, the demand for new methods that can handle high-throughput genome-wide data effectively is becoming stronger than ever before. Compressed sensing (CS) is an emerging approach in statistics and signal processing. With the CS theory, a signal can be uniquely reconstructed or approximated from its sparse representations, which can therefore better distinguish different types of signals. However, the application of CS approach to genome-wide data analysis has been rarely investigated. We propose a novel CS-based approach for genomic data classification and test its performance in the subtyping of leukemia through gene expression analysis. The detection of subtypes of cancers such as leukemia according to different genetic markups is significant, which holds promise for the individualization of therapies and improvement of treatments. In our work, four statistical features were employed to select significant genes for the classification. With our selected genes out of 7,129 ones, the proposed CS method achieved a classification accuracy of 97.4% when evaluated with the cross validation and 94.3% when evaluated with another independent data set. The robustness of the method to noise was also tested, giving good performance. Therefore, this work demonstrates that the CS method can effectively detect subtypes of leukemia, implying improved accuracy of diagnosis of leukemia.

Biquadratic TC-Bézier Curves with Shape Parameter

2011 ◽  
Vol 467-469 ◽  
pp. 57-62
Author(s):  
Xu Min Liu ◽  
Xian Peng Yang ◽  
Yan Ling Wu
Keyword(s):  
Shape Parameter ◽  
Free Form ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Curves And Surfaces ◽  
G1 Continuity ◽  
Shape Controlling

Shape controlling is a popular topic in curves and surfaces design with free form. In this paper, a new curve, to be called Biquadratic TC-Bézier curves with shape parameter , is constructed in the space . We show that such curves share the same properties as the traditional Bézier curves in polynomial spaces. The shape of new curves, representing circle and ellipse accurately, can be adjusted by changing the value of the parameter . Then we give the G1 continuity conditions of Biquadratic TC-Bézier curves with shape parameter and its application in surfaces modeling.

scholarly journals Error Bounds and Finite Termination for Constrained Optimization Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Wenling Zhao ◽  
Daojin Song ◽  
Bingzhuang Liu
Keyword(s):  
Constrained Optimization ◽  
Error Bound ◽  
Feasible Solution ◽  
Optimization Problems ◽  
Sufficient Conditions ◽  
Merit Function ◽  
Global Error Bound ◽  
Constrained Optimization Problems ◽  
Nonconvex Constrained Optimization ◽  
Necessary And Sufficient

We present a global error bound for the projected gradient of nonconvex constrained optimization problems and a local error bound for the distance from a feasible solution to the optimal solution set of convex constrained optimization problems, by using the merit function involved in the sequential quadratic programming (SQP) method. For the solution sets (stationary points set andKKTpoints set) of nonconvex constrained optimization problems, we establish the definitions of generalized nondegeneration and generalized weak sharp minima. Based on the above, the necessary and sufficient conditions for a feasible solution of the nonconvex constrained optimization problems to terminate finitely at the two solutions are given, respectively. Accordingly, the results in this paper improve and popularize existing results known in the literature. Further, we utilize the global error bound for the projected gradient with the merit function being computed easily to describe these necessary and sufficient conditions.

scholarly journals Generalized Fractional Bézier Curve with Shape Parameters

Mathematics ◽  
2021 ◽  
Vol 9 (17) ◽  
pp. 2141
Author(s):  
Syed Ahmad Aidil Adha Said Mad Said Mad Zain ◽  
Md Yushalify Misro ◽  
Kenjiro T. Miura
Keyword(s):  
Basis Functions ◽  
Bézier Curve ◽  
Free Form ◽  
Bezier Curve ◽  
Geometric Continuity ◽  
Shape Parameters ◽  
Complex Curves ◽  
Curves And Surfaces ◽  
New Type ◽  
Fractional Parameter

The construction of new basis functions for the Bézier or B-spline curve has been one of the most popular themes in recent studies in Computer Aided Geometric Design (CAGD). Implementing the new basis functions with shape parameters provides a different viewpoint on how new types of basis functions can develop complex curves and surfaces beyond restricted formulation. The wide selection of shape parameters allows more control over the shape of the curves and surfaces without altering their control points. However, interpolated parametric curves with higher degrees tend to overshoot in the process of curve fitting, making it difficult to control the optimal length of the curved trajectory. Thus, a new parameter needs to be created to overcome this constraint to produce free-form shapes of curves and surfaces while still preserving the basic properties of the Bézier curve. In this work, a general fractional Bézier curve with shape parameters and a fractional parameter is presented. Furthermore, parametric and geometric continuity between two generalized fractional Bézier curves is discussed in this paper, as well as demonstrating the effect of the fractional parameter of curves and surfaces. However, the conventional parametric and geometric continuity can only be applied to connect curves at the endpoints. Hence, a new type of continuity called fractional continuity is proposed to overcome this limitation. Thus, with the curve flexibility and adjustability provided by the generalized fractional Bézier curve, the construction of complex engineering curves and surfaces will be more efficient.

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