Dynamic Evaluation of Free-Form Curves and Surfaces

2017 ◽  
Vol 39 (2) ◽  
pp. B424-B441 ◽  
Author(s):  
Xunnian Yang ◽  
Jialin Hong
Keyword(s):  
Free Form ◽  
Dynamic Evaluation ◽  
Curves And Surfaces

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.

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.

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

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 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.

The Geometric Modeling and Interrogation System Praxiteles

1995 ◽  
Vol 11 (02) ◽  
pp. 117-132
Author(s):  
S. L. Abrams ◽  
L. Bardis ◽  
C. Chryssostomidis ◽  
N. M. Patrikalakis ◽  
S. T. Tuohy ◽  
...  
Keyword(s):  
Geometric Modeling ◽  
Data Exchange ◽  
Free Form ◽  
Mathematical Methods ◽  
Initial Shape ◽  
Marine Propellers ◽  
Curves And Surfaces ◽  
Design Requirements ◽  
And Performance

After establishment of design requirements and creation of an initial shape, the design process of free form shapes should include interrogation and fairing until a desired shape, with appropriate geometric and performance characteristics, is achieved. Afterwards, the quality of the manufactured product can be determined by comparing measured data with the design model. To permit automated design and manufacturing, mathematical methods and algorithms for the creation, interrogation, fairing, and inspection of curves and surfaces have been developed and integrated into a computer system called Praxiteles. The general layout of Praxiteles, along with a description of design capabilities, is presented. This description covers the areas of input, output, approximation and conversion for data exchange, a summary of some shape creation methods, and a description of some advanced interactive interrogation, fairing, and inspection methods for NURBS curves and surfaces. Examples illustrate some of the features of the system, as applied in the design and inspection of marine propellers. Recommendations for future development of the system are also presented.

Offset Curves and Surfaces With Continued Fractions

1999 ◽  
Author(s):  
Manhong Wen ◽  
Kwun-Lon Ting
Keyword(s):  
Continued Fractions ◽  
Free Form ◽  
New Approach ◽  
Planar Curves ◽  
Curves And Surfaces ◽  
Cad Cam ◽  
Offset Curves ◽  
Unified Algorithm ◽  
The Relationship ◽  
Spatial Curves

Abstract This paper provides a new approach to approximate the offset of free-form curves and surfaces with continued fractions. The properties of continued fractions are introduced. The convergence of the approximation is proved and the relationship between accuracy and approximate step is established. It shows that the approximation converges fast and reliably, the error of approximation is easily estimated and controlled, and a unified algorithm can be used to generate the rational offset of non-rational and rational planar curves, spatial curves, and surfaces in CAD/CAM industry.

Free-form curves and surfaces

1987 ◽  
pp. 28-52
Author(s):  
Nadia Magnenat-Thalmann ◽  
Daniel Thalmann
Keyword(s):  
Free Form ◽  
Curves And Surfaces

Aesthetic Curves and Surfaces in Computer Aided Geometric Design

2014 ◽  
Vol 8 (3) ◽  
pp. 304-316 ◽  
Author(s):  
Kenjiro T. Miura ◽  
◽  
R. U. Gobithaasan ◽  
Keyword(s):  
Free Form ◽  
Design Quality ◽  
Aesthetic Design ◽  
3D Objects ◽  
Curves And Surfaces ◽  
Viable Solution ◽  
Main Components ◽  
2D And 3D ◽  
Free Form Surfaces ◽  
Complex Curvature

Aesthetic shapes are usually actualized as 3D objects represented by free-form surfaces. The main components used to achieve aesthetic surfaces are 2D and 3D curves, which are the elements most basic for determining the shapes and silhouettes of industrial products. Bézier, B-Spline and NURBS are types of flexible curves developed for various design intents. These curves, however produce complex curvature functions that may undermine the formulation of shape aesthetics. A viable solution to this problem is to formulate aesthetic curves and surfaces from well-defined curvatures to improve aesthetic design quality. This paper advocates formalizing aesthetic curve and surface theories to fill the gapmentioned above, which has existed since the 1970s. This paper begins by reviewing on fair curves and surfaces. It then extensively discusses on the technicalities of Log-Aesthetic (LA) curves and surfaces and touches on industrial design applications. These emerging LA curves have a high potential for being used as standards to generate, evaluate and reshape aesthetic curves and surfaces, thus revolutionizing efficiency in developing curve and shape aesthetics.

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