scholarly journals Least Squares Fitting of Piecewise Algebraic Curves

2007 ◽  
Vol 2007 ◽  
pp. 1-11 ◽  
Author(s):  
Chun-Gang Zhu ◽  
Ren-Hong Wang
Keyword(s):  
Least Squares ◽  
Algebraic Curve ◽  
Algebraic Curves ◽  
Scattered Data ◽  
Energy Term ◽  
Bivariate Spline ◽  
Piecewise Algebraic Curve ◽  
Piecewise Algebraic Curves ◽  
The Given

A piecewise algebraic curve is defined as the zero contour of a bivariate spline. In this paper, we present a new method for fittingC1piecewise algebraic curves of degree 2 over type-2 triangulation to the given scattered data. By simultaneously approximating points, associated normals and tangents, and points constraints, the energy term is also considered in the method. Moreover, some examples are presented.

scholarly journals An Algorithm for Isolating the Real Solutions of Piecewise Algebraic Curves

2011 ◽  
Vol 2011 ◽  
pp. 1-11 ◽  
Author(s):  
Jinming Wu ◽  
Xiaolei Zhang
Keyword(s):  
Iterative Algorithm ◽  
Algebraic Curve ◽  
Spline Function ◽  
Algebraic Curves ◽  
The Real ◽  
Real Solutions ◽  
Bivariate Spline ◽  
Piecewise Algebraic Curve ◽  
Piecewise Algebraic Curves

The piecewise algebraic curve, as the set of zeros of a bivariate spline function, is a generalization of the classical algebraic curve. In this paper, an algorithm is presented to compute the real solutions of two piecewise algebraic curves. It is primarily based on the Krawczyk-Moore iterative algorithm and good initial iterative interval searching algorithm. The proposed algorithm is relatively easy to implement.

scholarly journals An Upper Bound of the Bezout Number for Piecewise Algebraic Curves over a Rectangular Partition

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Feng-Gong Lang ◽  
Xiao-Ping Xu
Keyword(s):  
Upper Bound ◽  
Algebraic Curve ◽  
Algebraic Curves ◽  
Zero Set ◽  
Bivariate Spline ◽  
Common Intersection ◽  
Piecewise Algebraic Curve ◽  
Piecewise Algebraic Curves ◽  
The Common ◽  
Intersection Points

A piecewise algebraic curve is a curve defined by the zero set of a bivariate spline function. Given two bivariate spline spaces (Δ) over a domainDwith a partition Δ, the Bezout number BN(m,r;n,t;Δ) is defined as the maximum finite number of the common intersection points of two arbitrary piecewise algebraic curves (Δ). In this paper, an upper bound of the Bezout number for piecewise algebraic curves over a rectangular partition is obtained.

Application of Fitting Estimation Method in GPS Height Fitting

2013 ◽  
Vol 694-697 ◽  
pp. 2545-2549 ◽  
Author(s):  
Qian Wen Cheng ◽  
Lu Ben Zhang ◽  
Hong Hua Chen
Keyword(s):  
Least Squares ◽  
Estimation Method ◽  
Least Square ◽  
Least Squares Collocation ◽  
Normal Height ◽  
Geodetic Height ◽  
The Given ◽  
Surveying And Mapping

The key point researched by many scholars in the field of surveying and mapping is how to use the given geodetic height H measured by GPS to obtain the normal height. Although many commonly-used fitting methods have solved many problems, they all value the pending parameters as the nonrandom variables. Figuring out the best valuations, according to the traditional least square principle, only considers its trend or randomness, which is theoretically incomprehensive and have limitations in practice. Therefore, a method is needed not only considers its trend but also takes randomness into account. This method is called the least squares collocation.

Curve Shape Modification and Fairness Evaluation

2007 ◽  
Author(s):  
Tetsuzo Kuragano ◽  
Akira Yamaguchi
Keyword(s):  
Least Squares ◽  
Least Squares Method ◽  
Curvature Vector ◽  
Radius Of Curvature ◽  
Curve Shape ◽  
Linear Quadratic ◽  
Nurbs Curve ◽  
First Derivative ◽  
Curvature Distribution ◽  
The Given

A method to generate a quintic NURBS curve which passes through the given points is described. In this case, there are four more equations than there are positions of the control points. Therefore, four gradients which are the first derivative of a NURBS equation are assigned to the given points. In addition to this method, another method to generate a quintic NURBS curve which passes through the given points and which has the first derivative at these given points is described. In this case, a linear system will be underdetermined, determined or overdetermined depending on the number of given points with gradients. A method to modify NURBS curve shape according to the specified radius of curvature distribution to realize an aesthetically pleasing freeform curve is described. The differences between the NURBS curve radius of curvature and the specified radius of curvature is minimized by introducing the least-squares method. A criterion for a fair curve is proposed. Evaluation whether the designed curve is fair or not is accomplished by a comparison of the designed curve to a curve whose radius of curvature is monotone. The radius of curvature is specified by linear, quadratic, and cubic function using the least-squares method. A curve whose radius of curvature is reshaped by one of these algebraic functions is considered as a fair curve. The curvature vector of the curve is used to evaluate the fairness. The comparison of unit curvature vectors is used to evaluate the directional similarity of the curve. The comparison of the curvature is used to evaluate the similarity of the magnitude of curvature vectors. If the directional similarity of the designed curve is close to the fair curve, and also the similarity of the curvature is close to the fair curve, the designed curve can be judged as a fair curve.

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