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# 数学代写|曲线和曲面代写Curves And Surfaces代考|MATH322 Removing knots and algebra of splines

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## 数学代写|曲线和曲面代写Curves And Surfaces代考|Removing knots and algebra of splines

If a spline curve of degree $n$ is represented in B-spline basis defined for a knot sequence $\hat{u}0, \ldots, \hat{u}{N+1}$, with a knot $\hat{u}{k+1}$ of multiplicity $r+1$, such that $\hat{u}_k \leq \hat{u}{k+1}<\hat{u}{k+2}$ and $n{k+1}$, then this knot is removable, i.e., there exists a representation of this curve with the knot sequence $\left(u_0, \ldots, u_N\right)=\left(\hat{u}0, \ldots, \hat{u}_k, \hat{u}{k+2}, \ldots, \hat{u}_{N+1}\right)$.

Algorithm A.9 is based on a linear dependency between the points $\boldsymbol{d}0, \ldots, \boldsymbol{d}{N-n-1}$ and the points $\hat{\boldsymbol{d}}0, \ldots, \hat{\boldsymbol{d}}{N-n}$ obtained by knot insertion:
\begin{aligned} &\boldsymbol{d}i=\hat{\boldsymbol{d}}_i \quad \text { for } i \leq k-n, \ &\frac{u{i+n}-\hat{u}{k+1}}{u{i+n}-u_i} \boldsymbol{d}{i-1}+\frac{\hat{u}{k+1}-u_i}{u_{i+n}-u_i} \boldsymbol{d}i=\hat{\boldsymbol{d}}_i \quad \text { for } i=k-n+1, \ldots, k-r, \ &\boldsymbol{d}{i-1}=\hat{\boldsymbol{d}}i \text { for } i>k-r . \ & \end{aligned} Given the control points $\hat{\boldsymbol{d}}_i$ representing an arbitrary B-spline curve we can write these equations with the intention of finding unknown control points $\boldsymbol{d}_i$. The number of equations is the number of unknown control points plus 1 . This system of equations is consistent if the knot $\hat{u}_k$ is removable and the curve has a shorter representation. To obtain a good accuracy, it is best to compute the points $\boldsymbol{d}{k-n}, \ldots, \boldsymbol{d}_{k-r}$ by solving a linear least-squares problem.

## 数学代写|曲线和曲面代写Curves And Surfaces代考|Convergence of repeated knot insertion

If an infinite sequence of numbers $v_1, v_2, \ldots$ is dense in the interval $\left[u_n, u_{N-n}\right)$, then the process of inserting these numbers as new knots produces an infinite sequence of representations of a given curve $\boldsymbol{s}$, with the control polylines made of increasing numbers of shorter and shorter line segments. These polylines converge to the curve. It may be proved (see Cohen and Schumaker [1985]) that the distance between the polyline and the curve is estimated by the expression $C h^2$, where the constant $C$ depends on the curve and $h$ is the maximal distance between consecutive knots of its representation. This is a fast convergence; often it suffices to insert relatively few knots in order to obtain a polyline being a very good approximation of the spline curve. This fact is exploited by the methods of generating surfaces via mesh refinement discussed in Section A.6.

## 数学代写|曲线和曲面代写Curves And Surfaces代考|THE DNIDED DIFFERENCES ALGORITHM

$$p_0(x)=1, p_1(x) \quad=x-u_0, p_2(x)=\left(x-u_0\right)\left(x-u_1\right), \quad \vdots p_n(x)=\left(x-u_0\right)\left(x-u_1\right) \ldots\left(x-u_{n-1}\right) .$$

## MATLAB代写

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