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Approximating Bézier curves with least square polygons

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Abstract

Approximating Bézier curve by polygon is a traditional technique in computer-aided geometric design, and there are several methods for constructing such polygons. In this paper, we propose to approximate Bézier curve by the least square polygon (LSP). LSP was derived by least square fitting which minimizes the integral of approximating errors. We provide a closed formula for the vertices of LSP. Because of its optimization method, LSP has lower approximation errors than existing approximating polygons. This observation is verified by many numerical examples we tested and several typical examples are included in this paper.

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Data availability statement

We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted. The datasets generated during and analyzed during the current study are available from the corresponding author on reasonable request.

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Acknowledgements

The authors owe their gratitude to the anonymous referees for their valuable comments which have helped to improve the presentation of this manuscript. This work was supported by the National Natural Science Foundation of China (NSFC) under the project numbers 61872121.

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Authors and Affiliations

  1. School of Science, Hangzhou Dianzi University, Hangzhou, 310018, China

    Yajuan Li, Wenbiao Jin & Chongyang Deng

  2. Hangzhou Dianzi University Information Engineering College, Hangzhou, 310018, China

    Meng Zhang

Authors
  1. Yajuan Li
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  2. Meng Zhang
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  3. Wenbiao Jin
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  4. Chongyang Deng
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Correspondence to Chongyang Deng.

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Appendices

Appendix A. The computation of \(\textbf{c}_j\)

$$\begin{aligned} \textbf{c}_{ j }= & {} \int _{0}^{\frac{1}{n}} \left[ nt \textbf{P}\left( t+t_{ j -1}\right) + ( 1- nt) \textbf{P}\left( t+t_j \right) \right] {\textrm{d}}t \\= & {} \int _0 ^{\frac{1}{n}}\sum \limits _{g = 0 }^{k} B^k_g \left( t+t_{j -1}\right) nt{{\varvec{p}}}_g{\textrm{d}}t \\{} & {} + \int _0 ^{\frac{1}{n}}\sum \limits _{g = 0 }^{k} B^k_g \left( t+t_j \right) (1-nt){{\varvec{p}}}_g{\textrm{d}}t. \\= & {} \sum \limits _{g = 0 }^{k} {{\varvec{p}}}_g \left[ \int _0 ^{\frac{1}{n}} B^k_g\left( t+t_{j -1}\right) nt{\textrm{d}}t \right. \\{} & {} \left. + \int _0 ^{\frac{1}{n}} B^k_g \left( t+t_j \right) (1-nt){\textrm{d}}t\right] . \end{aligned}$$

Let \(t+t_{j-1} = u\) and \(t+t_j = u\) in the two integrals respectively, we have

$$\begin{aligned} \textbf{c}_{j}= & {} \sum \limits _{g = 0 }^{k} {{\varvec{p}}}_g \left[ \int _{t_{j-1}} ^{t_j} B^k_g(u) (nu-j+1){\textrm{d}}u \right. \\{} & {} \left. + \int _{t_j}^{t_{j+1}} B^k_g(u) (1-nu+j){\textrm{d}}u \right] \\= & {} \sum \limits _{g = 0 }^{k} {{\varvec{p}}}_g \left[ n \int _{t_{j-1}} ^{t_j} B^k_g(u) u {\textrm{d}}u +(1-j)\int _{t_{j-1}} ^{t_j} B^k_g(u) {\textrm{d}}u \right] \\{} & {} + \sum \limits _{g = 0 }^{k} {{\varvec{p}}}_g \left[ - n\int _{t_j}^{t_{i+1}} B^k_g(u) u {\textrm{d}}u + (1+j)\int _{t_j}^{t_{i+1}} B^k_g(u) {\textrm{d}}u\right] . \\= & {} \sum \limits _{g = 0 }^{k} {{\varvec{p}}}_g \frac{n(g+1)}{k+1} \left( \int _{t_{j-1}}^{t_j} B^{k+1}_{g+1}(u) {\textrm{d}}u - \int _{t_j}^{t_{j+1}} B^{k+1}_{g+1}(u) {\textrm{d}}u\right) \\{} & {} + \sum \limits _{g = 0 }^{k} {{\varvec{p}}}_g \left[ \int _{t_{j-1}}^{t_{j+1}} B^k_g(u) {\textrm{d}}u - j\int _{t_{j-1}}^{t_j} B^k_g(u) {\textrm{d}}u+ j\int _{t_j}^{t_{j+1}} B^k_g(u) {\textrm{d}}u \right] . \end{aligned}$$

By the equation \( \int _{0}^t B^k_g(u) {\textrm{d}}u = \frac{1}{k+1} \sum \nolimits _{i = g+1 }^{k+1} B^{k+1}_{i} (t) \) [13], \(\textbf{c}_{j }\) can be computed as

$$\begin{aligned} \textbf{c}_{j}= & {} \frac{n }{(k+1)(k+2)} \sum \limits _{g = 0 }^{k} {{\varvec{p}}}_g (g+1) \\{} & {} \sum \limits _{i=g+2}^{k+2} \left[ 2B^{k+2}_{i}\left( t_{j}\right) - B^{k+2}_{i}\left( t_{j-1}\right) -B^{k+2}_{i}\left( t_{j+1}\right) \right] \\{} & {} + \frac{1}{k+1 } \sum \limits _{g = 0 }^{k} {{\varvec{p}}}_g \sum \limits _{i=g+1}^{k+1} \left[ B^{k+1}_{i}\left( t_{j+1}\right) - B^{k+1}_{i}\left( t_{j-1}\right) \right] \\{} & {} + \frac{j}{k+1 }\sum \limits _{g = 0 }^{k} {{\varvec{p}}}_g \sum \limits _{i=g+1}^{k+1} \left[ B^{k+1}_{i}\left( t_{j+1}\right) - 2B^{k+1}_{i}\left( t_{j}\right) + B^{k+1}_{i}(t_{j-1}) \right] \\= & {} \frac{n }{(k+1)(k+2)} \sum \limits _{g = 0 }^{k} {{\varvec{p}}}_g (g+1) \\{} & {} \sum \limits _{i=g+2}^{k+2} \left[ 2B^{k+2}_{i}\left( t_{j}\right) - B^{k+2}_{i}\left( t_{j-1}\right) -B^{k+2}_{i}\left( t_{j+1}\right) \right] \\{} & {} + \frac{1}{k+1 } \sum \limits _{g = 0 }^{k} {{\varvec{p}}}_g \sum \limits _{i=g+1}^{k+1} \left[ (j+1)B^{k+1}_{i}\left( t_{j+1} \right) \right. \\{} & {} \left. - 2j B^{k+1}_{i}\left( t_{j}\right) +(j-1) B^{k+1}_{i}\left( t_{j-1}\right) \right] \end{aligned}$$

By \(B^{k+1}_{i}(t) = \frac{i+1}{t(k+2)}B^{k+2}_{i}(t)\), It’s easy to compute

$$\begin{aligned}{} & {} \frac{1}{k+1 } \sum \limits _{g = 0 }^{k} {{\varvec{p}}}_g \sum \limits _{i=g+1}^{k+1} \left[ (j+1)B^{k+1}_{i}\left( t_{j+1}\right) \right. \\{} & {} \qquad \left. - 2j B^{k+1}_{i}\left( t_{j}\right) +(j-1) B^{k+1}_{i}\left( t_{j-1}\right) \right] \\{} & {} \quad = \frac{n}{(k+1)(k+2)} \sum \limits _{g = 0 }^{k} {{\varvec{p}}}_g \sum \limits _{i=g+1}^{k+1} (i+1) \left[ B^{k+2}_{i+1}\left( t_{j+1} \right) \right. \\{} & {} \qquad \left. - 2 B^{k+2}_{i+1}\left( t_{j} \right) + B^{k+2}_{i+1}\left( t_{j-1} \right) \right] \\{} & {} \quad = \frac{n}{(k+1)(k+2)} \sum \limits _{g = 0 }^{k} {{\varvec{p}}}_g \sum \limits _{i=g+2}^{k+2} i \left[ B^{k+2}_{i}\left( t_{j+1} \right) \right. \\{} & {} \qquad \left. - 2 B^{k+2}_{i} \left( t_{j} \right) + B^{k+2}_{i}(t_{j-1}) \right] \end{aligned}$$

Denote \(\Delta _{j,i,n}^{k+2} = B^{k+2}_{i}\left( t_{j+1} \right) - 2 B^{k+2}_{i} \left( t_{j} \right) + B^{k+2}_{i}(t_{j-1}) \), we have

$$\begin{aligned} \textbf{c}_{j}= & {} \frac{n }{(k+1)(k+2)} \sum \limits _{g = 0 }^{k} {{\varvec{p}}}_g \sum \limits _{i=g+2}^{k+2} (i-g-1) \Delta _{j,i,n}^{k+2}. \end{aligned}$$

Appendix B. Examples

Example 1

\(\{ (3.60, 1.34); (-0.44. 7.28); ( 11.24,9.10); ( 7.41, 1.83 )\}\);

Example 2

\( \{ (3, 1); (1, 2); (2, 7); (5, 9); (9, 8); (10, 2); (8, 0); (5, 0) \}\);

Example 3

\( \{ (0, 0); (0, 4); (1, 8); (5, 10); (10, 10); (10, 6); (10, 2); (8, 0); (3, 0); (0, 0)\}\);

Example 4

\( \{(1,1); (2, 7); (6, 10); (9, 6); (4, 6); (3,3); (6, 5); (7, 5); (9, 4); (8, 1); (5, 1); (6, 4); (4, 2)\}\).

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Li, Y., Zhang, M., Jin, W. et al. Approximating Bézier curves with least square polygons. Vis Comput 40, 637–646 (2024). https://doi.org/10.1007/s00371-023-02806-0

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