Journal of Zhejiang University-SCIENCE A

, Volume 19, Issue 4, pp 304–314

# Curve smoothing using a continuous function

• Yu-fei Wu
• Liang He
• Zhi-dong Li
Article

## Abstract

Current curve smoothing technologies provide a smoothed curve by joining together separate curves that have certain degrees of continuity at junctions. These technologies have found many applications in science and engineering. However, none of them can provide a smoothed curve using a single continuous function for arbitrary segmental curves. This paper reports a new approach that can be used to construct a single continuous function that joins an arbitrary number of different segmental curves, with the required degree of continuity at all junctions. The smoothness of transition at different junctions can be controlled by separate parameters to suit different needs. The combined continuous function can approach the original segmental functions asymptotically or match the original segmental functions “exactly” inside each segment by adjusting the smoothness parameter. This new approach may also find application outside the scope of curve smoothing/curving fitting in the future.

## Keywords

Curve smoothing Curve fitting Transition curve Path planning Tool path

# 连续函数式曲线光滑法

## 概要

### 方法

1. 通过引入特殊的区域变量,并用该区域变量替 代原函数自变量的方法,将区域函数改造成在该 区域无限接近原函数而在区域外取值常数的函 数。2. 把所有的区域函数相乘得到一个连续函数 的方程。

### 结论

1. 由任意多非连续区域函数构成的函数可以改造 成一个连续函数。2. 该连续函数在原非连续边界 的光滑程度可以由各个边界上独立的参数按需 调整。3. 该方法产生的连续函数没有摆动现象, 其形状与原始区域函数无限接近。该方程在数学 上是连续的,同时无限接近原始非连续函数,包 括原来在边界上函数值的非连续。

TV5

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© Zhejiang University and Springer-Verlag GmbH Germany, part of Springer Nature 2018

## Authors and Affiliations

• Yu-fei Wu
• 1
• Liang He
• 2
• 3
• Zhi-dong Li
• 1
1. 1.School of EngineeringRMIT UniversityCarlton, VictoriaAustralia
2. 2.Guangzhou Metro Design and Research Institute Co. Ltd.GuangzhouChina
3. 3.Department of Architecture and Civil EngineeringCity University of Hong KongKowloon, Hong KongChina