# A general curve smoothing technique with a single continuous function that does not oscillate

### Significance

Many engineering practices utilize curve smoothing algorithms: algorithms that convert segmental curves with discontinuities at junctions to smooth ones, in product design and manufacture. Specifically, robot motion planning, machining, computer graphics and pharmacy are popular areas of application. Furthermore, the design of highway path planning in the field of civil engineering and mobile robot design demand perfectly smooth curves, as the opposite would result in slippage and over-actuation. Presently, many path smoothing algorithms have already been developed including, non-uniform rational basis spline, off-line post-processing, interpolation with a parametric curve such as Bezier and online number control interpolation, among others. These techniques generate smooth paths as a set of discrete data points or separate functions with certain degrees of continuity at junctions. Unfortunately, none of these techniques can provide a smoothed curve using a single continuous function for arbitrary segmental curves.

Professor Yu-fei Wu from RMIT University in Australia in collaboration with Liang He and Zhi-dong Li developed a new approach that could 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. In addition, the new technique asymptotically approached all original segments that were given by any continuous function, and lastly, their novel technique could be given by one single continuous function in the whole domain. Their work is currently published in the research journal, Journal of Zhejiang University-SCIENCE A (Applied Physics & Engineering).

The researchers commenced experimental investigations by introducing a regional variable that equaled the original variable in its sub-domain and mutated to a constant outside the region. Next, with the regional variable replacing the original, any function that took its original form inside its sub-domain later turned to a constant outside it to become a regional function. Lastly, the multiple-segment discontinuous curve is converted to a continuous function simply by multiplication of all the regional functions.

The authors observed that the smoothed continuous curve using the novel technique did not oscillate at all because of the smooth regional variable employed. Additionally, they noted that by utilizing regional variables, each segment of curve retained its original shape inside its own sub-domain. It also came to light that in between the two sub-domains, a smooth transition was provided with the required degree of smoothness controlled by a smoothness parameter for the junction under observation.

The Yu-fei Wu and colleagues study presented the development of a function multiplication technique that could convert any discontinuous segmental curve into a continuous one. The researchers attributed their success to the introduction of a regional variable that equaled the original variable in its sub-domain and became a constant outside the region. The developed technique is advantageous since it can bring convenience in engineering computation and computer programming to avoid dividing functions into sub-domains. Altogether, the technique presented provides a general mathematical tool that can be used in all scientific and engineering work.

### About the author

Yu-Fei Wu is currently a Professor in Infrastructure Engineering in the School of Engineering at RMIT University in Australia. He obtained his BSc in 1983 and MSc in 1986 from Zhejiang University, China. He received MEng from National University of Singapore in 1994 and completed his PhD in 2002 at the University of Adelaide, Australia. He has more than ten years of industry working experience in structural engineering as a professional engineer in consulting firms in China, Singapore and Australia and is a chartered professional engineer of New Zealand and Australia (FIEAust, CPEng, NER, MIPENZ). He worked in Shanghai Jiao Tong University from 1989 to 1992 as a lecturer. Prior to joining RMIT University, he was an Associate Professor in the Department of Architecture and Civil Engineering at City University of Hong Kong (2004-2015). His research interests lie in the broad field of structural engineering, including concrete structures, structural design, composite structures, FRP structures and structural rehabilitation.

Professor Wu has published more than 200 technical works, including 120 SCI indexed journal papers with an H index of 34 (Google Scholar) and 28 (ISI Web of Science). Professor Wu has received numerous research awards including the Moisseiff Award from American Society of Civil Engineers.

He is the inventor of numerous new structural technologies with six US patents, including (1) the theorems for flexural design of reinforced concrete, (2) the experimental method that can directly measure shear strength components in RC member tests, (3) the compression yielding structural design concept to increase the ductiliy of RC members, and (4) the experimental facility for concrete test under passive 3-dimentional confinement stresses.

### About the author

Liang He is currently an engineer in the Guangzhou Metro Design and Research Institute Co. Ltd. He received his BSc in 2012 from Southeast University, China. He completed his MS degree from Hunan University, China. He has studied at City University of Hong Kong as a research assistant from 2013 to 2015. His research interests include FRP strengthened concrete structures and steel-concrete composite structures.

### About the author

Zhidong Li is currently a first year PhD candidate in Civil and Infrastructure Engineering in the School of Engineering at RMIT University in Australia. He completes his Bachelor of Infrastructure and Civil Engineering (Honors) degree in 2017 from RMIT University. His research focuses on flexural behavior of reinforced concrete members and identification of constitutive relationship of concrete materials in the area of Structural Engineering.

### Reference

Yu-fei WU, Liang HE, Zhi-dong LI. Curve smoothing using a continuous function. Journal of Zhejiang University-SCIENCE A (Applied Physics & Engineering) 2018 volume 19(4):page 304-314