Geometric distance fitting of parabolas in $mathbb{R}^3$

Ik Sung Kim

doi:10.4134/CKMS.c210258

Communications of the
Korean Mathematical Society CKMS

ISSN(Print) 1225-1763 ISSN(Online) 2234-3024

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Commun. Korean Math. Soc. 2022; 37(3): 915-938

Online first article April 12, 2022 Printed July 31, 2022

https://doi.org/10.4134/CKMS.c210258

Geometric distance fitting of parabolas in $mathbb{R}^3$

Ik Sung Kim

Korea Maritime and Ocean University

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Abstract

We are interested in the problem of fitting a parabola to a set of data points in $mathbb{ R}^3 $. It can be usually solved by minimizing the geometric distances from the fitted parabola to the given data points. In this paper, a parabola fitting algorithm will be proposed in such a way that the sum of the squares of the geometric distances is minimized in~$mathbb{R}^3$. Our algorithm is mainly based on the steepest descent technique which determines an adequate number $ lambda $ such that $h ( lambda ) = Q ( u - lambda
abla Qigl( u igr) ) < Q ( u)$. Some numerical examples are given to test our algorithm.

Keywords: Fitting of parabolas, geometric distance, steepest descent

MSC numbers: Primary 65D18, 68U05