Algebraic curve fitting for multidimensional data with exact squares distance

This paper presents a new method for fitting algebraic curves to multidimensional data using the exact squares distance between data points and the curve. Fitting smooth curves is one of the most important themes in pattern recognition and data analysis. Simple regression analysis or multivariate regression analysis are in use for a data set consisting of observations on some variables which can be treated one of them as response variable and the others as explanatory variables. However, these analyses do not work well for a data set whose variables can not be distinguished between response and explanatory. We must prepare two algorithms to realize a method for fitting algebraic curves to data. The first is an algorithm for evaluating “distance” between data points and a given curve. The second is to find a fitting algebraic curve based on the “distance”. Taubin (1991) proposed an algorithm to find the algebraic curve such that the sum of the approximate squares distance between data points and the curve is minimum. The approximate squares distance does not always agree with exact squares distance. We develop an algorithm for evaluating the exact distance between them. The algorithm is based on the Newton-Rapson method, and the amount of computation is reasonable. We show the differences between the exact distance and the approximate distance with a numerical example. The partial derivatives of the sum of the exact squares distance are also shown for the algorithm to find the fitting curve based on the exact distances