### THE TWO STAGES CIRCLE FITTING METHOD

Nicolae URSU-FISCHER, Diana Ioana POPESCU, Iuliana Fabiola MOHOLEA

#### Abstract

The problem of circle fitting (and also of other conics, especially of ellipse) to a number of given points in plane is one of the great importance in computer graphics, image processing, quality control, metrology and other fields of science and engineering. Even if it is necessary to fit an entire circle or just an arc, the same methods are used. The solving of linear and nonlinear algebraic equations, the extensive use of well-known method of least-squares (for the first time discovered by Legendre [10]), the error analysis, are the main mathematical tools used in these procedures for perform the accurate fit of the points with a circle, if this is the demand. The first part of the paper summarizes the existing methods of algebraic and geometric fit, using a circle: the methods of Kåsa [8], Pratt [13] and Taubin [17], having in common the use of least-squares method. With these methods, the coordinates of the centre of the circle and its radius result simultaneously, after solving the system of nonlinear equations. The authors present a new original method for solving the same problem: the coordinates of the centre of the circle are obtained first as a consequence of approximate solving of an over-determined system of linear equations and, in the second stage, is determined the value of the radius by minimizing the objective function. The paper also contains two examples of fitting (with a complete circle and with an arc) solved with the new-imaged procedure.

Key words: Least-square method, circle fitting, algebraic and geometric fitting, Newton iterative method, nonlinear equations.

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