Ellipse Fitting Using Orthogonal Hyperbolae and Stirling's Oval

Abstract

Two methods for approximating the normal distance to an ellipse using (a) its orthogonal hyperbolae and (b) Stirling's oval are described. Analysis with a set of quantitative measures shows that the former provides an accurate approximation with few irregularities or biases. Its suitability is evaluated by comparing several approximations as error of fit functions and applying them to ellipse fitting.

Measurement of the distribution of red blood cell deformability using an automated rheoscope

“…Calculating this distance requires solving a quadratic equation, which is an inefficient procedure. A good approximation can be obtained by using orthogonal hyperbolae (25). Confocal ellipses and hyperbolae intersect orthogonally (Fig.…”

“…This property was exploited to find the confocal hyperbola that crosses a specific data point. The distance from the data point to the point of intersection between the ellipse and the hyperbola provided a good approximation of the shortest distance to the ellipse (25,26).…”

On serlio’s constructions of ovals

“…The circular arcs are sampled at approximately equally spaced points, and at each point the distance along the normal to the ellipse is approximated. See [26,27] for more details. Fixing 7 b B Q and using 1000 sample points in total the graph in figure 12 was generated.…”

Evaluating Harker and O’Leary’s distance approximation for ellipse fitting

“…In particular, we compare their method to one which was based on the ellipse and its confocal hyperbola that passes through P i [9]. Such confocal conics are mutually orthogonal, and since much of the hyperbola is relatively straight it is a good approximation to the normal from the point to the ellipse.…”

“…Such confocal conics are mutually orthogonal, and since much of the hyperbola is relatively straight it is a good approximation to the normal from the point to the ellipse. Computing the distance is then straightforward, and details are provided in [9].…”