Comment on Choi, M.G. Computing the Closest Approach Distance of Two Ellipsoids. Symmetry 2020, 12, 1302

Abstract

In this Comment, we point out that statements made in the paper ‘Computing the Closest Approach Distance of Two Ellipsoids’ by Min Gyu Choi are incorrect and misleading. We provide the needed corrections and suggest an explanation for the observations which gave rise to the incorrect claims.

Finding the distance of closest approach of two ellipses in 2D, or two ellipsoids in 3D, is an interesting problem in geometry. In addition, it is key to a broad range of important applications, ranging from liquid crystal displays and hard particle simulations to industrial problems where objects can be accurately approximated by ellipses or ellipsoids [1,2,3]. One example is the safe positioning of wellbores in drilling in petroleum production [4]. In this Comment, we point out that statements made in the paper entitled ‘Computing the Closest Approach Distance of Two Ellipsoids’ by Min Gyu Choi are incorrect and misleading. We provide the necessary corrections and suggest an explanation for the observations which gave rise to the incorrect claims.

In our 2007 paper [5], we derived, for the first time, an analytic expression for the distance of closest approach of the centers of two arbitrary hard ellipses in two dimensions as a function of their orientation. The essence of the approach is to map the two ellipses to a unit circle and an ellipse through a linear transformation, and then find the distance of closest approach in the transformed space. Applying the inverse transformation to the distance provides the solution. The closest distance of the centers of the circle and the transformed ellipse is the single real positive root of a quartic equation. Analytical solutions exist for quartic equations, and we determined the roots using the method developed by Ferrari [6]. Since the relevant root can be uniquely determined, after applying the inverse transformation, we were able to write down the solution for the distance explicitly.

An analytic result for the distance of closest approach of ellipsoids in 3D would be of great interest and utility. However, an analytic solution does not exist for ellipsoids because, in 3D, a sixth-order polynomial equation needs to be solved. Analytic expressions do not exist for the solutions of fifth and higher order polynomial equations; therefore, the solution for the distance of closest approach of two ellipsoids must be obtained numerically. In a subsequent paper [7] in 2009, we provided a numerical algorithm to calculate the distance of closest approach of two ellipsoids of arbitrary shape and orientation. Here, we considered the intersection of a plane, containing the line connecting the centers of two ellipsoids, and the two ellipsoids. The intersections of the plane and the ellipsoids are two ellipses. Therefore, we can use the analytic expression obtained in [5] to find the distance of closest approach of two ellipses in a plane. We showed that the maximum distance of closest approach of the ellipses as the plane rotates about the line connecting the centers of the two ellipsoids is the distance of closest approach of two ellipsoids. We employed the golden section method for finding the unique extremum of a unimodal function in our numerical algorithm. The golden-section method is a bracketing method where, starting with an initial interval, the interval is shrunk by a ratio ρ < 1 in each iteration until the size of the interval is smaller than the user-specified tolerance. This is a simple and effective method, but it is well known that such a bracketing method is much slower than derivative-based methods such as Newton’s method. We used the golden section method to demonstrate the principle of our distance-finding algorithm in 3D.

More recently, in their paper [8], the author, Min Gyu Choi, compared data from numerical experiments, computing the closest approach distance of two ellipsoids using four different methods. One of these, the cross-section method, is, in effect, our strategy described in [7], using the result presented in [5]. In discussing the results obtained from the various methods, Choi goes on to say:

The error in the analytic method for the closest approach distance of two ellipses increased as γ increased [5]. Correspondingly, the error in the cross-section search [7] also increased, regardless of the number of solver iterations in the golden section search that found the angle of the cross section resulting in the closest approach distance of two ellipsoids.

We do not agree with these statements. They are erroneous in describing the situation; they are also misleading and may be counterproductive to workers in the field. Our analytic method yields the exact solution; regardless of the values of γ and Γ, the analytic result has no error. Its validity can be readily checked with the resources we have provided [9]. Since there is no error in the analytic result, there can be no corresponding error in the cross-section search, nor a resulting iteration independent error. We assert that both our analytic result in [5] in 2D and our algorithm in 3D are error-free, do not produce errors, and may be used with full confidence.

We conjecture that the errors to which the author is referring are numerical truncation and approximation errors in their own calculation when evaluating our analytic formula for the distance of closest approach of two ellipses. Evaluating the analytic formula requires calculation of the cubes and cube roots of complex numbers. Accuracy in these calculations is key. In computers working with finite precision, there will be a loss of accuracy at each step, and the errors can accumulate during multi-step evaluations. This was even noted in [7] Section C: Computational details: We note that in our implementations there may be a loss of accuracy for ellipsoids with large aspect ratios e.g., ≥200 when using double precision. In the ellipse program, when the aspect ratio gets large, the ratios of the coefficients in the quartic equation get extremely large, and large number cancellations and/or rounding errors can lead to inaccurate results. If the aspect ratios of the ellipsoids are large, ≥200, quadruple precision should be used. We believe, therefore, that the errors to which Choi refers are due to computational inaccuracy, these may be remedied by increasing precision.

The purpose of this note is not to question the author’s results, but simply to point out that their statement quoted above is incorrect; the observed errors cannot result either from the exact results in [5] or from the algorithm in [7] using it.