4
$\begingroup$

Definition: The Hausdorff distance between two point sets is the greatest of all the distances from a point in one set to the closest point in the other set.

Question: Given a general triangle T, to find and characterize the ellipse E that minimizes the Hausdorff distance from T. It is not clear to me if E is unique.

Note 1: Some further questions/thoughts are recorded at https://nandacumar.blogspot.com/2023/01/approximating-triangles-by-ellipses.html and https://nandacumar.blogspot.com/2023/01/approximating-planar-convex-sets-by-n.html

Note 2: As has been pointed out by Matt F in comments below, the special case of the ellipse being a circle is itself of nontrivial interest.

$\endgroup$
4
  • 2
    $\begingroup$ I think the right place to start is some easier questions: What circle minimizes this distance? This is an algebraic question, whose answer is probably a well-known triangle center. And even easier: What radius minimizes this distance for circles around the incenter? $\endgroup$
    – user44143
    Jan 17, 2023 at 8:43
  • $\begingroup$ Thanks! I had arrived at the question thinking about the steiner ellipses of a triangle and had overlooked the basic case of the ellipse being a circle. The broader point that one was trying to make with this question and also in the pages linked above was that although smallest containing x / largest contained x for a given y (where x and y are any two different kinds of shapes) kind of problems are well studied, closest x to a given y (closest in terms of Hausdorff distance) is a class of questions that seems relatively unexplored. $\endgroup$ Jan 17, 2023 at 17:12
  • 1
    $\begingroup$ For circles around the incenter, I think the optimal radius is $\frac12(1+\cot\frac\alpha2)$ times the inradius, where $\alpha$ is the smallest angle in the triangle. $\endgroup$
    – user44143
    Jan 17, 2023 at 19:53
  • $\begingroup$ Guess: At least in the case of the triangle-circle pair, if C is the circle that approximates (in the Hausdorff sense as in the present question) a given triangle T best, then, T might not be the triangle that approximates C best. $\endgroup$ Jan 18, 2023 at 3:58

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.