2
$\begingroup$

This post is related to Smallest 3-ellipses that contain triangles and tries to clarify a basic issue.

Question: Given a general triangle T, How does one find and characterize the ellipse of least perimeter that contains T?

Note 1: The ellipse we are looking for is in general not the Steiner circumellipsse (which is the least area containing ellipse for any triangle). Indeed, the Steiner ellipse has its center at the centroid of T and when T is a long and thin isosceles triangle, the least perimeter containing ellipse has center close to the mid point of the altitude of T and not the centroid.

Note 2: That there is no known closed form expression of the perimeter of an ellipse need not rule out a definite answer to above question. From triangles one can go to sets of points on the plane and ask for algorithms that give the least perimeter ellipses that contain the set (least area ellipse containers are addressed in https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.53.5019)

Note 3: Some experiments indicate that (https://arxiv.org/ftp/arxiv/papers/1802/1802.10447.pdf) the orientations of the least area and least perimeter containing ellipses are quite close for any triangle.

A further question: For any triangle, is the ratio between the perimeters of the smallest perimeter containing ellipse and the largest perimeter contained ellipse a constant (despite lack of closed expression for the perimeters themselves)?

$\endgroup$
4
  • $\begingroup$ A subquestion is: What are the possible lengths for the major and minor axes of ellipses containing $T$? Unfortunately, this seems to require an 8th-degree equation. $\endgroup$
    – user44143
    Aug 25, 2022 at 15:49
  • $\begingroup$ A good answer to this question would belong on Mathworld's list of circumconics (mathworld.wolfram.com/Circumconic.html), and it might yield a new entry for @ClarkKimberling's Encyclopedia of Triangle Centers. $\endgroup$
    – user44143
    Aug 25, 2022 at 18:38
  • $\begingroup$ Thanks. One would be surprised if the center of such an ellipse is yet to be documented. Look forward! $\endgroup$ Aug 26, 2022 at 10:42
  • $\begingroup$ It would surprise me — almost all the centers so far have formulas (though I have emailed about adding one without a formula), and I wouldn’t expect a formula for this one. $\endgroup$
    – user44143
    Aug 26, 2022 at 15:12

0

Your Answer

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