On polygons inscribed in ellipses
Question: Given any ellipse, any number n and a point P on the ellipse. We need to find the max area inscribed polygon in the ellipse that has P as one of its vertices.
As P moves around the boundary the area of the max area inscribed n-gon will change. I have no guesses as to where it will be a minimum - by "where" I mean "P lies at which value of theta, the ellipse parameter".
The value of theta for which the n-gon is minimal could depend on two things - n and the eccentricity of the ellipse. If so how?
The same question can be asked with perimeter replacing area. And further possible tweaks include 2 vertices P and Q of the n-gon specified and so on.
Answer: When C is an ellipse, the area of the max area inscribed triangle remains constant as P moves around C - at each position of P, the max area inscribed triangle is one with centroid coincident with the center of C and has C as its Steiner circumellipse.
However, if instead of area, we consider perimeter, it is found that when C is an ellipse, the variation in the perimeter of the max perimeter inscribed triangle is within around 10% (experimental result) even when C is highly eccentric
(one thus has a very real geometric function that grows extremely slowly. The ratio between max and min among the set of the max perimeter inscribed triangles with one vertex fixed goes from 1 (circle) to just about 1.1 as the eccentricity of the ellipse - with say area constant - goes from 0 all the way to infinity),
Question: Among all planar convex regions of given area and perimeter, is the ellipse the shape that minimizes the variation in the perimeter of the max perimeter inscribed triangle with one vertex constrained to be at P - as P varies around the boundary of the convex region?
As P moves around the boundary the area of the max area inscribed n-gon will change. I have no guesses as to where it will be a minimum - by "where" I mean "P lies at which value of theta, the ellipse parameter".
The value of theta for which the n-gon is minimal could depend on two things - n and the eccentricity of the ellipse. If so how?
The same question can be asked with perimeter replacing area. And further possible tweaks include 2 vertices P and Q of the n-gon specified and so on.
Answer: When C is an ellipse, the area of the max area inscribed triangle remains constant as P moves around C - at each position of P, the max area inscribed triangle is one with centroid coincident with the center of C and has C as its Steiner circumellipse.
However, if instead of area, we consider perimeter, it is found that when C is an ellipse, the variation in the perimeter of the max perimeter inscribed triangle is within around 10% (experimental result) even when C is highly eccentric
(one thus has a very real geometric function that grows extremely slowly. The ratio between max and min among the set of the max perimeter inscribed triangles with one vertex fixed goes from 1 (circle) to just about 1.1 as the eccentricity of the ellipse - with say area constant - goes from 0 all the way to infinity),
Question: Among all planar convex regions of given area and perimeter, is the ellipse the shape that minimizes the variation in the perimeter of the max perimeter inscribed triangle with one vertex constrained to be at P - as P varies around the boundary of the convex region?
posted by R.Nandakumar @ 1:27 AM
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