Minimal Ellipse

26/09/2007 - 13:41 von Narasimham | Report spam
Find the minimum area ellipse passing through four arbitrary given
points in a plane. (Find center, axes length and inclination to the
coordinate axes). Five points determine a conic, as is known. No given
point falls inside a triangle with the other three as vertices,
avoiding formation of a hyperbola.

Hint: It is relatively easy to prove the symmetric case of finding
minimum area ellipse through four corners of a rectangle. For an
important special derived case of five points where four given
points(A,B,C,D) are corners of a rectangle, the solution ellipse is
inscribed (symmetrically touching) in a larger similar (homothetically
placed) rectangular box scaled up sqrt(2) times, where the fifth given
point (E marked red) is constrained to move on the circumscribed
ellipse.

http://i22.tinypic.com/64mp0g

This is not assigned homework!

TIA,
Narasimham
 

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#1 hoffmann
27/09/2007 - 12:29 | Warnen spam
Narasimham schrieb:
Find the minimum area ellipse passing through four arbitrary given
points in a plane. (Find center, axes length and inclination to the
coordinate axes). Five points determine a conic, as is known. No given
point falls inside a triangle with the other three as vertices,
avoiding formation of a hyperbola.

Hint: It is relatively easy to prove the symmetric case of finding
minimum area ellipse through four corners of a rectangle. For an
important special derived case of five points where four given
points(A,B,C,D) are corners of a rectangle, the solution ellipse is
inscribed (symmetrically touching) in a larger similar (homothetically
placed) rectangular box scaled up sqrt(2) times, where the fifth given
point (E marked red) is constrained to move on the circumscribed
ellipse.

http://i22.tinypic.com/64mp0g

This is not assigned homework!

TIA,
Narasimham



Perhaps like this:
http://www.fho-emden.de/~hoffmann/ellipse08032004.pdf

Best regards --Gernot Hoffmann

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