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Square the Circle (Posted on 2004-10-28) Difficulty: 4 of 5
There is a family of curves on the Cartesian plane described by this form:

If n is equal to 2, then it describes an ellipse (if a = b, then it describes a circle).

If n is greater than 2, then this is a "superellipse" (if a = b, then this is a supercircle).

As n increases, the ellipse becomes more "rectangularish", and as n approaches ∞, the limit is a rectangle (or a square if a=b).

What value must n have such that the figure has an area exactly halfway between the associated ellipse (when n=2) and rectangle (when n=∞)?

The graphs below, calculated by varying n with a = b = 1, show this property. Note that as n approaches zero, the curve degenerates into two crossed lines along the x- and y-axes.

No Solution Yet Submitted by SilverKnight    
Rating: 3.0000 (4 votes)

Comments: ( You must be logged in to post comments.)
  Subject Author Date
sol using mathematicasunny2004-10-31 11:57:12
Riemann sumsCharlie2004-10-29 02:11:20
re: No Subject (numerically solved)Christian2004-10-28 21:00:30
Some ThoughtsNo SubjectChristian2004-10-28 20:37:23
re: Some thoughts--Numeric solutionCharlie2004-10-28 19:44:43
Some ThoughtsSome thoughtsFederico Kereki2004-10-28 18:56:10
re: Quad ICharlie2004-10-28 18:24:25
QuestionQuad Inikki2004-10-28 18:05:13
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