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I'm trying to define a circle that is sort of squashed in to a square-like shape with exponential(?) curved corners. Here is an image showing what I'm describing:

Notice it is a circle at a small size and grows more square-like as it gets larger. Is there an equation for this?

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You might want a supercircle with $x^k+y^k=1$. A circle has $k=2$. If you increase $k$, the corners move out. You would have to increase $k$ with size to get the "sharper corners" you refer to.

Alternately you can make a square and put tangent circles at each corner. Say you want the square to have a side of $2$ with circles of radius $\frac 14$ in each corner. The centers of the circle are then at $\left(\pm(2-\frac 14),\pm(2-\frac 14)\right)=(\pm \frac 74, \pm \frac 74). $ The sides are just $x= \pm2, -\frac 74 \le y \le \frac 74$ and the corresponding segments in $y$. The arcs are $(x\pm \frac 74)^2+(y\pm \frac74)^2=(\frac 14)^2$

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Thanks, this got me pointed in the right direction I think. I believe what I'm looking for is sometimes called a "supercircle" which is defined by: x = a cos^n(angle) y = a sin^n(angle) with n being the exponent that defines the shape. –  CR. Feb 8 '13 at 14:25
    
@CR.: Yes, a supercircle to a superellipse is like a circle to an ellipse. –  Ross Millikan Feb 8 '13 at 14:26
    
Alright, I got it to work using the Superellipse formula off Wikipedia. x(Θ)=|cosΘ|^2/m * a sgn(cosΘ) y(Θ)=|sinΘ|^2/n * b sgn(sinΘ) Thanks! –  CR. Feb 8 '13 at 16:17
    
@CR.: Note you should have parentheses around $2/m$ and $2/n$. Otherwise you would get $\frac 1m|cos \theta|^2$ because of the precedence of operations. –  Ross Millikan Feb 8 '13 at 16:27

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