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I get the overall idea, but why is the shape of these figures not a circle?

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Consider new frame with coordinate axes defined as $X=x^2$ and $Y=y^2$.

In this new frame of reference, the equation you mentioned will be a perfect circle with center at (2,2) and radius of $\sqrt 2$.

But every point on this new frame of reference will have, 4 copies in the original $x,y$ frame. Why?
A point $(X_0,Y_0)$ in new frame will have $4$ copies which will be $(\sqrt {X_0},\sqrt {Y_0}),(\sqrt {X_0},-\sqrt {Y_0}),(-\sqrt {X_0},\sqrt {Y_0}),(-\sqrt {X_0},-\sqrt {Y_0}) $.

Since the circle in new reference frame lies completely in 1st quadrant, therefore the circle will be replicated into $4$ figures in $4$ different quadrants in original frame of reference.

This can also be verified by viewing the graph here.

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  • $\begingroup$ But why are these "circles" pillow-shaped and not normal ones? $\endgroup$ Nov 8, 2017 at 21:21

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