I had to plot the graph of the implicitly defined function $\sin^2 x + \sin^2 y = 1$ in an exam. This is not particularly difficult, but it got me wondering what the graph would look like when the exponent is taken inside, viz.
$$\sin(x^2) + \sin(y^2) = 1$$
I found it difficult to figure this out, so I resorted to Desmos' graphing calculator. It looks like this:
I can explain some parts of this picture, but others elude me, and I think someone with more experience will do a better job of saying why this thing looks like it does.
I'd be particularly interested to know whether the figure in the middle is a special case of some other function, and similarly with the curlicues on the axes. (I have a good idea of what the polka dots are.)
Pre-emptive note: I had no trouble plotting the other thing mentioned ($\sin^2 x + \sin^2 y = 1$), so you needn't bother including that in your answer.
EDIT: After looking at @Jean Marie's answer below, I plotted the graph of
$$(x^2 + y^2) – \frac{x^6 + y^6}{6} = 1,$$
reasoning that near the origin a few terms of the Taylor series might help. The resulting graph was quite similar to the strange shape near the origin in thing above. Somewhat unexpectedly (at least for a callow neophyte like me), a much closer approximation (shapewise) was offered by
$$x^2 + y^2 – \frac{x^4 + y^4}{4} = 1$$
In case it is of any use, here is a picture (from Desmos) of the two plots I mentioned.
In the figure above, the blue is the sextic and the red the quartic.
Best Answer
The surface of equation $$z=\sin x+\sin y$$ has the shape of an "egg tray". It has maxima $z=2$ on a periodic grid, and this is why the level curves $z=1$ are regularly spaced approximate circles.
Now if we replace $x$ by $x^2$, we deform space horizontally so that $x^2$ increases faster and faster, giving a "compression effect".
By replacing $y$ with $y^2$, we get the effect on both axis.
Of course, the picture is symmetric by reflection, as the square function is even.
Remains to explain the "spikes" against the axis. If $y$ is small,
$$\sin x^2+\sin y^2=1\to y\approx\sqrt{1-\sin x^2}=\sqrt2\left|\cos\frac{x^2}2\right|$$
shows them.