In the problem:
Eliminate $\theta$ from the system of equations. $$x\sin\theta-y\cos\theta=-\sin4\theta$$ $$x\cos\theta+y\sin\theta=\frac52-\frac32\cos4\theta$$
it is stated in a previous answer that the resultant is
$$x^{10}+5 x^8 y^2+10 x^6 x^4 + 10 x^4 y^6+5 x^2 y^8+y^{10}-705 x^8+12180 x^6 y^2 -24230 x^4 y^4+12180 x^2 y^6-705 y^8+122560 x^6-112320 x^4 y^2 -112320 x^2 y^4 +122560 y^6+599040 x^4-1361920 x^2 y^2 +599040 y^4+327680 x^2+327680 y^2-1048576=0$$
(generated via Mathematica)
which ultimately turned out to be
$$\boldsymbol{(x+y)^{2/5}+(x-y)^{2/5}=2}$$
that was accomplished by the parametrization, $$x=\cos(\theta)\,(5-4\cos^4(\theta)),$$ $$y=\sin(\theta)\,(5-4\sin^4(\theta)).$$
That represents the cartesian equation of the evolute of an astroid.(?)
With the victory over that problem (thanks to @Claude Leibovici), I was attracted by another similar problem: Eliminate $\theta$ from $4x=5\cos\theta -\cos 5\theta$ and $4y=5\sin\theta -\sin 5\theta$, where the answer is the equation of an epicycloid.
In that question, one can find that $$x=\cos^3(\theta)\,(5-4\cos^2(\theta))$$ $$y=\sin^3(\theta)\,(5-4\sin^2(\theta))$$
which seems almost similar to the aforementioned approach.
Also as mentioned in one answer, the eliminant is $$-81 - 45 x^2 + 365 x^4 - 15 x^6 - 480 x^8 + 256 x^{10} - 45 y^2 - 2395 x^2 y^2 - 45 x^4 y^2 - 1920 x^6 y^2 + 1280 x^8 y^2 + 365 y^4 - 45 x^2 y^4 - 2880 x^4 y^4 + 2560 x^6 y^4 - 15 y^6 - 1920 x^2 y^6 + 2560 x^4 y^6 - 480 y^8 + 1280 x^2 y^8 + 256 y^{10}=0$$
(generated via Wolfram|Alpha)
So because of the similarity, there may be a neat solution as before.
I tried plugging some possible equations in Wolfram|Alpha and Desmos graphing calculator, seeking for a hint. Yet, I haven't found the exit.
Question. Can there be a nice equation for this curve?
If it helps, here's a table showing the coefficients of each term of each polynomial (for the ease of comparison).
| $(x+y)^{2/5}+(x-y)^{2/5}=2$ | unknown | |
|---|---|---|
| $x^{10},y^{10}$ | 1 | 256 |
| $x^8,y^8$ | -705 | -480 |
| $x^6,y^6$ | 122560 | -15 |
| $x^4,y^4$ | 599040 | 365 |
| $x^2,y^2$ | 327680 | -45 |
| $x^8y^2,x^2y^8$ | 5 | 1280 |
| $x^6y^4,x^4y^6$ | 10 | 2560 |
| $x^6y^2,x^2y^6$ | 12108 | -1920 |
| $x^4y^2,x^2y^4$ | -112320 | -45 |
| $x^2y^2$ | -1361920 | -2395 |
| $x^4y^4$ | -24230 | -2880 |
| constant | -1048576 | -81 |
Curves:
The red curve is the hypocycloid and the other is the epicycloid.
The symmetry of these curves is probably implying that they have well formed equations too.
