rational points on the quadrifolium $(x^2 + y^2)^3 = (x^2 - y^2)^2$ I have been reading the Wikipedia page on the Quadrifolium there are two of them:
\begin{eqnarray*}
r &=& \sin 2\theta \\ 
(x^2 + y^2)^3 &=& 4 x^2 y^2
\end{eqnarray*}
and it's $45^\circ$-rotated version, which is also a rose curve:
\begin{eqnarray*}
r &=& \cos 2\theta\\
(x^2 + y^2)^3 &=& (x^2 - y^2)^2 \\
\end{eqnarray*}
These equations also have a $(x,y)$-parameterization, which can be found with some algebra.
\begin{eqnarray*}
x &=& \cos k \theta \cos \theta = \tfrac{1}{4}\big[ e^{i(k+1)\theta} + e^{i(k-1)\theta} + e^{-i(k-1)\theta} + e^{-i(k+1)\theta} \big] \\
y &=& \cos k \theta \sin \theta
\,= \tfrac{1}{4}\big[ e^{i(k+1)\theta} - e^{i(k-1)\theta} + e^{-i(k-1)\theta} - e^{-i(k+1)\theta} \big] \\
\end{eqnarray*}
And $\cos k (\theta + \phi)$ for a rotated version.  Here $\theta = 0, \frac{\pi}{4}$ and $k = 2$.

Here is the picture of the quadrifolium, a kind of rose curve, and when it's rotated.



The reason I am going to look at this question again, is because I wanted to study the rational paramterization. Let $X$ be our curve, with a singular point at the origin $(0,0)$. We're trying to describe $X(\mathbb{Q})$. 
The curve on the left right has no rational points.
Naively, if I divide one equation by the other, I get an expression for the tangent function.  In fact, $\frac{x}{y} = \tan \theta$ for all values of $\theta \in [0, 2\pi]$ just as if it were a circle.  Let's define a map:
$$ \big[ (x,y) = (\cos \theta, \sin \theta) \big] \mapsto 
\Big[ ( (2x^2 - 1 )x, (1 - 2y^2) y) = ( (2 \cos^2 \theta - 1) \cos \theta , (1 - 2\sin^2 \theta)\sin \theta ) \Big] $$
And we have that $LHS \in \mathbb{Q}$ if $RHS \in \mathbb{Q}$.  Is this sufficient?  Is this map birational, is this enough for a paramerization?
Here is even another strategy, since we have Fourier series (even just trigonometric series because there are only 4 terms). 
$$ \left[ \begin{array}{c} x \\ y \end{array}\right] =
\left[ \begin{array}{c} 
e^{3i\theta} + e^{i\theta} + e^{-i\theta} + e^{-3i\theta} \\
e^{3i\theta} - e^{i\theta} + e^{-i\theta} - e^{-3i\theta}\end{array} \right]
= \left[ \begin{array}{rrrr} 1 & 1 & 1 & 1 \\
1 & -1 & 1 & -1 \end{array} \right]
\left[ \begin{array}{c} e^{3i\theta} \\ e^{i\theta} \\
e^{-i\theta} \\ e^{-3i\theta} \end{array} \right]  $$ 
Or we could even try a different approach using the triple-angle formulas of trigonometry:
\begin{eqnarray*}
\cos 3\theta &=& 4 \cos^3 \theta - 3 \cos \theta \\
\sin 3\theta &=& 3 \sin \theta - 4 \sin^3 \theta \\
\tan 3\theta &=& \frac{3 \tan \theta - \tan^3 \theta}{1 - 3 \tan^2 \theta}
\end{eqnarray*}
Now these are reading as a degree-map from the circle to another algebraic variety (or scheme).  In that case, why does the angle $3\theta$ appear in a 4-fold symmetric curve?

These singularities could be studied by Netwton's method (e.g. Puiseux series or resolution of singularities).  Unfortunatley I could never get the jargon straight about the exceptional divisor and so forth. E.g. the strategy here to rationalize the lemniscate seems to amount to computing a blow-up of the curve.  At this point I am going to consult an algebraic geometry textbook.
Example What is the "derivative" or tangent space at the origin $(0,0)$ ? 
 A: You may want to find a parametrization like $(x, y) = (f_1(t), f_2(t))$ for some  nonconstant rational functions $f_1(t), f_2(t)\in \mathbb{Q}(t)$. Until now, I can show you that there's no such parametrization with $f_1(t), f_2(t)\in \mathbb{Q}[t]$ (i.e. both are polynomial with rational coefficients). 
If such parametrization exists, we should have $x^{2} - y^{2} = f(t)^{3}$ and $x^{2} + y^{2} = f(t)^{2}$ for some $f(t)\in \mathbb{Q}[t]$. Then 
$$
y^{2} = \frac{1}{2}f(t)^{2}(1-f(t)), 
$$
so $\frac{1}{2}(1-f(t)) = g(t)^{2}\Leftrightarrow f(t) = 1-2g(t)^{2}$ for some $g(t)\in \mathbb{Q}[t]$. Then $$x^{2} = \frac{1}{2}(f(t)^{2} + f(t)^{3}) =(1-2g(t)^{2})^{2}(1-g(t)^{2})$$
Now $\mathrm{gcd}(1-2g(t)^{2}, 1-g(t)^{2}) = 1$, so we should have $1-g(t)^{2} = h_{1}(t)^{2}$ and $1-2g(t)^{2} =h_{2}(t)^{2}$ for some $h_{1}, h_{2}\in \mathbb{Q}[t]$. By the way, this give $2h_{1}(t)^{2} -h_{2}(t)^{2} = 1$, which is impossible if we see the first coefficients of $h_{1}, h_{2}$, unless they are constant. 
I think we can use the same argument in case of rational functions, too. (I'm not sure)
