A one-dimensional complex torus can be described as the quotient of $\mathbb{C}/\{m_1 \omega_1 + m_2 \omega_2 \}$, where $m_i \in \mathbb{Z}$ and the $\omega_i$ are complex numbers which form a basis for a lattice $\Lambda$.
From this we can construct the Weierstrass $\wp$ function,
$$ \wp(z; \omega_1, \omega_2) = \frac{1}{z^2} + \sum_{m_1, m_2}^{\prime} \left( \frac{1}{(z- m_1 \omega_1 - m_2 \omega_2)^2} - \frac{1}{(m_1 \omega_1 + m_2 \omega_2)^2} \right) $$ (where the prime means $(m_1, m_2) \neq (0,0)$).
We can then map this to a cubic curve in $\mathbb{P}^2$ by using the relation $$ (\wp^{\prime}(z) )^2= 4 \wp(z)^3 - g_2 \wp(z) - g_3 $$ and letting $(x,y) \in \mathbb{C}^2$ be $x =\wp(z)$, $y = \wp^{\prime}(z)$, and then projectivizing by letting $x = z_1 / z_0$, $y = z_2 / z_0$.
I want to go the other way around, starting with an equation that is in Weierstrass form, and then arrive at a map to a real $T^2$ surface in $\mathbb{R}^3$, e.g. a hypersurface of the form $$ (x^2 + y^2 + z^2 + R^2 - r^2) = 4 R^2 (x^2 + y^2) $$ with $(x,y,z) \in \mathbb{R^3}$ and $R > r \in \mathbb{R}^+$.
Is there a clear and systematic way to do this?
EDIT: Thought about it a bit more, and I recall that for different choices of the complex structure $\tau = \omega_2 / \omega_1$, we have different complex manifolds, but these are all diffeomorphic homeomorphic (can they be diffeomorphic?) to each other, i.e. the underlying real manifold $T^2$ does not change for different values of $\tau$. So in going from a Weierstrass model to a quartic hypersurface in $\mathbb{R}^3$ there is some "forgetfulness" involved. In any case, it would be nice to have a map that is consistent in the sense that nearby points on the cubic in $\mathbb{P}^2$ are mapped to nearby points in the quartic hypersurface in $\mathbb{R}^3$ .
It is $4\wp(z)^3-g_2 \wp(z)-g_3$ and it is unclear why you want a torus embedded in $\Bbb{R}^3$ as a (smooth) real manifold.
Any complex elliptic curve is isomorphic (as a complex manifold) to $\Bbb{C}/L$ where $L$ is the lattice obtained by integrating some non-zero holomorphic 1-form on closed-loops. For $E:y^2=x^3+ax+b$ it will be $\omega = c\frac{dx}y$. The isomorphism will be $P\in E\to \int_O^P \omega\in \Bbb{C}/L$ which will send $\omega$ to some multiple of $dz$.
When identifying $\Bbb{C}/L$ with $\Bbb{C}/cL$ for all $c\in \Bbb{C}^*$ and identifying two elliptic curves whenever they are isomorphic then this is a one to one correspondence.