Converting a point in a finite field to its real (x, y) coordinate Let curve $A = y^2 = x^3 + 3$ and curve $B = y^2 \equiv x^3 + 3 \pmod{19}$
Let $G$ be the positive y-valued point in the curve where $x = 2$
Let $r$ be a random scalar integer, for example, $r = 5$
Compute the point $G*r$ in both curves $A$ and $B$
Now, assume I give you the point $G*r$ in curve $B$, can you find what my point $G*r$ in curve $A$ is? You don't know the value for $r$, but you do know all other parameters. You can't bruteforce the curve.

In the question (as in rev. 3), it can be shown by enumeration that there are 12 solutions to $y^2 \equiv x^3 + 3\pmod{19}$ :
$$\begin{array}{}
(1,2),&(2,7),&(3,7),&(7,2),&(11,2),&(14,7)\\
(1,17),&(2,12),&(3,12),&(7,17),&(11,17),&(14,12)
\end{array}$$
Thus with the neutral element $\infty$ (aka point at infinity), the order of curve $B$ is $12+1=13$. The point $B$ has the same $x=2$ coordinate on curves $A$ and $B$. We get $y=\sqrt{11}$ on $A$, and¹ $y=7$ on $B$.
Point addition $R\gets P+Q$ can go by the same formulas for $A$ and $B$, with addition, multiplication, division², and equality in the field $\mathbb R$ for $A$, the field $\mathbb F_{19}$ for $B$:

*

*If $P=\infty$, $R\gets Q$.

*Otherwise, if $Q=\infty$, $R\gets P$.

*Otherwise, if $x_P\ne x_Q$
$$\begin{align}
\lambda&\gets\frac{y_Q-y_P}{x_Q-x_P}\\
x_R&\gets\lambda^2-x_P-x_Q\\
y_R&\gets\lambda(x_P-x_R)-y_R
\end{align}$$

*Otherwise, if $P=Q$ (that is $x_P=x_Q$ and $y_P=y_Q$)
$$\begin{align}
\lambda&\gets\frac{3{x_P}^2}{2y_P}\\
x_R&\gets\lambda^2-2x_P\\
y_R&\gets\lambda(x_P-x_R)-y_R
\end{align}$$

*Otherwise (that is $x_P=x_Q$ and $y_P\ne y_Q$), $R\gets\infty$.

In order to compute $G*5$ on each curve, we can compute $G_2\gets G+G$, $G_4\gets G_2+G_2$, $G*5\gets G_4+G$ per these formulas. Try it online!. We get
$$\begin{align}
G*r&=\left(\frac{60503882}{151321},-\frac{141898736429 \sqrt{11}}{58863869}\right)&\text{ on curve }A\\
&\approx(399.838,-7995.14)\\
\\
G*r&=(7,2)&\text{ on curve }B
\end{align}$$
In general, we don't know how to compute $G*r$ on curve $A$ from $G*r$ on curve $B$ other than by finding $r$ first. If we could, that would³ allow finding $r$, which would solve the discrete logarithm problem on a (non-singular) elliptic curve on $\mathbb F_p$ in time polynomial w.r.t. $\log p$. That feat is widely conjectured impossible with a classical Turing machine, and the basis of the conjectured security of Elliptic Curve cryptography, e.g. on secp256k1.

¹ Finding a modular square root of $11$ and taking "positive" as $\in[0,9]$. Square roots in the field $\mathbb F_p$ can be efficiently computed even when prime $p$ is large, see the Tonelli–Shanks algorithm.
² Division $\frac uv$ in field $\mathbb F_{19}$ can be computed as $v^{-1}\,u$ where the modular inverse $v^{-1}$ can be found using the extended Euclidean algorithm.
³ If we get the coordinates of $G*r$ accurately enough.
