I am having trouble finding the Lamé coefficients for the following curves. I have specified all the necessary theory, however I am not sure how to practically compute it. $$ \begin{cases} x = \frac{1}{2}(v_1^2 - v_2^2), & v_1 = \text{const} \\ y = v_1v_2, & v_2 = \text{const} \end{cases} $$
Here is some background: A point $P$ in $3$-D space (or its $\mathbf{r}$) can be defined using Cartesian coordinates $(x,y,z)$ by $\mathbf{r} = x \mathbf{e}_x + y\mathbf{e}_y + z\mathbf{e}_z$, where $\mathbf{e}_x,\mathbf{e}_y, \mathbf{e}_z$ are the standard basis vectors.
In the Cartesian system, the standard basis vectors can be derived from the derivative of the location of point $P$ with respect to the local coordinate $$ \mathbf{e}_{x} = \frac{\partial\mathbf{r}}{\partial x}; \quad \mathbf{e}_{y} = \frac{\partial\mathbf{r}}{\partial y}; \quad \mathbf{e}_{z} = \frac{\partial\mathbf{r}}{\partial z}. $$ Applying the same derivatives to the curvilinear system locally at point $P$ defines the natural basis vectors: $$ \mathbf{h}_1 = \frac{\partial\mathbf{r}}{\partial q^1}; \quad \mathbf{h}_2 = \frac{\partial\mathbf{r}}{\partial q^2}; \quad \mathbf{h}_3 = \frac{\partial\mathbf{r}}{\partial q^3}. $$ we define the Lamé coefficients (after Gabriel Lamé) by $$ h_1 = |\mathbf{h}_1|; \quad h_2 = |\mathbf{h}_2|; \quad h_3 = |\mathbf{h}_3|. $$
Additional note: Lamé coefficients are also known as scale factors.