I have a surface that is defined using a two dimensional polynomial:
$$z = f(x) + g(y)$$
I want to offset the curve in the $XY$ plane from a point on the surface $\left(x_0, y_0, z_0\right)$ to a new point $\left(x_1, y_1, z_0\right)$.
This is essentially equivalent to the following.
$$z_0 = f(x_0) + g(y_0) \to z_0 = f(x_1) + g(x_1) = f(x_0 + dx) + g(y_0 + dy)$$
I am hoping to avoid a situation where I have to transform the points that were used to create the polynomial based on dx and dy and then regenerate the curve.
I interpret this as a question of recentering a polynomial. The formula $$ \sum_{k=0}^p a_k \left(z+z'\right)^k = \sum_{\ell=0}^p \left(\sum_{k=l}^p \binom{k}{\ell} (z')^{k-\ell} a_k \right) z^\ell=\sum_{\ell=0}^p b_{\ell}\, z^\ell $$ can be derived using Taylor series.