Curve fitting under different regions and stitching

Question

Is there a way to fit a 2D curve under the following conditions:

1. The curve is defined by 2 functions for x>a, and x<a
2. Prefer a fit that is continuous and differentiable at x=a

Answer 1

I hope I understood you correctly and that the following thoughts help you a bit.

Reference point: Univariate curve fitting using splines

With a univariate function $f(x)$ you can perform 1D spline interpolation and require for each (inner) $x_i$-node that: $$ \begin{align} \left.f_{i-1}(x)\right|_{x=x_i}&=\left.f_i(x)\right|_{x=x_i} \quad \mathrm{continuity}\\ \left.\frac{\partial ^k f_{i-1}(x)}{\partial x^k}\right|_{x=x_i}&=\left.\frac{\partial ^k f_{i}(x)}{\partial x^k}\right|_{x=x_i} \quad k\mathrm{th \ order \ differentiability}\\ \end{align} $$

And the $K$th order 1D-spline function is identified by

$$f_i(x)=\sum_{j=0}^K\beta_{i,j}x^j \quad \mathrm{for} \quad x_i\leq x<x_{i+1} $$

For an $K-1$th-order spline you get $K$ degrees of freedom per function. Say you have a total of $N$ test points $\{x_i,y_i\}_{i=0\ldots{N-1}}$. This nets you a total of $N-1$ functions to fit and hence $K\times(N-1)$ parameters. Say you select a cubic spline ($K=3$, four degrees of freedom per function), then you could, for example, produce a spline such that $$ \begin{align} \beta_{i-1,0}+\beta_{i-1,1}x_i+\beta_{i-1,2}x_i^2+\beta_{i-1,3}x_i^3&=\beta_{i,0}+\beta_{i,1}x_i+\beta_{i,2}x_i^2+\beta_{i,3}x_i^3 \quad \mathrm{equality}\\ \beta_{i-1,1}+2\beta_{i-1,2}x_i+3\beta_{i-1,3}x_i^2&=\beta_{i,1}+2\beta_{i,2}x_i+3\beta_{i,3}x_i^2 \quad \mathrm{f'}\\ 2\beta_{i-1,2}+6\beta_{i-1,3}x_i&=2\beta_{i,2}+6\beta_{i,3}x_i \quad \mathrm{f''}\\ \beta_{i-1,3}&=\beta_{i,3} \quad \mathrm{f'''} \end{align} $$ holds at all (inner) nodes. In total, you have thus identified $N + 3(N-2)$ of your parameters. Adding an assumption on the behavior of $f_0‘(x_0)$ and $f_{N-1}^‘(x_{N-1})$ usually yields the remaining parameters.

Your case: Bivariate extension

In the bivariate spline case, each point is now bounde by a surface segment (surrounded by 4 test points):

$$ \begin{align} x_{i-1}\leq x < x_i, y_{j-1}\leq y < y_j \end{align} $$ and the spline function is now commonly given by:

$$ f_{i,j}(x,y)=\sum_{k=0}^K\sum_{l=0}^K\beta_{i,j,k,l}x^ky^l $$

As you can see, a $K$th order 2D-spline requires (canonically) $(K+1)^2$ parameters per segment. Hence, we need more test points to enter the spline function, and we have a quadratic increase in conditions (continuity, differentiability...). This may be a good starting point for your endeavour.