# Curve fitting under different regions and stitching

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
• In the benefint of a more concrete discussion can you provide an example of numerical data. Or alternatively can you give an example of 2 functions that you commonly enconter in your fitting calculus. Apr 29, 2021 at 15:30

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

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.

\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.