Vtyjkyuk

My aim is to smooth the terrain in a video game. Therefore I contrived an algorithm that makes use of Bézier curves of different orders. But this algorithm is defined in a two dimensional space for now.

To shift it into the third dimension I need to somehow combine the Bézier curves. Given two curves in each two dimensions, how can I combine them to build a surface?

I thought about something like a curve of a curve. Or maybe I could multiply them somehow. How can I combine them to cause the wanted behavior? Is there a common approach?

In the image you can see the input, on the left, and the output, on the right, of the algorithm that works in a two dimensional space. For the real application there is a three dimensional input.

The algorithm relays only on the adjacent tiles. Given these it will define the control points of the mentioned Bézier curve. Additionally it marks one side of the curve as inside the terrain and the other as outside.

You can define a bezier-area by
$$x(lambda,mu) = sum^m_i=0sum^n_j=0B_im(lambda)B_jn(mu)p_ij$$

with
$lambda,muin[0,1]$, $B_im$ are the Bernstein polynomials and $p_ij$ are the given points.

Note: For this approach you need a grid of points.

I would not use Bezier curves for this. Too much work to find the end-points and you end up with a big clumsy polynomial.

I would build a linear least squares problem minimizing the gradient (smoothing the slopes of hills).

First let's split each pixel into $6times 6$ which will give the new smoothed resolution (just an example, you can pick any size you would like).

Now the optimization problem $$bf v_o = min_bf v _F^2+$$

where $bf d$ is the initial pixely blocky surface, and $bf v$ is the vector of smoothing change you want to do.

Already after 2 iterations of a very fast iterative solver we can get results like this:

After clarification from OP, I realized it is more this problem we have ( but in 3D ).

Now the contours (level sets) to this smoothed binary function can be used to create an interpolative effect: