I wanted to use Monotone cubic interpolation, but the site only provide explanation for 2D case. How can I extend it to 3D?
Ok, monotonic interpolation depends on what you are monotonic about. For a simple 1D function interpolation monotonicity is easy to define. But for a 2D and 3D dataset its not so self evident what the situation would be.
First you could interpolate along a independent variable t in which case your monotonicity is most probably in relation to t. This is the same as interpolating each direction separately.
Second you could interpolate along one of the axes, so that your interpolant becomes a function of position on that axis. Reducing a 2D case into a 1D case and a 3D case to two separate 2D cases.
Third you could interpolate on some other spatial variable.
So you see the question is a bit open ended and is hard to say for sure.