triple_r wrote:
> Fairly simple question, but potentially complex:
> It works to an extent, but is the pictured scheme entirely correct for
> tricubic interpolation?  I know there's no problem for linear interpolation
> like this, but I'm not quite sure whether it can be extended to cubic.  I
> think it would be possible to use the basis functions to find each point's
> contribution to the interpolated value, but that can overshoot the data
> (for a cfd simulation) so it is better to interpolate along a line if
> possible - since it's easier to prevent overshooting with four points than
> with 64.  Maybe (quite possibly) there's a better way yet.  Thanks.
>  - Ricky
> 
> 
> ------------------------------------------------------------------------
> 
I am not sure if I understand your question. A tricubic interpolation
has 64 parameters. Any set of 64 independent points can fix those.
Often a regular distribution like in your drawing is used, but
technically that is not necessary. If you use these points you can
either choose to fix them in your volume so that when you move a
point the volume deforms with it or you can use them as a control
point in the Bezier sense where you can be sure that the volume
will stay within the control points, that the 'direction' of the
surface will be in the direction of the next point etc.
Such control points will avoid overshooting but you can not fix
the surfaces to go through one specific point. Besides these
common approaches to specify a volume, you can also specify
at each corner the derivatives in three orthogonal directions
plus the three mixed second order derivatives in UV, VW and UW
directions plus the third order derivative in the UVW direction.
That will also fix the 64 parameters. And there are a whole lot
of yet again different approached to fix them.

I don't get what you mean by interpolation along a line. Even if you
can prevent overshoots along that line, nothing prevents overshoots
in any of the two othogonal directions.

To cut a long story short, I think that: you either specify the exact
position of a point after some deformation and then you will get
overshoots, or you use a bezier approach and then the points will
not be exactly in the right place but at least you get a nice
volume where you can easily control the continuity along the outer
surfaces of the 'cubes' you use to describe your volume.

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