Mike Raiford wrote:

> The article speaks of derivatives, so calculus must be involved. I'm 
> gathering that the formulae in the algorithm were probably found by 
> applying calculus to a 3rd order polynomial.
> 
> http://en.wikipedia.org/wiki/Cubic_spline

Or they were derived by calculating which coefficients will yield 
continuity in the first derivative.  And maybe the second.

I am presently note-scribbling for a scene-building feature for my 
modeler, and I want something that is simple and C2 continuous when the 
user specifies continuity at a point.  I've decided to use the limit 
curves of a subdivided border (end points stay still, new mid points are 
the average of the start and end of the edge, and new interior points 
are calculated on a 1-6-1 mask).  The result is a cubic spline with C0, 
C1, and C2 continuity.  It does not necessarily pass through the control 
points, but I have a work-around for that.

C2 continuity is important in animation; if the two spline ends meet in 
the middle of a curve, and the object moving along the curve is tilted 
to show its acceleration, then any discontinuity at the join will result 
in the sudden change in the tilt of the object.  This is usually undesired.

Regards,
John

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