Hi again,

as you can see in the screenshots, XLS does definitely not use simple CR splines to interpolate data when using its "smooth curve" feature. The XLS spline definitely is cubic because the second derivative changes linearly. From looking at the screenshot showing the second derivatives, it seems to me as if two additional nodes (around 3) have been "magically" added and one node (2) has been "magically" shifted by XLS. By "magically" I mean "in a manner as to have the resulting spline run through all the given data points", which indeed is a typical feature of CR splines. Because of the added nodes, the third derivative in the first and last segment, resp., cannot be set to zero and consequently, the XLS cubic spline does not run out parabolically on either end as is true in the case of AlgLib's Catmull-Rom splines. Concluding, I cannot tell at the moment which kind of spline XLS uses in its "smooth curve" procedure, but it is definitely not a CR spline having the data points as nodes. Perhaps, it is a CR spline on the set of given plus "magically" added nodes, but I have not checked that yet.

Cheers

Georg

P.S.: I used the freeware "Engauge Digitizer 4.1" to digitise the XLS smoothed curve using a 24MP graphics file obtained by copying a large chart to the clibboard as picture (no more than 32MP allowed with that software!). There is a website on which the XLS procedure secrets seem to be disclosed:

http://www.xlrotor.com/resources/files.shtml, look for "Smooth_curve_bezier_example_file.xls". And I've learned that CR splines are famous in the animation business and implemented in MS-DirectX:

http://www.mvps.org/directx/articles/catmull/ .