I’ve got some time so I thought I try my hand at paraphrasing the information in those two links.
A Spline curve forms a continuous interpolation between the control points(CP). So, any point on a curve is the result of processing several nearby CPs. The two notable exceptions are the Start- and EndPoint.
Due to the condition that several CPs influence any particualr point on the curve, the mathematical processing requires a method of ‘weighting’ each CP along the entire span (parametic range). The knots divide the full range, and help define when and how much each CP is contributing.
If we consider a degree 3 spline with 7 CPs, and a knot sequence 0 0 0 1 2 3 4 4 4 we can see that the knots match the technical condition where the quantity = Degree + N – 1 ( 9 = 3 + 7 – 1). I should point out that AutoCAD uses the alternate algorithm that uses Degree + N +1. I’m using this example because it matches the information contained within the links.
Even though there are 9 knots, there are only 4 specific intervals (0-1, 1-2, 2-3, 3-4). Typically, 4 CPs are involved through any interval evaluation, and that evaluation continually modifies how much each CP contributes to the position of a point on the curve. Where an interval is bordered by high multiplicity sequence, 0 0 0 and 4 4 4 for instance, the position uses that associated CP more time in the calculations. This way the Start- and EndPoint contribute 100% at that point in the interval.
Given that the whole purpose for a Spline is as a continuous interpolation scheme, finding evidence of when and where a knot comes into play is tough. Degree 3 splines are G2 continuous throughout the entire span – which is to say there is no point where the curvature changes abruptly. The rate of change of the curvature, however, does change abruptly at the change from one knot interval to the next.
The attached picture is a curve analysis diagram from rhino. The attached DXF is that same spline.
