Part 5 of this series introduced the mathematical foundation for a quadratic Hermite spline with segments comprised of individual quadratic Hermite curves (C-1 continuity at the joins). Since the system of equations for the start tangent of each curve is underdetermined, a start tangent for the first curve is required.
Selection of an initial tangent is entirely arbitrary. For the Degrafa spline, I intend to have a method that provides an automatic selection. This allows the spline to fit a set of knots without any further user interaction. It is important to understand that no matter what algorithm is chosen, there is no theory that makes any one method ‘better’ than any other. It’s entirely a method of designer preference.
This is a feature that I really like about this type of spline. Because of the ‘coupling’ in the upper bi-diagonal system shown in part 5, the selection of a start tangent for the first segment ‘ripples’ through the entire curve. This allows a wide variety of interesting curves to be created under the control of a single parameter.
To see what I mean, two screenshots are provided from the program I’m using to test the base utility on top of which the Degrafa spline will be created.
Two dramatically different curves are obtained from the same set of knots by adjusting one parameter; the start tangent of the first quad. Hermite segment.
The Algorithmist 