Because these techniques have a graphical foundation it makes them easier to understand how they work and under what conditions they should be applied. This graphical foundation also inspires new observations and constructions.
The algorithms present here for the calculus for tabular functions were developed from a set of simple concepts:
- Work directly on the tabular functions. Do not attempt to curve fit tabular functions and then operate on the curve fits. Let the data speak for itself.
- Focus on the local behavior of the curve as represented by a grouping of three points. Clearly three points is the smallest number of points that can capture the local shape of the tabular function.
- An empirical arc length has proven to be a very useful parameterization for a tabular function and is very important step in building these algorithms.
- Do not enforce curvature continuity between adjacent segments of the tabular function.
- Avoid the evaluation of coefficients. Graphical construction methods are applied completely by passing coefficient evaluations.