Relative Squared Distances to a Conic

Abstract

The midpoint method or technique is a measurement and as each measurement it has a tolerance, but worst of all it can be invalid, called Out-of-Control or OoC. The core of all midpoint methods is the accurate measurement of the difference of the squared distances of two points to the polar of their midpoint with respect to the conic. When this measurement is valid, it also measures the difference of the squared distances of these points to the conic, although it may be inaccurate, called Out-of-Accuracy or OoA. The primary condition is the necessary and sufficient condition that a measurement is valid. It is comletely new and it can be checked ultra fast and before the actual measurement starts. Modeling an incremental algorithm, shows that the curve must be subdivided into piecewise monotonic sections, the start point must be optimal, and it explains that the 2D-incremental method can find, locally, the global Least Square Distance. Locally means that there are at most three candidate points for a given monotonic direction; therefore the 2D-midpoint method has, locally, at most three measurements. When all the possible measurements are invalid, the midpoint method cannot be applied, and in that case the ultra fast OoC-rule selects the candidate point. This guarantees, for the first time, a 100% stable, ultra-fast, berserkless midpoint algorithm, which can be easily transformed to hardware.