RELATIVE SQUARED DISTANCES TO A CONIC

Valere Huypens, Belgium 

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. The new algorithm is on average (26.5±5)% faster than Mathematica, using the same resolution and tested using 42 different conics. Both programs are completely written in Mathematica and only ContourPlot[] has been replaced with a module to generate the grid-points, drawn with Mathematica’s Graphics[Line{gridpoints}] function. . 

Index Terms 

Midpoint method, two-point method, incremental curve algorithms, squared Euclidean distance, Mathematica, conic, QSIC, generation of CNC-grid points, Bresenham.

More Details  : http://airccse.org/journal/ijcga/papers/5115ijcga02.pdf