The constant feedrate 3D RMD algorithms avoid most of the constraints introduced by the sampled-data system (chord error, inaccuracy, jerk). The main bottlenecks were the computations of the intersections (QSICs) of the implicit quadrics, the determination of the candidate point with the minimum distance to the QSIC (the best incremental step), the small execution time per step and the fast determination of the small 3D-offsets.The Twopoint RMD IPO is the extension to implicit polynomials of degree higher than two.

The well-known Bresenham and the midpoint algorithm of Pitteway and Van Aken belong to the RMD algorithms. The RMD theorem proves why they have the highest accuracy and the fastest real time execution time. Nowadays the RMD’s have constant feedrate and they even measure the rounded arclength in real time. That was until now a main property of the Pythagorean-hodograph curves of Farouki, which also have applications in offsetting curves. The RMD algorithms implement the basic offset curves from the connected points, the implementation with the high degree PH-curves is complex. This paper proves that everyone can generate a “classic” OoA error and that everyone can correct that OoA error at an amazingly straightforward way, and furthermore the OoARange is inversely proportional with the rounded radius. Therefore, rejecting RMD algorithms for the sake of OoA is hair-splitting.

The paper presents three new 26-connected constant feedrate incremental step algorithms that can be used in practical situations in CNC machining tools. The 1st, the perfect 3D line IPO is 100% incremental, the word “perfect” means that the accuracy can be much better than the accuracy of Bresenham’s 3D line (e.g. accuracy can be 37% worse). The simplified state diagram computes one perfect major axis points and possibly a perfect non-major axis point. The selection criterion uses the real 3D distance to the line. The 2nd, the perfect 3D curve IPO is a QSIC-algorithm (intersection of two quadrics). The selection criterion uses the “Relative Curve Measurement Theorem” extended to quadrics and QSICs. The consequences of this theorem are crucial, it means that one must not calculate the time-consuming distance to the 3D curve, but it suffices to calculate the RMDPL or the relative minimal distance of two candidate points to the polar line of the QSIC with respect to the midpoint of the candidate points. As the midpoints are close to the curve, the polar lines enclose and inclose the curve. Theoretical, the RMDPL is fundamental, it is the core of all the successful 2D incremental step algorithms and the paper proves that it is the core of the 3D incremental step algorithms or the 3D reference pulse IPOs. Thanks to the RMDPL, the paper represents QSICs in a unique way comparable with 3D-lines. The 3rd, Bresenham’s imperfect 3D curve IPO is less accurate but super-fast and can be used in many practical situations as the maximum error (MaxErr) is bounded to 0.707. The curve algorithms can have singular points, but that problem is simple solved. Each curve is a sub-segment of a monotonic curve from the starting extreme point to the ending extreme point. All the extreme points and the singular points are offline precomputed as the intersection points of three quadrics. The constant feedrate of sampled-data curves is clear when the arc length is known, but the real time calculation of the arc length of incremental step curves was until now an open problem. The former paper used the super-fast PRM-cs algorithm for 3D-lines and 2D curves and the same constant feedrate algorithm (actually, a real time length algorithm) can be used even in integer form. The implementation of the constant feedrate algorithm to a 26-connected curve with high accuracy turns out to be piece of cake in contrast to the sampled-data curves. All IPOs can be converted to constant feedrate listSIM-IPOs which can be used in real time in rigid simplified CNC machine tools.

Corrected version of 19-12-10 (YR-Month-Day) of Part I 2D IPO’s for constant speed Lines, Curves, NURBS. Mainly corrected typos and some clearer formulations. The manuscript (44 pages) “Constant Speed Lines – Curves – NURBS Reference Pulse IPOs (Part I)” of Valere Huypens has been accepted for publication in “The International Journal of Advanced Manufacturing Technology” of Springer

Current constant speed IPO’s, usually, use Sampled-data IPO’s and constant speed lines use the wrong initialized software DDA-ipo’s, which make these IPO’s unusable. The Bresenham- and midpoint IPO’s are non-constant speed reference pulse IPO’s with bounded inaccuracy. By adding an ultra-fast 3-lines algorithm “PRM-cs” to the actual midpoint or Bresenham algorithms, we convert these midpoint-ipo’s to very fast, constant speed, reference pulse IPO’s. This applies to 2D-lines, 3D-lines, 2D-curves and 2D-NURBS. The PRM-cs measures, in real-time, the length of the discrete curve and the PRM-cs is completely new. We define the best IPO, the major axis principle and the LSD-priority. The major axis principle holds for the actual 3D-line IPO’s. These IPO’s are, generally, inaccurate, but they can be updated to constant speed 3D-line IPO’s, when the production manager agrees. The Digital Geometric Geometry (DAG) defines the discrete lines globally, but this global definition of a discrete 3D-line, gives discrete 3D-lines whose accuracy is much less than the accuracy of the best discrete 3D-lines (e.g. 37% worse). We describe the three causes of the inaccurate (imperfect) discrete 3D-lines. All existing pulse-rate or PRM-ipo’s use a wrong initialization, which deteriorates the accuracy. We determine the right initialization for the new PRM-cs and the updated PRM-ipo. We propose the benchmark-ipo “listSIM-ipo”. This constant speed IPO can, also, be used in real- time for every 2D- and 3D-curve. The 3rd-degree Trident NURB shows that the constant speed reference pulse method is much better than the existing sampled-data methods.