Perfect 3D-curve RMDPL-IPOs& Bresenham’s 3D-Curve Algorithm (Part 2 &
3)

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.