# 2D IPO's for Constant Speed Lines, Curves, NURBS

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.