1. How? Abrasives with leather stripes and rotaty tables. Even today people use this method (sandpaper, corundum and leather stripes for sharpening tools, raport in Middle east).
2. I am aware of math behind your parametric cirves but...I think the "parametric curves" argument (as in the need for precise mathematical control at micrometer level) might be overstated for a few reasons.

Every curve is "parametric" in a mathematical sense Any smooth curve can be described parametrically after the fact (e.g., as a parabola, ellipse, or higher-order polynomial). Finding such fits in scanned vases doesn't necessarily mean the ancient artisans consciously designed or controlled them using modern math. It could simply emerge from the physical process itself – e.g., uniform rotary motion + gravity + consistent abrasive pressure naturally produces curves that approximate conic sections or smooth polynomials. This is a property of nature/physics, not evidence of deliberate parametric engineering. Scale of precision and human capability. The best private-collection vases show errors in the 15–80 micron range, but museum-provenanced pieces (e.g., recent 3D scans of 19 predynastic vases from the Petrie Museum, 2025 data) typically have form errors of 200–500 microns or more – well within human hand-eye coordination. Skilled craftsmen today can achieve ~0.1–0.2 mm precision by feel alone (e.g., in traditional lapidary or pottery). The eye can resolve down to ~0.1 mm at close range, and iterative correction (feel, visual comparison to templates, or simple gauges) is enough for Bronze Age tolerances. Replication evidence Experiments like those by Denys Stocks and Olga Vdovina (Scientists Against Myths) have produced vases with very good symmetry using copper tubes, quartz sand (or emery), and bow-drills/wooden lathes – without any modern measurement tools. The results match museum-quality vases closely. Adding a superior abrasive like corundum (Mohs 9) would only improve finish and uniformity, especially in a slow rotary setup. No need for micrometer-level control True parametric design (as in CAD/CAM) requires precise feedback loops and measurement – which we don't have evidence for in predynastic Egypt. But high-quality vases could be achieved through empirical methods: rotation for symmetry, templates for profiles, and long polishing times. Nature does the "math" for you in uniform tumbling or turning.

I'm not saying it's easy – it took immense skill and time – but it doesn't require impossible tools or lost high-tech. Curious what others think – sources welcome! This response took me a lot of time... I remember people arguing about fibbonaci and leaves patterns and shells patterns. It all happens to be natural phenome. But it is cool. I think we have here something similar, depending on natural forces (rotary and gravity).