Dini surface: tractrix helicoid with constant negative curvature — Classical Chaos

Ulisse Dini’s surface (1866) is the helicoid of a tractrix: constant negative curvature, the pseudospherical kind. Revolve a tractrix and you get a band of the pseudosphere; add a helical climb and that band corkscrews. The tractrix itself is the classic “dog leash” curve — one short nod to the picture, then leave it.

On Classical Chaos the drawing is not a lit mesh. It is a UV isoline particle wire: constant- and constant- curves as points. Color walks coral → amber → teal along UV so the parameter directions stay readable while the cage grows in.

Radius, pitch, and the climb

Two numbers shape the sheet: radius and helical pitch . A classic parametrization is

The term is the tractrix height. The term is the climb that makes the surface Dini instead of a plain revolution band.

When , the wire is a band of the pseudosphere. When 0'} />, meridians start the helical climb. Defaults sit near , : enough twist to read as a helicoid without collapsing into an unreadable spring.

Presets and the full stage

On the full Dini stage, twist and , try the presets , flat band , taller twist , quieter climb , scrub the reveal, and drag to rotate (or leave auto-rotate on).