A

The rulings of a ruled surface

The Gaussian curvature at a regular point of a ruled surface is

A ruling is

A ruling is

A ruled surface is

A ruled surface with a finite number of torsal lines are called a

If a scroll is a quadric (the hyperbolic paraboloid or the hyperboloid of one sheet) it is a double generated ruled surface - through each point on the surface two rulings pass. The rulings define the asimptotic directions.

All other algebraic ruled surfaces are simple generated - through each regular point on it exactly one ruling passes (3rd degree algebraic scrolls, 4th degree algebraic scrolls). The asimptotic directions at regular points of such a scroll are defined by the rulings and the tangent lines of the intersection curves of tangent planes and a scroll. For n-order algebraic ruled surfaces these intersections are (n-1)-order algebraic curves.

At a regular point the principal directions bisect the angle between the asymptotic directions.

3.1 EXAMPLE (the hyperbolic paraboloid)

3.2 EXAMPLE (the 4th degree conoid)

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