For the regular ruled surface , put . The second-form coefficient in the ruling direction is zero, and the mixed coefficient is . The determinant formula for Gaussian curvature therefore gives the displayed nonpositive expression. Its vanishing is equivalent to the zero scalar triple product, only at regular points.
On , an axial plane such as cuts out two disjoint meridian geodesics. The tangent plane cuts out the two straight ruling lines , which are geodesics meeting orthogonally at .
For a unit-speed geodesic on , Clairaut's constant givesIf , the geodesic has a turning point at and remains entirely in one of the regions or .
The waist circle is a geodesic. Its Gaussian curvature is , whereasoff the waist. Since isometries preserve Gaussian curvature, every isometry preserves this circle setwise.
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A **ruled surface** is a type of surface in three-dimensional space that can be generated by moving a straight line (the ruling) continuously along a path. In a more technical sense, a ruled surface can be defined as a surface that can be represented as the locus of a line segment in space, meaning that for every point on the surface, there exists at least one straight line that lies entirely on that surface.