Kurvenfläche

Kurvenfläche is a German term that translates to "ruled surface" in English. It refers to a surface that can be generated by the continuous movement of a straight line. This moving line is called the generatrix, and its path defines the shape of the surface. In simpler terms, a ruled surface is composed of an infinite number of straight lines.

Examples of ruled surfaces include developable surfaces and skew surfaces. Developable surfaces are ruled surfaces that can be unrolled or flattened onto a plane without stretching or tearing. Think of the surface of a cylinder or a cone; these can be flattened into a rectangle and a sector of a circle, respectively. Skew surfaces, on the other hand, are ruled surfaces that cannot be flattened onto a plane without distortion. The hyperboloid of one sheet and the hyperbolic paraboloid are classic examples of skew ruled surfaces.

The mathematical description of a ruled surface often involves a parametric equation where one of the parameters controls the position along the generatrix and the other parameter traces the path of the generatrix. The study of ruled surfaces is a fundamental concept in differential geometry, with applications in computer-aided design, architecture, and engineering, particularly in the design of structures and objects that require smooth, continuous surfaces.