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If a = b, an elliptic paraboloid is a circular paraboloid or paraboloid of revolution. It is a surface of revolution obtained by revolving a parabola around its axis. A circular paraboloid contains circles. This is also true in the general case (see Circular section). From the point of view of projective geometry, an elliptic paraboloid is an ...
A surface is doubly ruled if through every one of its points there are two distinct lines that lie on the surface. The hyperbolic paraboloid and the hyperboloid of one sheet are doubly ruled surfaces. The plane is the only surface which contains at least three distinct lines through each of its points (Fuchs & Tabachnikov 2007).
The coordinate surfaces of the former are parabolic cylinders, and the coordinate surfaces of the latter are circular paraboloids. Differently from cylindrical and rotational parabolic coordinates, but similarly to the related ellipsoidal coordinates , the coordinate surfaces of the paraboloidal coordinate system are not produced by rotating or ...
In the second case (−1 in the right-hand side of the equation): a two-sheet hyperboloid, also called an elliptic hyperboloid. The surface has two connected components and a positive Gaussian curvature at every point. The surface is convex in the sense that the tangent plane at every point intersects the surface only in this point.
If both curves are contained in a common plane, the translation surface is planar (part of a plane). This case is generally ignored. ellipt. paraboloid, parabol. cylinder, hyperbol. paraboloid as translation surface translation surface: the generating curves are a sine arc and a parabola arc Shifting a horizontal circle along a helix. Simple ...
The sign of the Gaussian curvature at a point determines the shape of the surface near that point: for K > 0 the surface is locally convex and the point is called elliptic, while for K < 0 the surface is saddle shaped and the point is called hyperbolic. The points at which the Gaussian curvature is zero are called parabolic.
The red paraboloid corresponds to τ=2, the blue paraboloid corresponds to σ=1, and the yellow half-plane corresponds to φ=-60°. The three surfaces intersect at the point P (shown as a black sphere) with Cartesian coordinates roughly (1.0, -1.732, 1.5).
Ellipsoidal coordinates are a three-dimensional orthogonal coordinate system (,,) that generalizes the two-dimensional elliptic coordinate system. Unlike most three-dimensional orthogonal coordinate systems that feature quadratic coordinate surfaces , the ellipsoidal coordinate system is based on confocal quadrics .