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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 ...
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 ...
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).
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 ...
This is in contrast to the previous section, which was about 2-dimensional elliptic geometry. The quaternions are used to elucidate this space. Elliptic space can be constructed in a way similar to the construction of three-dimensional vector space: with equivalence classes. One uses directed arcs on great circles of the sphere.
In geometry, the elliptic coordinate system is a two-dimensional orthogonal coordinate system in which the coordinate lines are confocal ellipses and hyperbolae. The two foci F 1 {\displaystyle F_{1}} and F 2 {\displaystyle F_{2}} are generally taken to be fixed at − a {\displaystyle -a} and + a {\displaystyle +a} , respectively, on the x ...
In mathematics, a quadric or quadric surface (quadric hypersurface in higher dimensions), is a generalization of conic sections (ellipses, parabolas, and hyperbolas).It is a hypersurface (of dimension D) in a (D + 1)-dimensional space, and it is defined as the zero set of an irreducible polynomial of degree two in D + 1 variables; for example, D = 1 in the case of conic sections.
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.