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From the point of view of projective geometry, an elliptic paraboloid is an ellipsoid that is tangent to the plane at infinity. Plane sections. The plane sections of an elliptic paraboloid can be: a parabola, if the plane is parallel to the axis, a point, if the plane is a tangent plane. an ellipse or empty, otherwise.
For example, a cone is formed by keeping one point of a line fixed whilst moving another point along a circle. 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.
In geometry a conoid (from Greek κωνος 'cone' and -ειδης 'similar') is a ruled surface, whose rulings (lines) fulfill the additional conditions: (1) All rulings are parallel to a plane, the directrix plane. (2) All rulings intersect a fixed line, the axis. The conoid is a right conoid if its axis is perpendicular to its directrix ...
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 ...
A conic is the curve obtained as the intersection of a plane, called the cutting plane, with the surface of a double cone (a cone with two nappes).It is usually assumed that the cone is a right circular cone for the purpose of easy description, but this is not required; any double cone with some circular cross-section will suffice.
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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 examples: Right circular cylinder: is a circle (or another cross section) and is a line.
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