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For a function ... of three variables, ... , ... , and ... , the level surface of level ... is defined as the set of points in ... that are solutions of ... . A quadratic surface or quadric is a surface that is given by a second-order polynomial equation in the three variables ... , ... , and ... . Let ... , ... , and ... be nonzero constants. We plot level surfaces for quadratic functions in three variables, which give some well-known quadratic surfaces: ⋄ ... gives ellipsoids; when ... , this is a sphere centered at the origin of radius ... . ⋄ ... or ... give elliptical cylinders with symmetry axes along the ... axis and ... axis, corresponding to ... and ... . ⋄ ... gives elliptic paraboloids, opening up or down as ... or ... . ⋄ ... and ... , with ... , give elliptic cones. For ... , the level surfaces are hyperboloids of one sheet. ⋄ ... ( ... ) and ... ( ... ) give hyperboloids of two sheets.

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