Level Surfaces and Quadratic Surfaces
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 .[more]
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.[less]