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 .

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.