Level Surfaces and Quadratic Surfaces

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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.


Contributed by: Ana Moura Santos and João Pedro Pargana (Instituto Superior Técnico) (March 2011)
Open content licensed under CC BY-NC-SA



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