quadratic surfaces

The common equation of all quadratic surfaces is


where A,B,C,A,B,C,A′′,B′′,C′′,D are constants and at least one of the six first does not vanish.  The different non-degenerate kinds are as follows; we give also the simplest equation.

This classification is based on examining the signaturePlanetmathPlanetmathPlanetmath (http://planetmath.org/SylvestersLaw) of the quadratic formMathworldPlanetmath


and the signature of the form


Note that, because of the fact that the equation describes the same surface if we simultaneously change the signs of all the coefficients, we obtain the same type of surface if we change all the signs in both signatures.

Surfaces without midpointsMathworldPlanetmathPlanetmathPlanetmath (http://planetmath.org/Midpoint3):



a) Elliptic paraboloid,  x2a2+y2b2=2z

Signatures: [++0], [+++-] (or [--0], [+---])



b) Hyperbolic paraboloid,  x2a2-y2b2=2z;  it is a doubly ruled surface.

Signatures: [+-0], [++--]



c) Parabolic cylinder,  x2=2pz;  it is a developable surfaceMathworldPlanetmath.

Signatures: [+00], [++-0] (or [-00], [+--0])

Surfaces with one midpoint:



a) EllipsoidMathworldPlanetmath,  x2a2+y2b2+z2c2=1

Signature: [+++], [+++-] (or [---], [+---])



b) One-sheeted hyperboloid,  x2a2+y2b2-z2c2=1;  it is a doubly ruled surface.

Signatures: [++-], [++--] (or [+--], [++--])



c) Two-sheeted hyperboloid,  x2a2-y2b2-z2c2=1

Signature: [++-], [+++-] (or [+--], [+---])




d) ,  x2a2+y2b2-z2c2=0;  it is a developable surface.

Signatures: [++-], [++-0] (or [+--], [+--0])

Surfaces with infinitely many midpoints



a) Hyperbolic cylinder,  x2a2-y2b2=1;  it is a developable surface.

Signatures: [+-0], [+--0] (or [+-0], [++-0])



c) Elliptic cylinder,  x2a2+y2b2=1;  it is a developable surface.

Signatures: [++0], [++-0] (or [--0], [+--0])

b) Two intersecting planes,  x2a2-y2b2=0
Signatures: [+-0], [+-00]

d) Two parallel planesMathworldPlanetmath,  x2=a2
Signatures: [+00], [+-00] (or [-00], [+-00])

e) Double plane,  x2=0
Signatures: [+00], [+000] (or [-00], [-000])

Algebraically, there are other possibilities for the signatures, such as [+++] and [++++].  However, these give rise to equations which have no real solutions, hence they have been ignored.

Title quadratic surfaces
Canonical name QuadraticSurfaces
Date of creation 2013-03-22 14:59:40
Last modified on 2013-03-22 14:59:40
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 53
Author pahio (2872)
Entry type Topic
Classification msc 51N20
Synonym surfaces of second degree
Related topic TangentPlaneOfQuadraticSurface
Related topic Ellipsoid
Related topic SurfaceOfRevolution2
Related topic GeneratricesOfOneSheetedHyperboloid
Related topic GeneratricesOfHyperbolicParaboloid
Related topic AnalyticGeometry
Related topic IntersectionOfQuadraticSurfaceAndPlane
Defines elliptic paraboloid
Defines hyperbolic paraboloid
Defines parabolic cylinder
Defines ellipsoid
Defines one-sheeted hyperboloid
Defines two-sheeted hyperboloid
Defines cone surface
Defines hyperbolic cylinder
Defines elliptic cylinder