1 Cone with sections

2 Paraboloid

3 Ellipsoid

4 Hyperbolic paraboloid

5 Elliptic hyperboloid 1

6 Elliptic hyperboloid 2

7 Elliptic cylinder

8 Circular cylinder

9 Parabolic cylinder

10 Hyperbolic cylinder

11 Circular cone

12 Circular hyperboloid

13 Elliptic paraboloid

14 Ellipsoid with sections

A quadratic (or quadric) surface is a surface in three-space defined by an equation of degree two. Thus it is the three-dimensional analog of a conic section, which is a curve in two-space defined by an equation of degree two. The plane cross-sections of a quadratic surface must be conic sections, and the names for quadratic surfaces refer to the different types of plane cross-sections it possesses. The models in this group show the various types.

Surfaces of revolution are obtained by revolving a conic section about one of its axes. The surface of revolution obtained from an ellipse is called an ellipsoid, and that obtained from a parabola is called a paraboloid. A hyperbola gives rise to different surfaces of revolution, depending on whether it is revolved about the conjugate axis (which passes between the two branches of the hyperbola) or the transverse axis (which crosses the two branches). The first gives a hyperboloid of two sheets, the second a hyperboloid of one sheet. Surfaces of revolution have circular cross-sections perpendicular to the axis of revolution. More general surfaces have elliptic or hyperbolic cross-sections: thus one obtains elliptic and hyperbolic paraboloids, and elliptic hyperboloids of one or two sheets.

Degenerate quadratic surfaces occur when all the plane cross-sections through a given straight line are degenerate conics: either a pair of parallel straight lines, giving rise to a cylinder, or a pair of intersecting straight lines, giving rise to a cone.