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Elementary Surfaces

Ellipsoid, sphere, hyperboloid, cone and more

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The simplest of surfaces are the unbound planes. The next complexity are the second degree or quadric surfaces.

Non-degenerate Quadrics


x2/a2 + y2/b2 + z2/c2 = 1                       (1)


a, b, c = intrinsic parameters


Sphere, a special ellipsoid, can be expressed as

x2 + y2 + z2 = a2                            (2)


a = radius

Hyperboloid of one sheet

x2/a2 + y2/b2 - z2/c2 = 1                            (3)


a, b, c = intrinsic parameters

Hyperboloid of two sheets

x2/a2 + y2/b2 - z2/c2 = -1                            (4)


a, b, c = intrinsic parameters

Quadric Cone

x2/a2 + y2/b2 - z2/c2 = 0                             (5)


a, b, c = intrinsic parameters

Elliptic Paraboloid

x2/a2 + y2/b2 - 2 z = 0                                (6)


a, b = intrinsic parameters

Elliptic Cylinder

x2/a2 + y2/b2 = 1                                (7)


a, b = intrinsic parameters


x2 + y2 = a2                                (8)


a, b = intrinsic parameters

Hyperbolic Cylinder

x2/a2 - y2/b2 = 1                                 (9)


a, b = intrinsic parameters

Parabolic Cylinder

y2 - 2 l x = 0                             (10)


l = intrinsic parameters

Degenerate Quadrics

Parallel Planes

x2 - a2 = 0                             (11)


a = half the distance between the parallel planes

Intersecting Planes

x2/a2 - y2/b2 = 0                          (12)


tan-1(b/a) = half the angle between the intersecting planes

Coincident Planes

x2 = 0                            (13)

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  • en: elementary surfaces ellipsoid sphere hyperboloid cone elliptic cylinder
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  • Engineering ToolBox, (2009). Elementary Surfaces. [online] Available at: [Accessed Day Mo. Year].

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