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# Elementary Surfaces

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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

#### Ellipsoid

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

where

a, b, c = intrinsic parameters

#### Sphere

Sphere, a special ellipsoid, can be expressed as

x2 + y2+ z2= a2                            (2)

where

a = radius

#### Hyperboloid of one sheet

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

where

a, b, c = intrinsic parameters

.

#### Hyperboloid of two sheets

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

where

a, b, c = intrinsic parameters

#### Quadric Cone

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

where

a, b, c = intrinsic parameters

#### Elliptic Paraboloid

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

where

a, b = intrinsic parameters

#### Elliptic Cylinder

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

where

a, b = intrinsic parameters

#### Cylinder

x2 + y2 = a2                                (8)

where

a, b = intrinsic parameters

#### Hyperbolic Cylinder

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

where

a, b = intrinsic parameters

#### Parabolic Cylinder

y2- 2 l x = 0                             (10)

where

l = intrinsic parameters

.

### Degenerate Quadrics

#### Parallel Planes

x2 - a2 = 0                             (11)

where

a = half the distance between the parallel planes

#### Intersecting Planes

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

where

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

#### Coincident Planes

x2 = 0                            (13)

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