Elementary Surfaces
Ellipsoid, sphere, hyperboloid, cone and more.
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)
