Quadratic Surfaces -- Rectangular, Cylindrical and Spherical Coordinates
In a Nut Shell: Besides planes in three dimensions there exist a family of cylinders
and quadric surfaces that are of interest in mathematics. A quadric surface is the
graph of a second-degree equation in three variables x, y, and z. You may be asked
to identify and draw figures of these cylinder and quadric surfaces.
Definition of Cylinders
A cylinder is a surface that consists of all lines (called rulings) that are parallel to a
given line and pass through a given plane curve.
Three examples include:
x2 + y2 = 1 a cylinder with its axis along the z-axis of radius 1
y2 + z2 = 1 a cylinder with its axis along the x-axis of radius 1
z = x2 a parabolic cylinder with rulings parallel to the y-axis
Definition of Quadric Surfaces
The family of quadric surfaces include the ellipsoid, the elliptic paraboloid, the
elliptic cone, the hyperboloid of one sheet, the hyperboloid of two sheets, and
the hyperbolic paraboloid.
(x/a)2 + (y/b)2 + (z/c)2 = 1 ellipsoid
(x/a)2 + (y/b)2 = (z/c) elliptic paraboloid
(x/a)2 + (y/b)2 = (z/c)2 elliptical cone
(x/a)2 + (y/b)2 - (z/c)2 = 1 hyperboloid of one sheet
(z/c)2 - (x/a)2 - (y/b)2 = 1 hyperboloid of two sheets
(y/b)2 - (x/a)2 = z/c c > 0 hyperbolic paraboloid
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