




In three dimensions, a single equation typically can determine only one coordinate
given the other two. It therefore usually defines a surface and not a curve.
Thus x^{2} + y^{2 }= 4 defines an infinite cylindric surface
or tube of radius 2 with the origin on its axis of symmetry.To describe a curve
or path by equations you typically need two of them. On the other hand, the
parametric description works equally well in two or three dimensions;
the only difference is in three dimensions you must describe the third coordinate,
z, in terms of the parameter as well as x and y.
A surface in three dimensions is often presented by a defining equation; one
can also describe it by parameters, but one requires two of them: varying one
parameter can only produce a one dimensional figure, some sort of curve.Thus
a circular tube of radius 2 around the origin stretching from z = 1 to z =
1 can be described by the parameters z and
in the obvious way. The surface of a sphere of radius a with the origin as center
can be described by the angle parameters of spherical coordinates, and
:
r = x i + y j + z k,
x = a sincos, y = a coscos, z = acos,
0 , 0 2.