Surface algorithms using bounds on derivatives

308 D. Filip et al. / <strong>Surface</strong> <strong>algorithms</strong>
Three special cases frequently used are
(1) Translational Sweep. A 2½-D surface results when D(v) is a linear curve and k(v) is a
constant function.
(2) 3-D Sweep. Let k(v)= 1. Then S(u, v) can be thought of as C(u) being swept along
D(v) or visa-versa.
(3) Spherical Product [Barr '81]. Let da(v ) = d2(v ) = c3(u ) = 0. So
x(u,v)=k(v)q(u),
y(u, v)=k(v)c2(u ),
z(u, v)=da(v ).
Then C(u)= [ca(u) c2(u) 0] is a curve in the x-y plane and E(v)= [k(v) d3(v)] can be
considered a curve in the x-z plane. Any quadric or superquadric [Barr '81] can be parame-
trized as a spherical product of conics or superconics. If C(u) is some parameterization of the
unit circle, then S(u, v) is a surface of revolution with cross section E(v) and axis [0 0 1].
Since in the general case all three component functions look identical, we just consider the
X(U, V) = k(V)Cl(U ) + dl(V ) component:
a x(u, o)
OU 2
= k(v)
d2q(u)
du 2