Surfaces in Computer Graphics - Computer Graphics and Geometric Modeling - page 519

12.15 Composite Surfaces and Geometric Continuity

This section looks at conditions that ensure that a collection of surface patches that
meet along boundary curves will define a globally smooth surface. For more infor-
mation and references to work in this area see [FauP79], [Mort85], [Greg89],
[HosL93], or [Fari97].
The idea here is basically the same as it was for curves. As a set, the union S of
the sets that are traced out by the individual C k parameterizations should be a smooth
surface. Of course, since there are now two parameters, things are somewhat more
involved computationally. Now C k continuity would mean that the parameterization
of the union S induced by the individual parameterizations be C k . This requirement
is again stronger than needed.

Definition. Two parameterized surface patches p(u,v) and q(u,v) meet along their
boundary with kth-order geometric continuity , or G k continuity , if there is a repar-
ameterization r(u,v) equivalent to p(u,v) so that r(u,v) and q(u,v) meet with C k
continuity.

One can show that G 1 continuity simply means that the two patches have the same
tangent planes at the points where they meet, so that G 1 continuity is sometimes
referred to as tangent plane continuity . Since the analog of tangent lines for curves is
tangent planes for surfaces, this is certainly a natural first condition for patches to
meet in a way that is visually smooth. We mentioned earlier that one can get G 1 con-
tinuity with Gregory patches. Here we shall consider some simple conditions that will
achieve this for tensor product and triangular patches. Section 15.2 discusses addi-
tional curvature-related criteria.
First of all, let p(u,v) and q(u,v) be two rectangular Bézier patches with domain
[a,b] ¥ [c,d] and [a,b] ¥ [d,e] and control points p ij , 0 £ i £ m, 0 £ j £ n, and p ij , 0 £ i
£ m, n £ j £ 2n, respectively. The condition that they meet with G 1 continuity along
the boundary curve defined by the points p in , 0 £ i £ n, is that the three points p i,n-1 ,
p in , p i,n+1 are collinear for 0 £ i £ n. See Figure 12.25. The reason is that the normal
for the tangent plane is the cross-product of the partials in the u- and v-direction.
Since the boundary curve has the same control points for both patches, we only need
the tangent vector in the v-direction to be parallel. For C 1

continuity, collinearity is

Figure 12.25. Collinearity condition for

smoothly meeting Bézier
patches.