Geometric primitives87Figure 4.14 Rational b-spline curve (some homogeneous coordinates are not 1.0)Note in Figure 4.14 that the b-spline curve passes through the very first CV. This is because in our RIB command we repeated that CV coordinates thrice, giving that CV a multiplicity (redundancy) of 3. For a cubic spline curve (the ones supported by RenderMan), repeating a CV thrice will force the curve to pass through that CV. This is further illustrated in Figure 4.15 where the CV numbered 6 and the last CV (number 12) belonging to a b-spline are repeated thrice. CV number 9 is only repeated twice, so the curve does not quite pass through that CV but passes close to it instead. By repeating CVs sequentially for a b-spline as shown in figures 4.13 through 4.15 to increase their multiplicity, we are faking the effect of what are called knots whose direct specification is not supported in RenderMan Curve command. Knots however are supported (indeed, required) for the NuPatch NURBS surface command, so we will discuss them in detail in the section on patches (Section 4.7). The RIB code used to generate Figure 4.15 is:# Wee using Pw with w=1 everywhere, which makes the # curve a polynomial one. We could have used P # instead, without the extra w values Basis "b-spline" 1 Curves "cubic" [18] "nonperiodic" "Pw" [ 0 0 0 1 1 0 0 1 1 -2 0 1 -2 -2 0 1 -2 2 0 1 3 2 0 1 # The following CV occurs thrice, forcing the curve to # pass through it 3 -4 0 1 3 -4 0 1
