By R. Barnhill
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Extra resources for Computer Aided Geometric Design
Microfiche hard-copy from this device was used to produce Figures 4 and 5. Secondly we have been dissatisfied with attempts to create an illusion by means of two-dimensional pictures, of three-dimensional objects. Similarly, the folded curves of Resch and Cohen*^ can best be ap preciated as physical models. We have had many instances where design faults could not be or were not discovered by looking at dynamic displays, stereo projections, and other visualizations. As a result, 2 3 4 and following Bezier's example ' ' we have built a three-dimensional model making machine^ (which pro duced the model, Figure 8, of the surface drawn by the laser display in Figure 5).
We consider linear interpolation operators P. along parallels to each side E. of T. Figure 5 is a symbolic representation of these interpolants. 3) P3F(x,y) = (^) F(0,x+y) F(x+y,0) + (^) 49 ROBERTE. BARNHILL Each of these projectors is a "lofting11 interpolant, and Ρ^ίχ,Ι-χ) * F(x,l-x). 3) are removable, as are the singularities of all the rational interpolants that occur in this paper. 4) sum of two projectors and Q is P®QEP+Q-PQ. 5) (P. )F = F Example. P on can be verified by direct sub implies that 3T, 1 < i,j < 3, i φ j.
F1~1(0,0)), from interpolants and this idea is presented in the paper by Gregory in these Proceedings. 16) does not hold, then (P Φ P«) F can be "corrected" so as to remove the incompatibility. ) F also interpolates to dT - E.. 1), which is a special case of the cubic case. Interpolation to F along the side E. implies interpolation to the (tangential) derivative of F along E and so the only problem is interpolation to the normal derivative of F along E . This is cor rected by the addition of a suitable rational function, as the following theorem states.
Computer Aided Geometric Design by R. Barnhill