4.2 Maintaining Discontinuities 4 Extensions 4.1 Periodic Closed Curves X I =0 3 Manipulation of Multiresolution Decompositions 2 Least Squares Multiresolution Decomposition Multiresolution Control for Nonuniform B-spline Curve Editing

(1992). 12 approach should be combined with precise designing tools. Adding the ability to merge curves using Boolean operations will allow one to merge a circle with the tip of the wing, making it feasible to precisely design the cross-section of the missile. The techniques presented herein with the aid of the NUBS representation can be easily enhanced to support these features. The method presented here is incapable of providing a continuous resolution control. The use of linear interpolation between two adjacent levels alleviates this somewhat, but does not solve it. A complete continuous control would require introducing new knot sequences into the modied curve, knot sequences that would change dynamically as the user employs the continuous resolution control. This issue deserves separate investigation. The development of an ecient m ultiresolution decomposition for rational NURBs curves is an intriguing question, considering the complexity that will be introduced into the algebraic summation, so easily computable for polynomials. Providing such a decomposition will make m ultiresolution methods applicable to the vast majority of curve representations used by contemporary solid modeling systems. References [1] Celniker, G., and Gossard, D. Deformable curve and surface nite-elements for free-form shape design. 11 Figure 5: Three curves all derived from the same airplane cross-section, freely editing the fuselage while not crossing the C 1 discontinuity to the wings. curve to the original curve directly. The multiresolution decomposition need not be computed explicitly and only the subspaces of the decomposition, induced by the decimated knot sequences , need be dened. Such an approach does not allow the display of the multiresolution decomposition of the edited curve and, as such, its usefulness should be investigated. 5 The Videotape The VHS/NTSC videotape accompanying this paper [5] demonstrates the methods proposed here, which w e implemented in an interactive X/Motif-based curve editing system. The videotape shows multiresolution manipulation of four curves, three of which appear in Figures 1 (star shaped curve), 2 (signature curve), and 3 (cross-section of an airplane). For the star shaped curve and the signature curve, we show the dierent l o w-resolution curves obtained on a continuous resolution scale, and the eect obtained by editing at various resolutions. In the airplane curve a s w ell as the fourth curve (a cross-section of a face), we also show adaptive renement and local control capabilities. 6 Conclusion We h a v e present e d a s c …