Freeform Surface Blends K . K . , G

12 Figure 8: Blending using two diierent magnitudes of T 2 i (t) = dC(t) dt n(u(t); v(t)). (a) (b) (c) Figure 9: Spout connection to the Utah teapot rounding using T 2 i (t) = dC(t) dt n(u(t); v(t)). 5 i (t) = (S u (u(t); v(t))r(t) + S v (u(t); v(t))s(t)). method, which is guaranteed to be perpendicular to the rail curves in Euclidean space. Figure 9 shows the original teapot (a), the lleted teapot (b), and the lleted region enhanced (c). 4 Conclusions The symbolic computation method proved to be a powerful tool in creating blend surfaces within machine accuracy and with a geometrically meaningful control. With the ability to symbolically compute and normalize cross tangent curves, the symbolic approach for the construction of blend surfaces can be automated. This approach is unique in that it provides an accurate tangent plane continuity for the entire boundary of the blend surface, as opposed to accurate tangent plane continuity at only a nite set of locations along this domain. Furthermore , once the tangent curves are normalized, results can be interacted with in a geometrically meaningful way. It is our opinion that the symbolic blend surface construction, as a promising way toward a robust and fast computation of blend surfaces for freeform based models, will nd its way into current solid modeling systems. 2 i (t) = dC(t) dt n(u(t); v(t)). Figure 6: Blending using T 4 i (t) = K i (t) ? hKi(t);ni(u(t);v(t))i hni(u(t);v(t));ni(u(t);v(t))i n i (u(t); v(t)). Figure 4: Blending using T 2 i (t) = dC(t) dt n(u(t); v(t)). 3 Examples Figure 4 provides a simple example for T 2 i (t) cross boundary tangent curve selection. Figure 5 is another, more complex, example of the use of T 2 i (t). The user, once selecting the T 2 i (t) method, can control the magnitude of the cross boundary tangent curves, by specifying a scaling curve k(t) as discussed Section 2.3. In Figure 5, the cross boundary tangent magnitude varies along the blend surface, larger at the wide blend surface region and smaller at the narrow blend surface region. By specifying two guiding curves, K i (t), and projecting then onto the tangent planes of the surfaces along the two rail curves, the vector eld curves T 3 i (t) are deened. By deening two independent guiding curves, K i (t), one for each …