Representing Cuts in Elastic Surfaces

A. Goshtasby


Abstract-The ability to render realistic elastic objects that contain cuts requires the ability to not only represent free-form surfaces of desired smoothness appearances but also to represent cuts in such surfaces. In this paper, elastic surfaces represented by rational Gaussian surfaces will be discussed and a method to render cuts in these surfaces will be introduced. Examples of surfaces of different elasticities and cuts of different sizes will be presented also.

Fig. 1. RaG surfaces with control points and nodes shown in Table 1. The weights at all the points are equal to 1. (a) -- (d) RaG surfaces with smoothnesses equal to 0.15, 0.2, 0.3, and 0.4, respectively. The small spheres show the control points. Top views of the surfaces are shown in the bottom row and the side views of the surfaces are shown in the top row.

Fig. 2. Introducing cuts in a parametric surface. (a) The uv space of a parametric surface before a cut. a, b, and c are nodes of the surface. (b) Defining a cut in the parameter space along u=0.5 from v=0 to v=0.5. (c) Partitioning of the parameter space into regions as a result of the cut. Surface points with parameters in region 3 are determined using control points associated with nodes a, b, and c, surface points with parameters in region 1 are determined using control points associated with nodes a and c, and surface points with parameters in region 2 are determined using control points associated with nodes b and c.

Fig. 3. Scattered 3-D points obtained by a range scanner while scanning the leg of a person around the knee area.

Fig. 4. (a) -- (d) RaG surfaces with smoothnesses equal to 0.05, 0.075, 0.1, and 0.2, respectively, fitting the Franke data set.

Fig. 5. (a) -- (d) RaG surfaces with smoothnesses equal to 0.02, 0.03, 0.05, and 0.075, respectively, fitting the leg data set.

Fig. 6. RaG surfaces of smoothness 0.1 fitting the Franke data set after being cut. (a) The RaG surface before the cut. RaG surfaces in (b) -- (l) show cuts along v=0.5 in the uv space. Cuts in these figures are from u=0.0 to u= 0.05, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, and 1.0.

Fig. 7. (a) -- (d) RaG surfaces with smoothnesses equal to 0.05, 0.075, 0.1, and 0.2, respectively, showing a cut from (u=0.0, v=0.5) to (u=0.5,v=0.5).

Fig. 8. Examples of cuts inside a RaG surface. Cuts in (a) -- (f) are at v=0.5 with u in the range 0.49 -- 0.51, 0.475 -- 0.525, 0.45 -- 0.55, 0.35 -- 0.65, 0.25 -- 0.75, and 0.15 -- 0.85, respectively.

Fig. 9. (a) -- (d) An interior cut in RaG surfaces with smoothnesses equal to 0.05, 0.075, 0.1, and 0.2, respectively. The cuts are from (u=0.45, v=0.5) to (u=0.45, v=0.5).

Fig. 10. Cuts in RaG surfaces of smoothness 0.075 fitting the leg data set. Cuts in (a) -- (d) are along u=0.5 with v in the range 0.49 -- 0.51, 0.475 -- 0.525, 0.45 -- 0.55, and 0.4 -- 0.5, respectively.

Fig. 11. (a) -- (d) RaG surfaces of smoothnesses equal to 0.02, 0.03, 0.05, and 0.075, respectively, fitting the Franke data set with a cut from (u=0.5, v=0.475) to (u=0.5, v=0.525).


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For more information contact A. Goshtasby (agoshtas@cs.wright.edu).

Last modified: 3/13/98.