*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.