| A function is called implicit for a given
set of points if it implicitly defines a set of points by
virtue of being equal to some particular value (usually zero)
on these points and nowhere else. Such implicit functions
may be obtained from any logical geometric construction.
Constructed implicit functions may behave
exactly as a distance to the boundary. In this case we call
them normal functions. However distance functions may be difficult
to construct and they may not be differentiable. In contrast,
normalized functions are smooth functions that approximate
distance functions in the neighboorhood of the boundary and
can be constructed automatically from most geometric representations
using theory of R-functions.
Normalized functions can be constructed from
Constructive Solid Geometry (CSG) models (Figure
1), boundary representations (Figures 2,
3), heterogeneous in dimension cell complexes
(Figure 4). Figure 5
shows parametric curve and isolines of the corresponding normalized
function. One normalized function may be used for modification
of another normalized function. For example, Figure
6(a) shows the zero set of the normalized function describing
surface of a pawn and Figure 6(b) shows zero set of the normalized
function defining the letters "RFM". Figure
6(c) shows zero set of the modified normalized function
which maps the letters "RFM" onto the surface of the pawn.
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(a)
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(b)
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Figure 1:(a) Domain described as Constructive
Solid Geometry model; (b) corresponding normalized function
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(a)
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(b)
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Figure 2: (a) Polygon; (b) function whose
zero set defines the boundary of the polygon
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(a)
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(b)
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Figure 3: A normalized function representing
boundary of the state of Wisconsin (20,748 edges)
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(a)
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(b)
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Figure 4: (a) Heterogeneous union of point
sets; (b) function implicitly describing the set in Figure
3(a)
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(a)
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(b)
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Figure 5: (a) A parametric curve and (b)
isolines of the corresponding normalized function
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(a)
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(b)
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(c)
Figure 6: (a) A normalized function describing
the surface of pawn; (b) a normalized function defining
letters "RFM"; (c) modification of normalized function shown
in Figure 6(a) by normalized function shown in Figure 6(b)
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| [1]
V. Shapiro and I. Tsukanov, Implicit Functions with Guaranteed
Differential Properties, In Proceedings of the Fifth ACM Symposium
on Solid Modeling and Applications, June 1999, Ann Arbor,
MI
[2]
V. Shapiro and I. Tsukanov, Meshfree Simulation of Deforming
Domains, Computer-Aided Design , Vol. 31, No. 7, 1999, pp.
459-471
[3]
V. L.Rvachev, T.I.Sheiko, V.Shapiro and I.Tsukanov, On Completeness
of RFM Solution Structures, Computational Mechanics, special
issue on meshfree methods, Vol. 25, 2000, pp. 305-316
[4]
V. L. Rvachev, T.I. Sheiko, V. Shapiro, I. Tsukanov, Transfinite
Interpolation Over Implicitly Defined Sets, Computer Aided
Geometric Design, Vol. 18, No. 4, 2001, pp.195-220 |
