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The term transfinite
interpolation has been often used to describe the problem of constructing
a surface that passes through a given collection of curves, i.e. the surface
must interpolate infinitely many points. In a more general setting, the
interpolation problem requires constructing a single function f(x)
that takes on prescribed values and/or derivatives on some collection of
point sets. In this sense, transfinite interpolation is a special type
of a boundary value problem. The sets of points may contain isolated points,
bounded or unbounded curves, as well as surfaces and regions of arbitrary
topology.
The inverse
distance weighting method [1] has been used to interpolate
functions over a discrete set of scattered points. In his 1967 book [2],
Rvachev observed that similar weighting functions may be constructed for
boundaries of more general sets using normalized functions.
Figure
1 shows two boundaries which are described by normalized functions
w1 and w2 (Figures
2(a), 2(b)). We use these functions for interpolation of the prescribed
functional values given on the boundaries. Figure 2(c)
shows a continuous function interpolating functions taking on the same
values at point A and B (Figure 1). If the prescribed
functions at common points of the boundary's pieces have different values
the interpolating function has discontinuities at those points (Figure
2(d)).
Our technique
is particularly appealing for transfinite interpolation problems because
it inherits the main advantage of the inverse distance weighting method:
it does not place any restriction on incidence, regularity, or the topology
of the sets being interpolated. For example, function shown in Figure
3(b) interpolates given functions and normal derivatives over heterogeneous
in dimension objects represented in Figure 3(a). Derivative
information may be prescribed in direction different from the normal. For
example, Figure 4 shows a function that interpolates
the values of derivatives prescribed not in normal directions to the geometrical
objects.
Figure 1: Functions w1
and w2 define the distance to
the boundaries
Figure 2: (a) The function implicitly defining the boundary
w1 (Figure 1); (b) the function
implicitly defining the boundary w2 (Figure 1);
(c) the continuous interpolating function; (d) the interpolating function
with discontinuities at the points A and B (Figure 1)
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(a)
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(b)
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Figure 3: (a) Heterogeneous in dimension regions and (b)
function interpolating given values of function and normal derivatives
Figure 4: Function interpolating given values and directional
derivatives over heterogeneous in dimension objects shown in Figure 3(a)
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