Selçuk Journal
of
Applied Mathematics
SummerAutumn, 2002
Volume 3
Number 2
Research Center
of
Applied Mathematics

SJAM
SummerAutumn 2002, Volume 3  Number 2

Locally
exact smooth
reconstruction of
lines, circles,
planes, spheres,
cylinders and
cones by
blending successive
circular
interpolants 
Richard Liska^{1},
Mikhail Shashkov^{2} and Blair Swartz^{2} 
^{1 }Faculty of
Nuclear Sciences and Physical Engineering, Czech Technical
University in
Prague,
Břehová 7, 115 19 Prague 1, Czech Republic;
email:
liska@siduri.fjfi.cvut.cz
^{2 }Group T7, Los Alamos National Laboratory, Los Alamos,
NM 87545, USA;
email:
shashkov@lanl.gov
email:
bks@lanl.gov
Received: July 16, 2002^{
} 
Summary
G^{1}smooth
curves and
surfaces are
developed to
span a given
logically cuboid
distribution of
nodes. Given
appropriate data, they
locally reconstruct
the curves
and surfaces
of spherical or
cylindrical
coordinates. Thus,
if a set of nodes
consists of a
contiguous subset of a
tensor product
grid of points
associated with
a (possibly nonuniform)
set of coordinate
values of some
rectangular,
cylindrical, or
spherical
coordinate system;
then the
appropriate
coordinate curves (linear
or circular
segments) and
coordinate surfaces
(segments of planes,
cylinders, spheres
and cones)
that interpolate
the subset
are reconstructed
exactly. The
underlying
construction uses
four successive
nodes to
define a curve
spanning the
middle pair as
follows: One
interpolates each
of the two
successive triples
of nodes with
the segment
of a circle or
straight line
going through
these three
points. Then
one blends
the two
segments
continuously between
the middle
pair of nodes.
The blend is
relatively linear
in terms of arclength
along each
segment. The
union of such
successive curvesections
forms a G^{1}
curve. Wireframes
of such curves
define cell edges.
Similar
intermediate curvilinear
interpolation of
the wires
defines cell
faces, and
their union
defines G^{1} coordinatelike
surfaces. The
surface generated
depends on the
direction one
interpolates the
wires. If
the nodes
are a tensor
product grid
associated with
a sufficiently
smooth reference
coordinate system,
then the
cell edges (and
probably also
the cell
faces) are
thirdorder
accurate. 

Key
words
smooth
reconstruction, line,
circle, plane,
sphere, cylinder,
cone, interpolation

2000 Mathematics
Subject
Classification : 65D15

Article
in PS format (2.35 Mb) 
Article
in ZIP format (492 Kb) 
