It is known that
Bezier curves and surfaces may have multiple
representations by different control polygons. The
polygons may have different number of control
points and may even be disjoint. Up to our
knowledge, Pekerman et al. (2005) were the first
to address the problem of testing two parametric
polynomial curves for coincidence. Their approach
is based on reduction of the input curves into
canonical irreducible form. They claimed that
their approach can be extended for testing tensor
product surfaces but gave no further detail. In
this paper we develop a new technique and provide
a comprehensive solution to the problem of testing
tensor product Bezier surfaces for coincidence. In
(Vlachkova, 2017) an algorithm for testing Bezier
curves was proposed based on subdivision. There a
partial solution to the problem of testing tensor
product Bezier surfaces was presented. Namely, the
case where the irreducible surfaces are of same
degree (n,m), n,m ∈ N, was resolved under certain
additional condition. The other cases where one of
the surfaces is of degree (n,m) and the other is of
degree either (n,n+m), or (n+m), or (n+m,n+m) remained open. We have implemented
our algorithm for testing tensor product Bezier
surfaces for coincidence using Mathematica
package. Experimental results and their analysis
are presented.
Abstract. We present
a new program package for interactive modeling and
visualization of smooth scattered data
interpolation using quartic triangular Bezier
patches (TBP).We implemented an optimized
algorithm (Vlachkova, 2021) based on quartic
smooth interpolation curve networks and splitting.
The algorithm allows to reduce the complexity of
the resulting surface and to improve its
smoothness. We have chosen the open-source data
visualization library Plotly.js as our main
implementation and visualization tool. This choice
ensures the platform independency
of our package and its direct use without
restrictions. The package can be used for
experiments with user's data sets since it works
with the host file system. The latter
allows wide testing, modeling, and editing of the
resulting interpolation surfaces.
We performed a large number of experiments using
data of increasing complexity.
Here we present and comment the results of our
work.
Abstract. We
consider the extremal problem of
interpolation of convex scattered data in ℝ3
by smooth edge convex curve networks with
minimal Lp-norm of the
second derivative for 1 < p ≤
∞. The problem for p = 2 was set
and solved by Andersson et al. (1995).
Vlachkova (2019) extended the results in
(Andersson et al., 1995) and solved the
problem for 1 < p < ∞. The
minimum edge convex Lp-norm
network for 1 < p < ∞ is
obtained from the solution to a system of
nonlinear equations with coeffificients
determined by the d ata. The solution in the
case 1 < p < ∞ is unique
forstrictly convex data. The approach used
in (Vlachkova, 2019) can not be applied to
the corressponding extremal problem for p
= ∞. In this case the solution is not
unique. Here we establish the existence of a
solution to the extremal interpolation
problem for p = ∞. This solution
is a quadratic spline function with at most
one knot on each edge of the underlying
triangulation. We also propose suffificient
conditions for solving the problem for p
= ∞.
Abstract. We present two new program packages
for modeling and visualization of scattered data
interpolation surfaces based on smooth
interpolation quartic curve networks that are
extended to surfaces comprising of quartic
triangular Bezier patches (TBP). Our work and
contributions are in the field of experimental
algorithmics and algorithm engineering. We have
chosen the open-source data visualization
libraries Plotly.js and Three.js as our main
implementation and visualization tools. This
choice ensures the platform independency of our
packages and their direct use without
restrictions. The packages can be used for
experiments with user's data sets since they
work with the host file system. The latter
allows wide testing, modeling, and editing of
the resulting interpolation surfaces. The
packages can be used both for research and
educational purposes. We experimented
extensively with our packages using data of
increasing complexity. The experimental results
are presented and analyzed.
Abstract. We
consider the extremal problem of interpolation
of scattered data in ℝ3 by smooth curve
networks with minimal Lp-norm
of
the second derivative for 1 < p
≤ ∞. The problem for p
= 2 was set and solved by Nielson [1]. Andersson
et al. [2] gave a new proof of Nielson's result
by using a different approach. Vlachkova [3]
extended the results in [2] and solved the
problem for 1 < p
< ∞. The minimum Lp-norm
network for 1 < p
< ∞ is obtained from the solution to a system
of nonlinear equations with coefficients
determined by the data. The solution in the case
1 < p
< ∞ is unique. We denote the corresponding
minimum Lp-norm
network
by Fp. In the
case where p
= ∞ we establish the existence of a solution of
the same type as in the case where 1 < p < ∞. This solution
on each edge of the underlying triangulation is
a quadratic spline function with at most one
knot. We denote this solution by F∞ and prove
that the minimum Lp-norm
networks
Fp converge
to the minimum L∞-norm
network
F∞ as p→∞.
Abstract:
We
consider the extremal problem of interpolation
of convex scattered data in ℝ3
by smooth edge convex curve networks with
minimal 𝐿𝑝-norm
of the second derivative for 1 < 𝑝 ≤ ∞. The problem for
𝑝
= 2 was set and solved by Andersson et al.
(1995). Vlachkova (2019) extended the results in
(Andersson et al., 1995) and solved the problem
for 1 < 𝑝 < ∞. The minimum
edge convex 𝐿𝑝-norm
network for 1 < 𝑝
< ∞ is obtained from the solution to a system
of nonlinear equations with coefficients
determined by the data. The solution in the case
1 < 𝑝
< ∞ is unique for strictly convex data. The
corresponding extremal problem for 𝑝
= ∞ remained open. Here we show that the
extremal interpolation problem for 𝑝
= ∞ always has a solution. We give a
characterization of this solution. We show that
a solution to the problem for 𝑝 = ∞ can be found by
solving a system of nonlinear equations in the
case where it exists.
Abstract: We consider the problem of
scattered data interpolation in R3 using curve
networks extended to smooth interpolation
surfaces. Nielson (1983) proposed a solution
that constructs smooth interpolation curve
network with minimal 𝐿2-norm of the second
derivative. The obtained minimum norm network
(MNN) is cubic. Vlachkova (2020) generalized
Nielson's result to smooth interpolation curve
networks with minimal 𝐿𝑝-norm of the second
derivative for 1 < 𝑝 < ∞. Vlachkova and
Radev (2020) proposed an algorithm that degree
elevates the MNN to quartic curve network and
then extends it to a smooth surface consisting
of quartic triangular Bezier surfaces. Here we
apply this algorithm to the following two curve
networks:
(i) the MNN which is degree elevated to quartic;
(ii) the minimum 𝐿𝑝-norm network for 𝑝 = 3/2
which is slightly modified to quartic.
We evaluate and compare the quality and the
shape of the obtained surfaces with respect to
different criteria. We performed a large number
of experiments using
data of increasing complexity. Here we present
and comment the results of our experiments.
Abstract: We revisit the problem of
interpolation of scattered data in R3 and
propose
a solution based on Nielson's minimum norm
network and triangular Bezier
patches.
We aimed at solving the problem using the least
number of polynomial patches of
the smallest possible degree. We propose an
alternative to the previously known
algorithms, see (Clough and Tocher, 1965) and
(Shirman and Sequin, 1987,1991).
Although conceptually similar, our algorithm
differs from the previous in all its steps.
As a result the complexity of the resulting
surface is reduced and its smoothness is
improved. We present results of our numerical
experiments.
Abstract: We consider the problem of
interpolation of data in R3 and propose a
solution based on Nielson's minimum norm network
and triangular Bezier patches. Our
algorithm uses splitting and constructs
G1-continuous bivariate interpolant consisting
of quartic triangular Bezier
surfaces. The algorithm is computationally
simple and produces visually pleasant smooth
surfaces. We have created a program packages for
implementation, 3D visualization and comparison
of our algorithm and the known Shirman and
Sequin's method which is also based on splitting
and quartic triangular Bezier patches. The
results of our numerical experiments are
presented and analysed.
Abstract: We consider the extremal problem of
interpolation of scattered data in R3 by smooth
curve networks with minimal Lp-norm of the
second derivative for 1<p<∞. The problem
for p=2 was set and solved by Nielson (1983).
Andersson et al. (1995) gave a new proof of
Nielson's result by using a different approach.
Partial results for the problem for 1<p<∞
were announced without proof in (Vlachkova,
1992). Here we present a complete
characterization of the solution for
1<p<∞. Numerical experiments are
visualized and presented to illustrate and
support our results.
Abstract: We
consider the extremal problem of interpolation
of scattered data in R3 by smooth curve networks
with minimal Lp-norm of the second derivative
for 1 < p < ∞. The problem for p = 2 was
set and solved by Nielson [7]. Andersson et al.
[1] gave a new proof of Nielson's result by
using a different approach. It allowed them to
set and solve the constrained extremal problem
of interpolation of convex scattered data in R3
by minimum L2-norm networks that are convex
along the edges of an associated triangulation.
Partial results for the unconstrained and the
constrained problems were announced without
proof in [8]. The unconstrained problem for 1
< p < ∞ was fully solved in [10]. Here we
present complete characterization of the
solution to the constrained problem for 1 < p
< ∞.
Abstract: It is known that Bezier curves and
surfaces may have multiple representations by
different control polygons. The polygons may
have different number of control points and may
even be disjoint. This phenomenon causes
difficulties in variety of applications where it
is important to recognize cases where different
representations define same curve (surface) or
partially coincident curves (surfaces). The
problem of finding whether two arbitrary
parametric polynomial curves are the same has
been addressed in Pekerman et al. (Are two
curves the same? Comput.-Aided Geom. Des. Appl.
2(1-4):85-94, 2005). There the curves are
reduced into canonical irreducible forms using
the monomial basis, then they are compared and
their shared domains, if any, are identified.
Here we present an alternative geometric
algorithm based on subdivision that compares two
input control polygons and reports the
coincidences between the corresponding Bezier
curves if they are present. We generalize the
algorithm for tensor product Bezier surfaces.
The algorithms are implemented and tested using
Mathematica package. The experimental results
are presented.
Abstract: Minimal surfaces with isothermal
parameters admitting Bezier
representation were studied by Cosın and
Monterde. They showed that, up to an affine
transformation, the Enneper surface is the only
bi-cubic isothermal minimal surface. Here we
study bi-quartic isothermal minimal surfaces and
establish the general form of their enerating
functions in the Weierstrass representation
formula. We apply an approach proposed by
Ganchev to compute the normal curvature and show
that, in contrast to the bi-cubic case, there is
a variety of bi-quartic isothermal minimal
surfaces. Based on the Bezier representation we
establish some geometric properties of the
bi-quartic harmonic surfaces. Numerical
experiments are visualized and presented to
illustrate and support our results.
Abstract: We consider the
problem of extremal interpolation of convex
scattered data in R3 and propose a feasible
solution. Using our previous work on edge
convex minimum Lp-norm interpolation curve
networks, 1 < p ≤ ∞, we construct a
bivariate interpolant F with the following
properties:
(i) F is G1-continuous;
(ii) F consists of tensor product Bezier
surfaces (patches) of degree (n, n)
where n ∈ N, n ≥ 4,
is priorly chosen;
(iii) The boundary curves of each patch are
convex;
(iv) Each Bezier
patch satisfies the tetra-harmonic equation
Δ4F = 0.
Hence F is an
extremum to the corresponding energy
functional.
Abstract: We consider the problem of extremal
scattered data interpolation in R3. Using our
previous work on minimum L2-norm interpolation
curve networks, we construct a bivariate
interpolant F with the following properties:
(i) F is G1-continuous,
(ii) F consists of triangular Bezier
surfaces,
(iii) each Bezier
surface satisfies the tetra-harmonic equation
Δ4F = 0.
Hence F is an
extremum to the corresponding energy functional.
We also discuss the case of convex scattered
data in R3.
Abstract: We present a new program package for
interactive implementation and 3D
visualization of three fundamental algorithms
for triangular surface subdivision.
Namely, these are Loop subdivision, Modified
Butterfly subdivision, and Kobbelt
\sqrt{3}-subdivision. Our work and contributions
are in the field of experimental algorithmics
and algorithm engineering. We have chosen Java
applet application and Java 3D as our main
implementation and visualization tools. This
choice ensures platform independency of our
package and its direct use without restrictions.
The applet can be used for experiments with
user's data sets since it works with host file
system. The latter allows its usage for research
and educational purposes. We have implemented
extensive experiments with our package using
meshes of di fferent type and increasing
complexity. We compared the behaviour of the
three algorithms with respect to various
criteria. The experimental results are presented
and analyzed.
Extremal
Scattered Data Interpolation in R3
Using Tensor Product Bezier
Surfaces
Krassimira Vlachkova
Constructive Theory of
Functions,
Sozopol 2013
(K. Ivanov, G. Nikolov and R. Uluchev,
Eds.), pp. 253-264
Prof. Marin Drinov Academic Publishing
House, Sofia, 2014 http://www.math.bas.bg/mathmod/Proceedings_CTF/CTF-2013/Proceedings_CTF-2013.html
Abstract: We consider the problem of extremal
scattered data interpolation in R3. Using our
previous work on minimum Lp-norm interpolation
curve networks, 1 < p < ∞, we
construct a bivariate interpolant F with the
following properties:
(i) F is G1-continuous;
(ii) F consists of tensor product Bezier
surfaces;
(iii) Each Bezier
surface satisfies the tetraharmonic equation Δ4F
= 0.
Hence F minimizes
the corresponding energy functional.
A
Comparison of Surface Subdivision Algorithms
for
Polygonal Meshes
Krassimira Vlachkova, Plamen Terziev
AIP Conference Proceedings 1487,
343-350 (2012) 10.1063/1.4758977
Abstract: We present a new program package for
interactive implementation and 3D visualization
of three fundamental algorithms for surface
subdivision. Namely, these are Doo-Sabin
algorithm, Catmull-Clark algorithm, and
Peters-Reif algorithm. Our work and
contributions are in the field of experimental
algorithmics and algorithm engineering.We have
chosen OpenGL and Qt graphics libraries as our
main implementation and visualization tools. Our
program analyzes the validity of the loaded mesh
and proceeds with valid meshes only.We provide a
user friendly interface so that users can load
their own data sets. The latter allows wide
testing and comparing the results from the
implementation of the three algorithms on
arbitrary polygonal meshes. The program has also
an option for creating new polygonal meshes. We
experimented extensively with our package. We
compared the behaviour of the three algorithms
based on different criteria and using meshes of
increasing complexity. The experimental results
are presented and analyzed.
Abstract: We
consider the problem of interpolating scattered
data in R3 assuming that the data are sampled
from a smooth bivariate function F = F(x, y).
For a fixed triangulation T associated with the
projections of the data onto the plane Oxy we
consider Nielson's minimum norm interpolation
network S defined in [6] and prove an estimate
of the form ||F−S||_{L2(T)} ≤C(T)
||F^{IV}||_{L2(T)}. The dependence of the term
C(T) on the triangulation T is analysed.
Abstract:
We present a new program tool for interactive 3D
visualization of some fundamental algorithms for
representation and manipulation of Bezier
curves. The program tool has an option for
demonstration of one of their most important
applications - in graphic design for creating
letters by means of cubic Bezier curves. We use
Java applet and JOGL as our main visualization
techniques. This choice ensures the platform
independency of the created applet and
contributes to the realistic 3D visualization.
The applet provides basic knowledge on the Bezier
curves and is appropriate for illustrative and
educational purposes. Experimental results are
included.
Abstract:
Given scattered data in R3 and an integer
k>=1 we consider the problem of
finding necessary and sufficient conditions for
k-times smoothness of interpolation curve
networks. Interpolation curve networks are
defined on the edges of a triangulation
associated with the projections of the data onto
a fixed plane and play an essential role in
construction of smooth interpolation surfaces in
R3. In previous work this problem has been
solved for k = 1 by the aid of a geometric
criterion which does not generalize for k >
1. In this paper we apply a di fferent approach
and obtain necessary and sufficient conditions
for k-times smoothness of interpolation curve
networks.
Abstract: Given convex scattered data
in R3 we consider the constrained interpolation
problem of finding a smooth, minimal Lp-norm (1 < p < ∞)
interpolation network that is convex along the
edges of an associated triangulation. In
previous work the problem has been reduced to
the solution of a nonlinear system of equations.
In this paper we formulate and analyse a
Newton-type algorithm for solving the
corresponding type of systems. The correctness
of the application of the proposed method is
proved and its superlinear (in some cases
quadratic) convergence is shown.
Abstract: In this paper a
characterization of the optimal (using the minimum
norm criterion) interpolant, convex along the
edges of a triangulation, usin data at the
vertices is obtained. We thereby generalize
results obtained by Nielson for the unconstrained
case.
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