pavel chalmoviansky - Academia.edu (original) (raw)

Papers by pavel chalmoviansky

Research paper thumbnail of Basis of quartic splines over triangulation

The modeling of complex shapes usually requires a well-based space of splines. The aim of this wo... more The modeling of complex shapes usually requires a well-based space of splines. The aim of this work is to give the construction method of such spline space basis over the chosen class of triangulations. This basis has several useful properties — local minimal support, low degree of polynomials. We also present several problems, that arise in lower-degree polynomials.

Research paper thumbnail of Fairness Criteria for Algebraic Curves

Geometric Modelling, 2004

We develop methods for the variational design of algebraic curves. Our approach is based on truly... more We develop methods for the variational design of algebraic curves. Our approach is based on truly geometric fairness criteria, such as the elastic bending energy. In addition, we take certain feasibility criteria for the algebraic curve segment into account. We describe a computational technique for the variational design of algebraic curves, using an SQP (sequential quadratic programming)-type method for constrained optimization. As demonstrated in this paper, the powerful techniques of variational design can be used not only for parametric representations, but also for curves in implicit form.

Research paper thumbnail of Visualisation of complex functions on Riemann sphere

The Visual Computer

The notion of a multi-valued function is frequent in complex analysis and related fields. A graph... more The notion of a multi-valued function is frequent in complex analysis and related fields. A graph of such a function helps to inspect the function, however, the methods working with single-valued functions can not be applied directly. To visualize such a type of function, its Riemann surface is often used as a domain of the function. On such a surface, a multi-valued function behaves like a single-valued function. In our paper, we give a quick overview of the proposed method of visualization of a single-valued complex function over its Riemann sphere. Then, we pass to the adaptation of this method on the visualization of a multi-valued complex function. Our method uses absolute value and argument of the function to create the graph in 3D space over the Riemann sphere. Such a graph provides an overview of the function behavior over its whole domain, the amount and the position of its branch points, as well as poles and zeros and their multiplicity. We have also created an algorithm f...

Research paper thumbnail of Basis of Quartic Splines over Triangulation

Journal for Geometry and Graphics

The modeling of complex shapes usually requires a well-based space of splines. The aim of this wo... more The modeling of complex shapes usually requires a well-based space of splines. The aim of this work is to give the construction method of such spline space basis over the chosen class of triangulations. This basis has several useful properties | local minimal support, low degree of polynomials. We also present

Research paper thumbnail of A predictor-corrector-type technique for the approximate parameterization of intersection curves

Applicable Algebra in Engineering Communication and Computing

We describe a method to approximate a segment of the inter- section curve of two implicitly dened... more We describe a method to approximate a segment of the inter- section curve of two implicitly dened surfaces by a rational parametric curve. Starting from an initial solution, the method applies predictor and corrector steps in order to obtain the result. Based on a preconditioning of the two given surfaces, the corrector step is formulated as an optimization problem, where the objective function ap- proximates the integral of the squared Euclidean distance of the curve to the intersection curve. An SQP-type method is used to solve the optimiza- tion problem numerically. Two dieren t predictor steps, which are based on simple extrapolation and on a dieren tial equation, are formulated. Error bounds are needed in order to certify the accuracy of the result. In the case of the intersection of two algebraic surfaces, we show how to bound the Hausdor distance between the intersection curve (an algebraic space curve) and its rational approximation.

Research paper thumbnail of A non-linear circle-preserving subdivision scheme

We describe a new method for constructing a sequence of refined polygons, which starts with a seq... more We describe a new method for constructing a sequence of refined polygons, which starts with a sequence of points and associated normals. The newly generated points are sampled from circles which approximate adjacent points and the corresponding normals. By iterating the refinement procedure, we get a limit curve interpolating the data. We show that the limit curve is G 1 , and that it reproduces circles. The method is invariant with respect to group of Euclidean similarities (including rigid transformations and scaling). We also discuss an experimental setup for a G 2 construction and various possible extensions of the method.

Research paper thumbnail of A predictor–corrector-type technique for the approximate parameterization of intersection curves

We describe a method to approximate a segment of the intersection curve of two implicitly defined... more We describe a method to approximate a segment of the intersection curve of two implicitly defined surfaces by a rational parametric curve. Starting from an initial solution, the method applies predictor and corrector steps in order to obtain the result. Based on a preconditioning of the two given surfaces, the corrector step is formulated as an optimization problem, where the objective function approximates the integral of the squared Euclidean distance of the curve to the intersection curve. An SQP-type method is used to solve the optimization problem numerically. Two different predictor steps, which are based on simple extrapolation and on a differential equation, are formulated. Error bounds are needed in order to certify the accuracy of the result. In the case of the intersection of two algebraic surfaces, we show how to bound the Hausdorff distance between the intersection curve (an algebraic space curve) and its rational approximation.

Research paper thumbnail of Approximate parameterization by planar rational curves

We describe a method for approximate parameterization of a planar algebraic curve by a rational B... more We describe a method for approximate parameterization of a planar algebraic curve by a rational Bézier (spline) curve. After briefly discussing exact methods for parameterization and methods for rational interpolation, we describe a new technique for rational parameterization. Our approach is based on the minimization of a suitable nonlinear objective function, which takes both the distance from the curve and the positivity of the weight function (i.e., the numerator of the rational parametric representation) into account. The solution is computed by using an SQP-type optimization technique. In addition, we use a region-growing-type approach in order to obtain a good initial solution, which is crucial for the convergence of the nonlinear optimization procedure.

Research paper thumbnail of http://link.springer.com/content/pdf/10.1007%2Fs00371-014-0928-3.pdf

The Visual Computer, 2014

Research paper thumbnail of A Sequential Coupling of Optimal Topology and Multilevel Shape Design Applied to 2Dimensional Nonlinear Magnetostatics

In this paper, a sequential coupling of 2-dimensional optimal topology and shape design is propos... more In this paper, a sequential coupling of 2-dimensional optimal topology and shape design is proposed so that a coarsely discretized and optimized topology is the initial guess for the following shape optimization. In be- tween, we approximate the optimized topology by piecewise Bezier shapes via least square fitting. For the topology optimization, we use the steep- est descent method. The

Research paper thumbnail of Approximate algebraic methods for curves and surfaces and their applications

Proceedings of the 21st spring conference on Computer graphics - SCCG '05, 2005

We report on approximate techniques for conversion between the implicit and the parametric repres... more We report on approximate techniques for conversion between the implicit and the parametric representation of curves and surfaces, i.e., implicitization and parameterization. It is shown that these techniques are able to handle general free-form surfaces, and they can therefore be used to exploit the duality of implicit and parametric representations. In addition, we discuss several applications of these techniques, such as detection of self-intersections, raytracing, footpoint computation and parameterization of scattered data for parametric curve or surface fitting.

Research paper thumbnail of Visualisation of complex functions on Riemann sphere

The Visual Computer, 2014

ABSTRACT The notion of a multi-valued function is frequent in complex analysis and related fields... more ABSTRACT The notion of a multi-valued function is frequent in complex analysis and related fields. A graph of such a function helps to inspect the function, however, the methods working with single-valued functions can not be applied directly. To visualize such a type of function, its Riemann surface is often used as a domain of the function. On such a surface, a multi-valued function behaves like a single-valued function. In our paper, we give a quick overview of the proposed method of visualization of a single-valued complex function over its Riemann sphere. Then, we pass to the adaptation of this method on the visualization of a multi-valued complex function. Our method uses absolute value and argument of the function to create the graph in 3D space over the Riemann sphere. Such a graph provides an overview of the function behavior over its whole domain, the amount and the position of its branch points, as well as poles and zeros and their multiplicity. We have also created an algorithm for adaptive grid which provides higher density of vertices in areas with higher curvature of the graph. The algorithm eliminates the alias in places where the branches are joined together.

Research paper thumbnail of Basic facts on hyperbolic geometry and its applications

Research paper thumbnail of Desingularization of ADE Singularities via Deformation

Spring Conference on Computer Graphics - SCCG '13, 2013

ABSTRACT The topology and structure of the ADE singularities in terms of their topological invari... more ABSTRACT The topology and structure of the ADE singularities in terms of their topological invariants are recalled. A representation of these curves as Riemann surfaces is used to propose a novel technique of visualization of multivalued complex functions. Here, not only the entire domain is displayed, but also the method of domain coloring is extended via utilizing a specific height function. The method is applied in order to show the structure of singularities and to resolve them. A sequence of 1-parameter deformations is used, each causing Milnor number to drop by one up to regularity. The changes in the internal structure are interpreted and the whole process is visualized via computer animation.

Research paper thumbnail of Approximate parameterization by planar rational curves

We describe a method for approximate parameterization of a planar algebraic curve by a rational B... more We describe a method for approximate parameterization of a planar algebraic curve by a rational Bézier (spline) curve. After briefly discussing exact methods for parameterization and methods for rational interpolation, we describe a new technique for rational parameterization. Our approach is based on the minimization of a suitable--nonlinear objective function, which takes both the distance from the curve and the positivity of the weight function (i.e., the numerator of the rational parametric representation) into account. The solution is computed by using an SQP-type optimization technique. In addition, we use a region--growing--type approach in order to obtain a good initial solution, which is crucial for the convergence of the nonlinear optimization procedure.

Research paper thumbnail of A graph-spectral method for surface height recovery

Pattern Recognition, 2005

This paper is concerned with computing graph edit distance. One of the criticisms that can be lev... more This paper is concerned with computing graph edit distance. One of the criticisms that can be leveled at existing methods for computing graph edit distance is that they lack some of the formality and rigor of the computation of string edit distance. Hence, our aim is to convert graphs to string sequences so that string matching techniques can be used. To do this, we use a graph spectral seriation method to convert the adjacency matrix into a string or sequence order. We show how the serial ordering can be established using the leading eigenvector of the graph adjacency matrix. We pose the problem of graph-matching as a maximum a posteriori probability (MAP) alignment of the seriation sequences for pairs of graphs. This treatment leads to an expression in which the edit cost is the negative logarithm of the a posteriori sequence alignment probability. We compute the edit distance by finding the sequence of string edit operations which minimizes the cost of the path traversing the edit lattice. The edit costs are determined by the components of the leading eigenvectors of the adjacency matrix and by the edge densities of the graphs being matched. We demonstrate the utility of the edit distance on a number of graph clustering problems.

Research paper thumbnail of Basis of Quartic Splines over Triangulation

The modeling of complex shapes usually requires a well-based space of splines. The aim of this wo... more The modeling of complex shapes usually requires a well-based space of splines. The aim of this work is to give the construction method of such spline space basis over the chosen class of triangulations. This basis has several useful properties | local minimal support, low degree of polynomials. We also present

Research paper thumbnail of Fairness Criteria for Algebraic Curves

Computing, 2004

We develop methods for the variational design of algebraic curves. Our approach is based on truly... more We develop methods for the variational design of algebraic curves. Our approach is based on truly geometric fairness criteria, such as the elastic bending energy. In addition, we take certain feasibility criteria for the algebraic curve segment into account. We describe a computational technique for the variational design of algebraic curves, using an SQP (sequential quadratic programming)-type method for constrained optimization. As demonstrated in this paper, the powerful techniques of variational design can be used not only for parametric representations, but also for curves in implicit form.

Research paper thumbnail of A predictor–corrector-type technique for the approximate parameterization of intersection curves

Applicable Algebra in Engineering, Communication and Computing, 2007

We describe a method to approximate a segment of the intersection curve of two implicitly defined... more We describe a method to approximate a segment of the intersection curve of two implicitly defined surfaces by a rational parametric curve. Starting from an initial solution, the method applies predictor and corrector steps in order to obtain the result. Based on a preconditioning of the two given surfaces, the corrector step is formulated as an optimization problem, where the objective function approximates the integral of the squared Euclidean distance of the curve to the intersection curve. An SQP-type method is used to solve the optimization problem numerically. Two different predictor steps, which are based on simple extrapolation and on a differential equation, are formulated. Error bounds are needed in order to certify the accuracy of the result. In the case of the intersection of two algebraic surfaces, we show how to bound the Hausdorff distance between the intersection curve (an algebraic space curve) and its rational approximation.

Research paper thumbnail of A non-linear circle-preserving subdivision scheme

Advances in Computational Mathematics, 2007

We describe a new method for constructing a sequence of refined polygons, which starts with a seq... more We describe a new method for constructing a sequence of refined polygons, which starts with a sequence of points and associated normals. The newly generated points are sampled from circles which approximate adjacent points and the corresponding normals. By iterating the refinement procedure, we get a limit curve interpolating the data. We show that the limit curve is G 1 , and that it reproduces circles. The method is invariant with respect to group of Euclidean similarities (including rigid transformations and scaling). We also discuss an experimental setup for a G 2 construction and various possible extensions of the method.

Research paper thumbnail of Basis of quartic splines over triangulation

The modeling of complex shapes usually requires a well-based space of splines. The aim of this wo... more The modeling of complex shapes usually requires a well-based space of splines. The aim of this work is to give the construction method of such spline space basis over the chosen class of triangulations. This basis has several useful properties — local minimal support, low degree of polynomials. We also present several problems, that arise in lower-degree polynomials.

Research paper thumbnail of Fairness Criteria for Algebraic Curves

Geometric Modelling, 2004

We develop methods for the variational design of algebraic curves. Our approach is based on truly... more We develop methods for the variational design of algebraic curves. Our approach is based on truly geometric fairness criteria, such as the elastic bending energy. In addition, we take certain feasibility criteria for the algebraic curve segment into account. We describe a computational technique for the variational design of algebraic curves, using an SQP (sequential quadratic programming)-type method for constrained optimization. As demonstrated in this paper, the powerful techniques of variational design can be used not only for parametric representations, but also for curves in implicit form.

Research paper thumbnail of Visualisation of complex functions on Riemann sphere

The Visual Computer

The notion of a multi-valued function is frequent in complex analysis and related fields. A graph... more The notion of a multi-valued function is frequent in complex analysis and related fields. A graph of such a function helps to inspect the function, however, the methods working with single-valued functions can not be applied directly. To visualize such a type of function, its Riemann surface is often used as a domain of the function. On such a surface, a multi-valued function behaves like a single-valued function. In our paper, we give a quick overview of the proposed method of visualization of a single-valued complex function over its Riemann sphere. Then, we pass to the adaptation of this method on the visualization of a multi-valued complex function. Our method uses absolute value and argument of the function to create the graph in 3D space over the Riemann sphere. Such a graph provides an overview of the function behavior over its whole domain, the amount and the position of its branch points, as well as poles and zeros and their multiplicity. We have also created an algorithm f...

Research paper thumbnail of Basis of Quartic Splines over Triangulation

Journal for Geometry and Graphics

The modeling of complex shapes usually requires a well-based space of splines. The aim of this wo... more The modeling of complex shapes usually requires a well-based space of splines. The aim of this work is to give the construction method of such spline space basis over the chosen class of triangulations. This basis has several useful properties | local minimal support, low degree of polynomials. We also present

Research paper thumbnail of A predictor-corrector-type technique for the approximate parameterization of intersection curves

Applicable Algebra in Engineering Communication and Computing

We describe a method to approximate a segment of the inter- section curve of two implicitly dened... more We describe a method to approximate a segment of the inter- section curve of two implicitly dened surfaces by a rational parametric curve. Starting from an initial solution, the method applies predictor and corrector steps in order to obtain the result. Based on a preconditioning of the two given surfaces, the corrector step is formulated as an optimization problem, where the objective function ap- proximates the integral of the squared Euclidean distance of the curve to the intersection curve. An SQP-type method is used to solve the optimiza- tion problem numerically. Two dieren t predictor steps, which are based on simple extrapolation and on a dieren tial equation, are formulated. Error bounds are needed in order to certify the accuracy of the result. In the case of the intersection of two algebraic surfaces, we show how to bound the Hausdor distance between the intersection curve (an algebraic space curve) and its rational approximation.

Research paper thumbnail of A non-linear circle-preserving subdivision scheme

We describe a new method for constructing a sequence of refined polygons, which starts with a seq... more We describe a new method for constructing a sequence of refined polygons, which starts with a sequence of points and associated normals. The newly generated points are sampled from circles which approximate adjacent points and the corresponding normals. By iterating the refinement procedure, we get a limit curve interpolating the data. We show that the limit curve is G 1 , and that it reproduces circles. The method is invariant with respect to group of Euclidean similarities (including rigid transformations and scaling). We also discuss an experimental setup for a G 2 construction and various possible extensions of the method.

Research paper thumbnail of A predictor–corrector-type technique for the approximate parameterization of intersection curves

We describe a method to approximate a segment of the intersection curve of two implicitly defined... more We describe a method to approximate a segment of the intersection curve of two implicitly defined surfaces by a rational parametric curve. Starting from an initial solution, the method applies predictor and corrector steps in order to obtain the result. Based on a preconditioning of the two given surfaces, the corrector step is formulated as an optimization problem, where the objective function approximates the integral of the squared Euclidean distance of the curve to the intersection curve. An SQP-type method is used to solve the optimization problem numerically. Two different predictor steps, which are based on simple extrapolation and on a differential equation, are formulated. Error bounds are needed in order to certify the accuracy of the result. In the case of the intersection of two algebraic surfaces, we show how to bound the Hausdorff distance between the intersection curve (an algebraic space curve) and its rational approximation.

Research paper thumbnail of Approximate parameterization by planar rational curves

We describe a method for approximate parameterization of a planar algebraic curve by a rational B... more We describe a method for approximate parameterization of a planar algebraic curve by a rational Bézier (spline) curve. After briefly discussing exact methods for parameterization and methods for rational interpolation, we describe a new technique for rational parameterization. Our approach is based on the minimization of a suitable nonlinear objective function, which takes both the distance from the curve and the positivity of the weight function (i.e., the numerator of the rational parametric representation) into account. The solution is computed by using an SQP-type optimization technique. In addition, we use a region-growing-type approach in order to obtain a good initial solution, which is crucial for the convergence of the nonlinear optimization procedure.

Research paper thumbnail of http://link.springer.com/content/pdf/10.1007%2Fs00371-014-0928-3.pdf

The Visual Computer, 2014

Research paper thumbnail of A Sequential Coupling of Optimal Topology and Multilevel Shape Design Applied to 2Dimensional Nonlinear Magnetostatics

In this paper, a sequential coupling of 2-dimensional optimal topology and shape design is propos... more In this paper, a sequential coupling of 2-dimensional optimal topology and shape design is proposed so that a coarsely discretized and optimized topology is the initial guess for the following shape optimization. In be- tween, we approximate the optimized topology by piecewise Bezier shapes via least square fitting. For the topology optimization, we use the steep- est descent method. The

Research paper thumbnail of Approximate algebraic methods for curves and surfaces and their applications

Proceedings of the 21st spring conference on Computer graphics - SCCG '05, 2005

We report on approximate techniques for conversion between the implicit and the parametric repres... more We report on approximate techniques for conversion between the implicit and the parametric representation of curves and surfaces, i.e., implicitization and parameterization. It is shown that these techniques are able to handle general free-form surfaces, and they can therefore be used to exploit the duality of implicit and parametric representations. In addition, we discuss several applications of these techniques, such as detection of self-intersections, raytracing, footpoint computation and parameterization of scattered data for parametric curve or surface fitting.

Research paper thumbnail of Visualisation of complex functions on Riemann sphere

The Visual Computer, 2014

ABSTRACT The notion of a multi-valued function is frequent in complex analysis and related fields... more ABSTRACT The notion of a multi-valued function is frequent in complex analysis and related fields. A graph of such a function helps to inspect the function, however, the methods working with single-valued functions can not be applied directly. To visualize such a type of function, its Riemann surface is often used as a domain of the function. On such a surface, a multi-valued function behaves like a single-valued function. In our paper, we give a quick overview of the proposed method of visualization of a single-valued complex function over its Riemann sphere. Then, we pass to the adaptation of this method on the visualization of a multi-valued complex function. Our method uses absolute value and argument of the function to create the graph in 3D space over the Riemann sphere. Such a graph provides an overview of the function behavior over its whole domain, the amount and the position of its branch points, as well as poles and zeros and their multiplicity. We have also created an algorithm for adaptive grid which provides higher density of vertices in areas with higher curvature of the graph. The algorithm eliminates the alias in places where the branches are joined together.

Research paper thumbnail of Basic facts on hyperbolic geometry and its applications

Research paper thumbnail of Desingularization of ADE Singularities via Deformation

Spring Conference on Computer Graphics - SCCG '13, 2013

ABSTRACT The topology and structure of the ADE singularities in terms of their topological invari... more ABSTRACT The topology and structure of the ADE singularities in terms of their topological invariants are recalled. A representation of these curves as Riemann surfaces is used to propose a novel technique of visualization of multivalued complex functions. Here, not only the entire domain is displayed, but also the method of domain coloring is extended via utilizing a specific height function. The method is applied in order to show the structure of singularities and to resolve them. A sequence of 1-parameter deformations is used, each causing Milnor number to drop by one up to regularity. The changes in the internal structure are interpreted and the whole process is visualized via computer animation.

Research paper thumbnail of Approximate parameterization by planar rational curves

We describe a method for approximate parameterization of a planar algebraic curve by a rational B... more We describe a method for approximate parameterization of a planar algebraic curve by a rational Bézier (spline) curve. After briefly discussing exact methods for parameterization and methods for rational interpolation, we describe a new technique for rational parameterization. Our approach is based on the minimization of a suitable--nonlinear objective function, which takes both the distance from the curve and the positivity of the weight function (i.e., the numerator of the rational parametric representation) into account. The solution is computed by using an SQP-type optimization technique. In addition, we use a region--growing--type approach in order to obtain a good initial solution, which is crucial for the convergence of the nonlinear optimization procedure.

Research paper thumbnail of A graph-spectral method for surface height recovery

Pattern Recognition, 2005

This paper is concerned with computing graph edit distance. One of the criticisms that can be lev... more This paper is concerned with computing graph edit distance. One of the criticisms that can be leveled at existing methods for computing graph edit distance is that they lack some of the formality and rigor of the computation of string edit distance. Hence, our aim is to convert graphs to string sequences so that string matching techniques can be used. To do this, we use a graph spectral seriation method to convert the adjacency matrix into a string or sequence order. We show how the serial ordering can be established using the leading eigenvector of the graph adjacency matrix. We pose the problem of graph-matching as a maximum a posteriori probability (MAP) alignment of the seriation sequences for pairs of graphs. This treatment leads to an expression in which the edit cost is the negative logarithm of the a posteriori sequence alignment probability. We compute the edit distance by finding the sequence of string edit operations which minimizes the cost of the path traversing the edit lattice. The edit costs are determined by the components of the leading eigenvectors of the adjacency matrix and by the edge densities of the graphs being matched. We demonstrate the utility of the edit distance on a number of graph clustering problems.

Research paper thumbnail of Basis of Quartic Splines over Triangulation

The modeling of complex shapes usually requires a well-based space of splines. The aim of this wo... more The modeling of complex shapes usually requires a well-based space of splines. The aim of this work is to give the construction method of such spline space basis over the chosen class of triangulations. This basis has several useful properties | local minimal support, low degree of polynomials. We also present

Research paper thumbnail of Fairness Criteria for Algebraic Curves

Computing, 2004

We develop methods for the variational design of algebraic curves. Our approach is based on truly... more We develop methods for the variational design of algebraic curves. Our approach is based on truly geometric fairness criteria, such as the elastic bending energy. In addition, we take certain feasibility criteria for the algebraic curve segment into account. We describe a computational technique for the variational design of algebraic curves, using an SQP (sequential quadratic programming)-type method for constrained optimization. As demonstrated in this paper, the powerful techniques of variational design can be used not only for parametric representations, but also for curves in implicit form.

Research paper thumbnail of A predictor–corrector-type technique for the approximate parameterization of intersection curves

Applicable Algebra in Engineering, Communication and Computing, 2007

We describe a method to approximate a segment of the intersection curve of two implicitly defined... more We describe a method to approximate a segment of the intersection curve of two implicitly defined surfaces by a rational parametric curve. Starting from an initial solution, the method applies predictor and corrector steps in order to obtain the result. Based on a preconditioning of the two given surfaces, the corrector step is formulated as an optimization problem, where the objective function approximates the integral of the squared Euclidean distance of the curve to the intersection curve. An SQP-type method is used to solve the optimization problem numerically. Two different predictor steps, which are based on simple extrapolation and on a differential equation, are formulated. Error bounds are needed in order to certify the accuracy of the result. In the case of the intersection of two algebraic surfaces, we show how to bound the Hausdorff distance between the intersection curve (an algebraic space curve) and its rational approximation.

Research paper thumbnail of A non-linear circle-preserving subdivision scheme

Advances in Computational Mathematics, 2007

We describe a new method for constructing a sequence of refined polygons, which starts with a seq... more We describe a new method for constructing a sequence of refined polygons, which starts with a sequence of points and associated normals. The newly generated points are sampled from circles which approximate adjacent points and the corresponding normals. By iterating the refinement procedure, we get a limit curve interpolating the data. We show that the limit curve is G 1 , and that it reproduces circles. The method is invariant with respect to group of Euclidean similarities (including rigid transformations and scaling). We also discuss an experimental setup for a G 2 construction and various possible extensions of the method.