Direct least-squares fitting of algebraic surfaces
SIGGRAPH '87 Proceedings of the 14th annual conference on Computer graphics and interactive techniques
On local implicit approximation and its applications
ACM Transactions on Graphics (TOG) - Special issue on computer-aided design
IEEE Transactions on Pattern Analysis and Machine Intelligence
Surface reconstruction from unorganized points
SIGGRAPH '92 Proceedings of the 19th annual conference on Computer graphics and interactive techniques
Higher-order interpolation and least-squares approximation using implicit algebraic surfaces
ACM Transactions on Graphics (TOG)
Implicit Curves and Surfaces in CAGD
IEEE Computer Graphics and Applications - Special issue on computer-aided geometric design
Implicitization using moving curves and surfaces
SIGGRAPH '95 Proceedings of the 22nd annual conference on Computer graphics and interactive techniques
On the validity of implicitization by moving quadrics for rational surfaces with no base points
Journal of Symbolic Computation
Mathematical Methods for Curves and Surfaces
Least-Squares Fitting of Algebraic Spline Curves via Normal Vector Estimation
Proceedings of the 9th IMA Conference on the Mathematics of Surfaces
Approximate parameterization by planar rational curves
Proceedings of the 20th spring conference on Computer graphics
Approximate algebraic methods for curves and surfaces and their applications
Proceedings of the 21st spring conference on Computer graphics
Approximate implicitization of triangular Bézier surfaces
Proceedings of the 26th Spring Conference on Computer Graphics
Approximate rational parameterization of implicitly defined surfaces
IMA'05 Proceedings of the 11th IMA international conference on Mathematics of Surfaces
Approximate µ-bases of rational curves and surfaces
GMP'06 Proceedings of the 4th international conference on Geometric Modeling and Processing
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We discuss methods for fitting implicitly defined (e. g. piecewise algebraic) curves to scattered data, which may contain problematic regions, such as edges, cusps or vertices. As the main idea, we construct a bivariate function, whose zero contour approximates a given set of points, and whose gradient field simultaneously approximates an estimate normal field. The coefficients of the implicit representation are found by solving a system of linear equations. In order to allow for problematic input data, we introduce a criterion for detecting points close to possible singularities. Using this criterion we split the data into segments and develop methods for propagating the orientation of the normals globally. Furthermore we present a simple fallback strategy, that can be used when the process of orentation propagation fails. The method has been shown to work successfully