Surface Intersections for Geometric Modeling

1990 ◽  
Vol 112 (1) ◽  
pp. 100-107 ◽  
Author(s):  
N. M. Patrikalakis ◽  
P. V. Prakash
Keyword(s):  
Geometric Modeling ◽  
Algebraic Curves ◽  
Solution Procedure ◽  
Rectangular Domain ◽  
Singular Points ◽  
Algebraic Surfaces ◽  
Surface Patches ◽  
Adaptive Subdivision ◽  
Rational Polynomial ◽  
Spline Surfaces

Evaluation of planar algebraic curves arises in the context of intersections of algebraic surfaces with piecewise continuous rational polynomial parametric surface patches useful in geometric modeling. We address a method of evaluating these curves of intersection that combines the advantageous features of analytic representation of the governing equation of the algebraic curve in the Bernstein basis within a rectangular domain, adaptive subdivision and polyhedral faceting techniques, and the computation of turning and singular points, to provide the basis for a reliable and efficient solution procedure. Using turning and singular points, the intersection problem can be partitioned into subdomains that can be processed independently and which involve intersection segments that can be traced with faceting methods. This partitioning and the tracing of individual segments is carried out using an adaptive subdivision algorithm for Bezier/B-spline surfaces followed by Newton correction of the approximation. The method has been successfully tested in tracing complex algebraic curves and in solving actual intersection problems with diverse features.

Method for intersecting algebraic surfaces with rational polynomial patches

1990 ◽  
Vol 22 (10) ◽  
pp. 645-654 ◽  
Author(s):  
G.A. Kriezis ◽  
P.V. Prakash ◽  
N.M. Patrikalakis
Keyword(s):  
Algebraic Surfaces ◽  
Rational Polynomial

scholarly journals Hierarchical Genetic Algorithm for B-Spline Surface Approximation of Smooth Explicit Data

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
C. H. Garcia-Capulin ◽  
F. J. Cuevas ◽  
G. Trejo-Caballero ◽  
H. Rostro-Gonzalez
Keyword(s):  
Genetic Algorithm ◽  
Geometric Modeling ◽  
Data Set ◽  
Surface Approximation ◽  
B Spline ◽  
B Splines ◽  
Spline Surface ◽  
Spline Surfaces ◽  
Surface Dimension ◽  
Hierarchical Genetic Algorithm

B-spline surface approximation has been widely used in many applications such as CAD, medical imaging, reverse engineering, and geometric modeling. Given a data set of measures, the surface approximation aims to find a surface that optimally fits the data set. One of the main problems associated with surface approximation by B-splines is the adequate selection of the number and location of the knots, as well as the solution of the system of equations generated by tensor product spline surfaces. In this work, we use a hierarchical genetic algorithm (HGA) to tackle the B-spline surface approximation of smooth explicit data. The proposed approach is based on a novel hierarchical gene structure for the chromosomal representation, which allows us to determine the number and location of the knots for each surface dimension and the B-spline coefficients simultaneously. The method is fully based on genetic algorithms and does not require subjective parameters like smooth factor or knot locations to perform the solution. In order to validate the efficacy of the proposed approach, simulation results from several tests on smooth surfaces and comparison with a successful method have been included.

scholarly journals Singular Points of Curves

2018 ◽  
Vol 25 (6) ◽  
pp. 692-710
Author(s):  
Artem D. Uvarov
Keyword(s):  
Iterative Process ◽  
Geometric Modeling ◽  
Global Minimum ◽  
Iterative Algorithms ◽  
Singular Points ◽  
Intersection Curve ◽  
Software Environment ◽  
Advantages And Disadvantages ◽  
Intersection Curves ◽  
Complex Cases

In this paper, we consider the key problem of geometric modeling, connected with the construction of the intersection curves of surfaces. Methods for constructing the intersection curves in complex cases are found: by touching and passing through singular points of surfaces. In the first part of the paper, the problem of determining the tangent line of two surfaces given in parametric form is considered. Several approaches to the solution of the problem are analyzed. The advantages and disadvantages of these approaches are revealed. The iterative algorithms for finding a point on the line of tangency are described. The second part of the paper is devoted to methods for overcoming the difficulties encountered in solving a problem for singular points of intersection curves, in which a regular iterative process is violated. Depending on the type of problem, the author dwells on two methods. The first of them suggests finding singular points of curves without using iterative methods, which reduces the running time of the algorithm of plotting the intersection curve. The second method, considered in the final part of the article, is a numerical method. In this part, the author introduces a function that achieves a global minimum only at singular points of the intersection curves and solves the problem of minimizing this function. The application of this method is very effective in some particular cases, which impose restrictions on the surfaces and their arrangement. In conclusion, this method is considered in the case when the function has such a relief, that in the neighborhood of the minimum point the level surfaces are strongly elongated ellipsoids. All the images given in this article are the result of the work of algorithms on methods proposed by the author. Images are built in the author’s software environment.

scholarly journals Arithmeticity of the monodromy of some Kodaira fibrations

2019 ◽  
Vol 156 (1) ◽  
pp. 114-157
Author(s):  
Nick Salter ◽  
Bena Tshishiku
Keyword(s):  
Mapping Class Group ◽  
Class Group ◽  
Monodromy Group ◽  
Algebraic Curves ◽  
Algebraic Surfaces ◽  
Mapping Class ◽  
Complex Varieties

A question of Griffiths–Schmid asks when the monodromy group of an algebraic family of complex varieties is arithmetic. We resolve this in the affirmative for a class of algebraic surfaces known as Atiyah–Kodaira manifolds, which have base and fibers equal to complete algebraic curves. Our methods are topological in nature and involve an analysis of the ‘geometric’ monodromy, valued in the mapping class group of the fiber.

Invariant singular points of algebraic curves

1983 ◽  
Vol 34 (6) ◽  
pp. 962-963
Author(s):  
E. I. Shustin
Keyword(s):  
Algebraic Curves ◽  
Singular Points

GEOMETRIC MODELING OF ADAPTIVE ALGEBRAIC CURVES PASSING THROUGH PREDETERMINED POINTS

2021 ◽  
pp. 26-34
Author(s):  
E. V. Konopatskiy ◽  
I. V. Seleznev ◽  
O. A. Chernysheva ◽  
M. V. Lagunova ◽  
A. A. Bezditnyi
Keyword(s):  
Initial Data ◽  
Geometric Modeling ◽  
Algebraic Curves ◽  
Geometric Theory ◽  
Multidimensional Space ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Multidimensional Interpolation ◽  
Geometric Objects ◽  
Parameter Values

In this paper, the geometric theory of multidimensional interpolation was further developed in terms of modeling and using adaptive curves passing through predetermined points. A feature of the proposed approach to modeling curved lines is the ability to adapt to any initial data for high-quality interpolation, which excludes unplanned oscillations, due to the uneven distribution of parameter values, the source of which are the initial data. This is the improvement of the previously proposed method for constructing and analytically describing arcs of algebraic curves passing through predetermined points, obtained on the basis of Bezier curves, which are compiled taking into account the expansion coefficients of the Newton binomial. The paper gives an example of using adaptive algebraic curves passing through predetermined points for geometric modeling of the stress-strain state of membrane coatings cylindrical shells using two-dimensional interpolation. The given example an illustrative showed the advantages of the proposed adaptation of algebraic curves passing through predetermined points and obtained on the basis of Bezier curves for geometric modeling of multifactor processes and phenomena. The use of such adaptation allows not only to avoid unplanned oscillations, but also self-intersection of geometric objects when generalized to a multidimensional space. Adaptive algebraic curves can also be effectively used as formative elements for constructing geometric objects of multidimensional space, both as guide lines and as generatrix’s.

scholarly journals TOPOLOGY OF FAMILIES OF ALGEBRAIC CURVES CONTINUOUSLY DEPENDING ON A PARAMETER, AND APPLICATIONS

2013 ◽  
Vol 23 (07) ◽  
pp. 1591-1610
Author(s):  
JUAN G. ALCAZAR
Keyword(s):  
Real Function ◽  
Real Root ◽  
Algebraic Curves ◽  
Algebraic Surfaces ◽  
Real Interval ◽  
Real Roots ◽  
Level Curves ◽  
The Family ◽  
Critical Sets ◽  
La Rioja

Given a family of algebraic curves whose coefficients depend continuously on a parameter t ∈ U ⊂ ℝ (U a union of real intervals) we address the problem of computing the topology types in the family. Under certain conditions, we provide a method to compute a univariate real function R*(t), with the property that the topology of the family stays invariant along every real interval I ⊂ U not containing any real root of R*(t). So, if R*(t) has finitely many real roots the topology types in the family can be computed. We apply this result to provide bounds on the number of topology types of the family in certain cases, including the important case when the coefficients belong to a Pfaffian chain, and to analyze the existence of certain degeneracies in the real part of a surface whose family of level curves corresponds to the type studied here. The results of the paper can be seen as generalizations of previous works [Alcázar, Applications of level curves to some problems on algebraic surfaces, in Contribuciones Científicas en Honor de Mirian Andrés Gòmez, eds. Lambán, Romero and Rubio (Universidad de La Rioja, Servicio de Publicaciones, 2010), pp. 105–122, Alcázar, Schicho and Sendra, A delineability-based method for computing critical sets of algebraic surfaces, J. Symbolic Comput.42 (2007) 678–691] on the topology of families algebraically depending on a parameter t.

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