GEOMETRIC THEORY OF MULTIDIMENSIONAL INTERPOLATION

2020 ◽  
Vol 2020 (1) ◽  
pp. 9-16
Author(s):  
Evgeniy Konopatskiy
Keyword(s):  
Response Function ◽  
Algebraic Curves ◽  
Geometric Theory ◽  
Analytical Description ◽  
Geometric Interpretation ◽  
Affine Geometry ◽  
Mathematical Apparatus ◽  
Multidimensional Interpolation ◽  
Pass Through

The paper presents a geometric theory of multidimensional interpolation based on invariants of affine geometry. The analytical description of geometric interpolants is performed within the framework of the mathematical apparatus BN-calculation using algebraic curves that pass through preset points. A geometric interpretation of the interaction of parameters, factors, and the response function is presented, which makes it possible to generalize the geometric theory of multidimensional interpolation in the direction of increasing the dimension of space. The conceptual principles of forming the tree of the geometric interpolant model as a geometric basis for modeling multi-factor processes and phenomena are described.

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.

Kinematic Geometry of Curve and Its Application to Geometric Modeling Flat Gear

2017 ◽  
Vol 5 (2) ◽  
pp. 25-31
Author(s):  
Панчук ◽  
K. Panchuk ◽  
Ляшков ◽  
A. Lyashkov ◽  
Варепо ◽  
...  
Keyword(s):  
Geometric Modeling ◽  
Geometric Model ◽  
Plane Curve ◽  
Geometric Theory ◽  
Analytical Description ◽  
Geometric Interpretation ◽  
Transmission Performance ◽  
Proposed Model ◽  
Kinematic Geometry ◽  
Fixed System

The paper presents the results of investigations in the field of kinematic geometry of the spatial curve of a line. The basis of the research is the method of the movable trihedron of the curve. The components of the trihedron motion along the spatial curve are considered and it is shown that its resultant instantaneous motion is screw motion. This result differs from the representation of the motion of a trihedron known in geometry as a rotation described by the Darboux vector. An analytical description of the set of axes of instantaneous helical motions of a trihedron in a moving and fixed system of assigning a spatial curve is given. The possibility of applying the obtained general results to the investigation of a plane curve is shown. The paper proposes a flat tooth gearing model based on the geometric interpretation of the motions of a trihedron of a plane curve and known in the geometric theory of plane mechanisms of the construction of Bobillier. The geometric scheme of this construction is expanded due to the introduction of evolutes simulating instantaneous motions of trihedron of the corresponding construction curves. As a result, a geometric model is obtained, which is more complete in comparison with the known models of flat gearing. It allows to perform both direct and inverse tasks of profiling the teeth of the wheels while simultaneously obtaining the curvature of the desired profiles in the absence of such. The proposed model can be used as the basis for the development of gears with a planar gearing scheme by the condition of achieving the necessary transmission performance due to the geometric shape of the teeth of the wheels.

scholarly journals Correspondence theorems via tropicalizations of moduli spaces

2016 ◽  
Vol 18 (03) ◽  
pp. 1550043 ◽  
Author(s):  
Andreas Gross
Keyword(s):  
Moduli Spaces ◽  
Toric Variety ◽  
Algebraic Curves ◽  
Rational Curves ◽  
Tropical Curves ◽  
Algebraic Tori ◽  
Pass Through ◽  
General Correspondence ◽  
Correspondence Theorem ◽  
Generic Points

We show that the moduli spaces of irreducible labeled parametrized marked rational curves in toric varieties can be embedded into algebraic tori such that their tropicalizations are the analogous tropical moduli spaces. These embeddings are shown to respect the evaluation morphisms in the sense that evaluation commutes with tropicalization. With this particular setting in mind, we prove a general correspondence theorem for enumerative problems which are defined via “evaluation maps” in both the algebraic and tropical world. Applying this to our motivational example, we show that the tropicalizations of the curves in a given toric variety which intersect the boundary divisors in their interior and with prescribed multiplicities, and pass through an appropriate number of generic points are precisely the tropical curves in the corresponding tropical toric variety satisfying the analogous condition. Moreover, the intersection-theoretically defined multiplicities of the tropical curves are equal to the numbers of algebraic curves tropicalizing to them.

The even master system and generalized Kummer surfaces

1994 ◽  
Vol 116 (1) ◽  
pp. 131-142
Author(s):  
Jose Bertin ◽  
Pol Vanhaecke
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Algebraic Curves ◽  
Invariant Curves ◽  
Symmetric Form ◽  
Kummer Surface ◽  
Master System ◽  
Kummer Surfaces ◽  
Pass Through ◽  
Invariant Configuration

AbstractIn this paper we study a generalized Kummer surface associated to the Jacobian of those complex algebraic curves of genus two which admit an automorphism of order three. Such a curve can always be written as y2 = x6 + 2kx3 + 1 and k2 ╪ 1 is the modular parameter. The automorphism extends linearly to an automorphism of the Jacobian and we show that this extension has a 94 invariant configuration, i.e. it has 9 fixed points and 9 invariant theta curves, each of these curves contains 4 fixed points and 4 invariant curves pass through each fixed point. The quotient of the Jacobian by this automorphism has 9 singular points of type A2 and the 94 configuration descends to a 94 configuration of points and lines, reminiscent to the well-known 166 configuration on the Kummer surface. Our ‘generalized Kummer surface’ embeds in ℙ4 and is a complete intersection of a quadric and a cubic hypersurface. Equations for these hypersurfaces are computed and take a very symmetric form in well-chosen coordinates. This computation is done by using an integrable system, the ‘even master system’.

scholarly journals Approximation the geometric objects of multidimensional space using arcs of curves passing through the given points

2019 ◽  
Author(s):  
Евгений Конопацкий ◽  
Evgeniy Konopatskiy ◽  
Сергей Ротков ◽  
Sergey Rotkov
Keyword(s):  
Response Function ◽  
Algebraic Curves ◽  
Influence Factors ◽  
Geometric Object ◽  
Multidimensional Space ◽  
Global Coordinate System ◽  
Factors Affecting ◽  
Geometric Objects ◽  
Basic Ideas ◽  
The Given

The paper presents the basic ideas of geometric objects approximation in multidimensional space by means the arcs of algebraic curves passing through given points, which is as follows. A special network of points with a dimension one less than the dimension of the space in which the simulated geometric object is located is formed. Taking into account the special properties the arcs of algebraic curves passing through the given points, a linear relationship between the parameters of the geometric object and the influence factors corresponding to the axes of the global coordinate system is established. Next, the nodes of the network are calculated such values of the response function, which provide the minimum value of the quadratic residual function. The proposed method allows to perform the generalization the method of least squares in the direction of increasing space dimension and, consequently, the number of investigated factors affecting the response function, which is especially important for modeling and optimization of multifactorial processes and phenomena.

Linkage Synthesis Using Algebraic Curves

1986 ◽  
Vol 108 (4) ◽  
pp. 543-548 ◽  
Author(s):  
J. L. Blechschmidt ◽  
J. J. Uicker
Keyword(s):  
Nonlinear Equations ◽  
Algebraic Polynomial ◽  
Algebraic Curve ◽  
Algebraic Curves ◽  
Optimization Techniques ◽  
Synthesis Methods ◽  
Nonlinear Functions ◽  
Pass Through ◽  
Four Bar Linkage ◽  
Set Of Points

A method to snythesize four-bar linkages using the algebraic curve of the motion of the coupler point on the coupler link of the four-bar linkage is developed. This method is a departure from modern synthesis methods, most of which are based upon Burmester theory. This curve, which is a planar algebraic polynomial in two variables for the four-bar linkage, is a trinodal tricircular sextic (sixth order). These properties are used to determine the coefficients of the curve given a set of points that the coupler point of the coupler link is to pass through. The coefficients of this curve are nonlinear functions of the linkage parameters. The resulting set of nonlinear equations are solved using iterative/optimization techniques for the linkage parameters.

Geometric Mappings on Geometric Lattices

1971 ◽  
Vol 23 (1) ◽  
pp. 22-35 ◽  
Author(s):  
David Sachs
Keyword(s):  
Vector Space ◽  
Linear Algebra ◽  
Classical Result ◽  
Linear Independence ◽  
Closure Operator ◽  
Geometric Interpretation ◽  
Affine Geometry ◽  
Algebraic Methods ◽  
Linear Transformations ◽  
Geometric Lattices

It is a classical result of mathematics that there is an intimate connection between linear algebra and projective or affine geometry. Thus, many algebraic results can be given a geometric interpretation, and geometric theorems can quite often be proved more easily by algebraic methods. In this paper we apply topological ideas to geometric lattices, structures which provide the framework for the study of abstract linear independence, and obtain affine geometry from the mappings that preserve the closure operator that is associated with these lattices. These mappings are closely connected with semi-linear transformations on a vector space, and thus linear algebra and affine geometry are derived from the study of a certain closure operator and mappings which preserve it, even if the “space” is finite.

scholarly journals Assessment of the impact of the duration of construction on the effectiveness of the investment project

2019 ◽  
Vol 135 ◽  
pp. 04021 ◽  
Author(s):  
Elena Mikhaylova
Keyword(s):  
Economic Efficiency ◽  
Capital Investment ◽  
Investment Project ◽  
Mathematical Expression ◽  
Analytical Description ◽  
Mathematical Apparatus ◽  
Investment Projects ◽  
Investment Process ◽  
Quantitative Indicators ◽  
The Impact

The article discusses the relevance and necessity of assessing the impact of the duration of construction on the economic efficiency of investments. Using a mathematical expression, the profit received by the investor during the life cycle of the direct investment process is described. The peculiarity of investment projects involving the creation of capital investment (real estate), is the period of time during which profit is impossible. This period of time is equal to the duration of construction, including installation of technological equipment and commissioning. An analytical description of the degree of influence of the duration of construction on the indicators of economic efficiency of the investment project (profitability and profitability of property) is proposed. On the basis of the learned expressions, a numerical experiment was performed and graphs were constructed. The results of the research prove the possibility of analytical description of the degree of influence of the duration of construction on the quantitative indicators describing the investment project (profitability (Bank interest), profitability of the created property). It seems that the described approach allows to carry out forecasts which reliability surpasses the results obtained, for example, by means of expert estimations. The proposed mathematical apparatus is much simpler than the methods of probability theory and fuzzy set theory.

An Analytic Solution for Minimum-Norm Rate Coordination in Redundant Manipulators

1991 ◽  
Author(s):  
Hareendra Varma ◽  
Ming Z. Huang
Keyword(s):  
Solution Space ◽  
Geometric Interpretation ◽  
Alternative Formulation ◽  
Redundant Manipulators ◽  
Minimum Norm ◽  
Redundancy Resolution ◽  
Coordination Problem ◽  
Efficient Alternative ◽  
Novel Method ◽  
Pass Through

Abstract In this paper, we present a novel method which results in efficient minimum norm solution for the rate coordination problem in redundant manipulators. The theory is developed based on a geometric interpretation that, for minimum norm criterion, vectors orthogonal to constraint space should pass through the origin of the solution space. It is shown that for any spacial manipulator with 1, 2 or 3 degrees of redundancy, the minimum norm rate solution can be derived in analytic closed form. The method offers an equivalent but much more efficient alternative to using the pseudoinverse in redundancy resolution and, in fact, is applicable to any underdetermined linear system. An alternative formulation of pseudo-inverse arrived at in the course of the development is also presented.

GEOMETRIC MODELING OF MULTI-FACTOR PROCESSES BASED ON VARIABLE POINT ALGORITHMS

2021 ◽  
pp. 29-38
Author(s):  
E. V. Konopatskiy ◽  
I. V. Seleznev ◽  
M. V. Lagunova ◽  
A. A. Bezditniy
Keyword(s):  
Geometric Modeling ◽  
Geometric Theory ◽  
Physical And Mechanical Properties ◽  
Multidimensional Space ◽  
Fine Grained ◽  
Variable Approach ◽  
Multidimensional Interpolation ◽  
Geometric Objects ◽  
Variable Point ◽  
Current Point

In this paper, the geometric theory of multidimensional interpolation was further developed. It has been established that the geometric models of multivariate processes obtained using multidimensional interpolation are characterized by variability, which is a consequence of the multiplicity of choice of reference lines in the process of developing a geometric modeling scheme. At the same time, all possible variations of geometric interpolants fully satisfy the initial experimental and statistical data, but have different curvature between the node points of the interpolation. As the dimension of the space increases, the number of variations increases significantly. The variable approach to geometric modeling of multifactorial processes generates a number of scientific problems that require further research, such as: comparison of geometric objects of multidimensional space, development of criteria for choosing the best solutions, construction of averaged geometric objects as one of the tools for optimizing the results of modeling, etc. The article also presents the results of a computational experiment on geometric modeling of the dependence of the physical and mechanical properties of fine-grained concrete on the composition of the combined aggregate based on variable point algorithms with the subsequent construction of an averaged response surface, the current point of which is the center of gravity of a multidimensional tetrahedron, for which the dimension of space depends on the amount possible interpolation options.

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