Dynamic Programming as a Method of Spline Approximation in the CAD Systems of Linear Constructions

Karpov, D. A.; Struchenkov, V. I.

doi:10.32362/2500-316x-2019-7-3-77-88

Cited by 10 publications

(7 citation statements)

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“…Smooth curves are one of the basic design tools applied in contemporary CAD. For example, the reference [4] describes the application of Spline Approximation in the CAD Systems for the Linear Constructions. The Bezier curves represent the classic type of smooth curves are of the greatest interest in this current research.…”

Section: Previous Workmentioning

confidence: 99%

“…Bezier spline is the classical type of smooth curves represented parametrically. The construction of Bezier curve is carried out by means of Bernstein polynomials [4]:…”

Section: Bezier Curvementioning

confidence: 99%

“…where ( )the amount of combinations of n choose i, nexponent of the polynomials, i -consecutive number of the anchor vertex. At the same time, abscissa and ordinate are described parametrically by the system of equations (4).…”

Section: Bezier Curvementioning

confidence: 99%

“…Fig.4 illustrates the implementation of such an approach: It's possible to draw a conclusion that the application of this algorithm also doesn't allow to obtain an appropriate image of the Bezier curve in consequence of the curvature peculiar to it. Speaking about the Bezier curve, we cannot help but mention the De Casteljau's algorithm (Fig.5) [4]. On the basis of this algorithm lies the finding the Bezier curve tangents at each value of t-parameter.…”

Section: Bezier Curvementioning

confidence: 99%

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Construction of the Functional Voxel Model for a Spline Curve

Tolok

Sycheva

2020

Proceedings of the 30th International Conference on Computer Graphics and Machine Vision (GraphiCon 2020). Part 2

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Analytical models are the most accurate method of geometric information representation. Parameterized smooth curves cannot be used in the field of analytical geometry, which explains the necessity for finding of analytical representation of such curves. The article considered the construction of a smooth curve presented in an analytical form and some approaches to finding an analytical model for a parametric Bezier curve. А presentation of a function in the form of its functional areas was chosen as prototype of the analytical model. The selected representation formed on the basis of the De Casteljau's method of constructing the Bezier curve and set-theoretic modeling. The Rvachev functions (R-functions) are used as the mathematical apparatus of set-theoretic operations on function areas. The functional-voxel method makes it possible to simplify the computation of R-functional procedures. An algorithm for constructing the functional area of the Bezier curve is developed on the basis of the presented combined R-voxel approach. The obtained results allow for the conclusions about the adequacy of this approach and its development protentional to construct more complicated structures.

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Section: Previous Workmentioning

confidence: 99%

“…Bezier spline is the classical type of smooth curves represented parametrically. The construction of Bezier curve is carried out by means of Bernstein polynomials [4]:…”

Section: Bezier Curvementioning

confidence: 99%

Section: Bezier Curvementioning

confidence: 99%

Section: Bezier Curvementioning

confidence: 99%

See 2 more Smart Citations

Construction of the Functional Voxel Model for a Spline Curve

Tolok

Sycheva

2020

Proceedings of the 30th International Conference on Computer Graphics and Machine Vision (GraphiCon 2020). Part 2

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“…In the simplest case of using a first-order spline, the task is to transform the original broken line (ground profile) into another broken line that satisfies a number of constraints: on the slopes of elements and the difference in slopes of adjacent elements, on the minimum length of the element, on ordinates of individual points and zones (height constraints) [6,7,8]. Due to the smallness of the design slopes, the length of the element and the difference between the abscissas of its ends practically coincide; the difference in the slopes of adjacent elements is identified with the angle of rotation, and the slope is identified with the angle of the element with the OX axis.…”

Section: Introductionmentioning

confidence: 99%

Two-Stage Spline-Approximation with an Unknown Number of Elements in Applied Optimization Problem of a Special Kind

Struchenkov¹,

Karpov²

2021

Self Cite

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Being a continuation of the paper published in Mathematics and Statistics, vol. 7, No. 5, 2019, this article describes the algorithm for the first stage of splineapproximation with an unknown number of elements of the spline and constraints on its parameters. Such problems arise in the computer-aided design of road routes and other linear structures. In this article we consider the problem of a discrete sequence approximation of points on a plane by a spline consisting of line segments conjugated by circular arcs. This problem occurs when designing the longitudinal profile of new and reconstructed railways and highways. At the first stage, using a special dynamic programming algorithm, the number of elements of the spline and the approximate values of its parameters that satisfy all the constraints are determined. At the second stage, this result is used as an initial approximation for optimizing the spline parameters using a special nonlinear programming algorithm. The dynamic programming algorithm is practically the same as in the mentioned article published earlier, with significant simplifications due to the absence of clothoids when connecting straight lines and curves. The need for the second stage is due to the fact that when designing new roads, it is impossible to implement dynamic programming due to the need to take into account the relationship of spline elements in fills and in cuts, if fills will be constructed from soils of cuts. The nonlinear programming algorithm is based on constructing a basis in zero spaces of matrices of active constraints and adjusting this basis when changing the set of active constraints in an iterative process. This allows finding the direction of descent and solving the problem of excluding constraints from the active set without solving systems of linear equations in general or by solving linear systems of low dimension. As an objective function, instead of the traditionally used sum of squares of the deviations of the approximated points from the spline, the article proposes other functions, taking into account the specifics of a specific project task.

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