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A mathematical method for modeling the shape of apples. Part 1. Description of the method

Mieszkalski L., Wojdalski J.

ECONTECHMOD : An International Quarterly Journal on Economics of Technology and Modelling Processes

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2017

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Vol. 6, No 3

97--104

EN

This study proposes a mathematical method for modeling the shape of apples, locules and pericarps with the use of Bézier curves. The concave and convex parts of apples cv. Ligol were described with three smoothly-joined Bézier curves. Contours were described based on images of an apple rotated at intervals of 36° relative to its natural axis of symmetry. A 3D model was formed by Bézier curves positioned along the apple’s meridians. The shape of the locule and the pericarp was described with the use of two smoothly-joined Bézier curves each, rotated relative to the apple’s longitudinal axis.

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Matematyczne modelowanie kształtu cytryn

Mieszkalski L.

Postępy Techniki Przetwórstwa Spożywczego

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2016

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Nr 2

62--66

PL

W artykule przedstawiono propozycję metody matematycznego modelowania kształtu cytryn z wykorzystaniem krzywych Béziera. Kontur cytryny leżący na jej południku opisano trzema połączonymi krzywymi Béziera. Podstawą do opisu konturów były fotografie 10 położeń cytryny obracanej co 36o względem jej naturalnej osi symetrii. Rozmieszczone wzdłuż południków cytryny krzywe Béziera są jej modelem 3D.

EN

The article proposes the method of mathematical modeling of the shape of the lemons behind using Bezier curves. Outline lemon lying on the meridian described three connected Bezier curves. The basis for the description of contours is photos of 10 positions lemon rotated at 36° relative to its natural axis of symmetry. 3D model are the arranged along the meridians lemon Bézier curve.

3

Genetic algorithm coupled with Bézier curves applied to the magnetic field on a solenoid axis synthesis

Ziolkowski M., Gratkowski S.

Archives of Electrical Engineering

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2016

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Vol. 65, nr 2

361--370

EN

Electromagnetic arrangements which create a magnetic field of required distribution and magnitude are widely used in electrical engineering. Development of new accurate designing methods is still a valid topic of technical investigations. From the theoretical point of view the problem belongs to magnetic fields synthesis theory. This paper discusses a problem of designing a shape of a solenoid which produces a uniform magnetic field on its axis. The method of finding an optimal shape is based on a genetic algorithm (GA) coupled with Bézier curves.

4

Fast multidimensional Bernstein-Lagrange algorithms

Kapusta J., Smarzewski R.

Annales Universitatis Mariae Curie-Skłodowska. Sectio AI, Informatica

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2012

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Vol. 12, no. 2

7--18

EN

In this paper we present two fast algorithms for the Bézier curves and surfaces of an arbitrary dimension. The first algorithm evaluates the Bernstein-Bézier curves and surfaces at a set of specific points by using the fast Bernstein-Lagrange transformation. The second algorithm is an inversion of the first one. Both algorithms reduce the initial problem to computation of some discrete Fourier transformations in the case of geometrical subdivisions of the d-dimensional cube. Their orders of computational complexity are proportional to those of corresponding d-dimensional FFT-algorithm, i.e. to O (N logN) + O (dN), where N denotes the order of the Bernstein-Bézier curves.