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1

Mannheim curves and their partner curves in Minkowski 3-space E13

Elsharkawy Ayman, Elshenhab Ahmed M.

Demonstratio Mathematica

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2022

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Vol. 55, nr 1

798--811

EN

The modified orthogonal frame is an important tool to study analytic space curves whose curvatures have discrete zero points. In this article, by using the modified orthogonal frame, Mannheim curves and their partner curves are investigated in Minkowski 3-space. Some characterizations according to the curvatures and torsions of the curves are given. Finally, some relations under the conditions for Mannheim curves and their partner curves to be generalized helices are presented. All the possible cases for the partner curves to be spacelike and timelike are considered in the whole of the article.

2

Drogi i pojazdy - aspekty geometryczne i kinematyczne

Icha Andrzej

Modelowanie Inżynierskie

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2019

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T. 39, nr 70

32--42

PL

Niech będzie dana droga w postaci krzywej prostowalnej określonej parametrycznie jako j(t) = (x(t), y(t)), gdzie x(t) jest funkcją rosnącą, x(0) = 0, y(t) ≤ 0. Przez pojazd odpowiadający drodze rozumie się krzywą, opisywaną równaniem ρ = ρ(Θ), która toczy się bez poślizgu po drodze. Kinematyka pojazdu jest opisywana przez funkcję Θ = Θ(t) określającą ruch pojazdu po drodze w zakresie zmienności od j(0) do j(t). Klasycznym przykładem zagadnienia droga - pojazd jest toczenie się okręgu (pojazdu) bez poślizgu po linii prostej (drodze). Torem mchu punktu położonego na promieniu toczącego się okręgu jest cykloida. Celem pracy jest przedstawienie wybranych zagadnień geometrycznych i kinematycznych dotyczących toczenia się elipsy (pojazdu) po odpowiednio dobranej (wyliczonej) drodze (krzywej cosinusoidalnej) oraz zagadnienie dobom kształtu pojazdu do zadanej periodycznej drogi, opisywanej pierwszymi wyrazami szeregu Fouriera. Rozważania uzupełniono stosownymi animacjami, wyko¬nanymi w językach programowania TeX i PostScript przy wykorzystaniu pakietu makr PSTricks.

EN

Let j(t) = (x(t), y(t)) be a road in the form of a rectifiable curve, where x(t) is increasing, x(0) = 0, y(t) ≤ 0. By the wheel corresponding to the road is meant a curve ρ = ρ(Θ) that will roll smoothly on the road. The kinematics of the wheel is characterized by a function Θ = Θ(t) that describes the amount of angular rotation for the wheel to roll from j(0) to j(t). The famous example of the roads - wheels problem is the classical round wheel rolling on a straight road, which leads to the cycloid as the path of a point on a wheel. The purpose of the paper is to present the selected geometrical and kinematical problems concerning the elliptical wheel rolling on a road defined using the cosine function. The problem of what shape the wheels for the prescribed Fourier approximations of roads, is also analysed. The considerations are supplemented with suitable animations made in TeX and PostScript programming languages, with the use of the PSTricks macro package.

3

Spatial cyclographic modeling on Naumovich hyperdrawing

Panchuk K., Lyubchinov E.

Journal Biuletyn of Polish Society for Geometry and Engineering Graphics

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2018

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Vol. 31

69--78

EN

The present paper shows the capability of application of three-dimensional drawing of space R4 proposed by N.V. Naumovich in 1958, in the field of cyclographic modeling of non-linear objects: curves and surfaces of space R4, models of which are widely applied in modern CAD/CAM and Computer Aided Geometric Design (CAGD) systems. The possibility of implementing spatial cyclography on the Naumovich hyperdrawing is illustrated by the example of modeling a canal surface as a cyclographic image of a spatial curve of R4 space.

PL

W niniejszej pracy przedstawiono możliwości zastosowania trójwymiarowego rysunku przestrzeni R4, zaproponowanego przez N.V. Naumowicza w 1958 roku, w dziedzinie cyklicznego modelowania obiektów nieliniowych: krzywych i powierzchni przestrzeni R4, których modele są szeroko stosowane w nowoczesnych systemach CAD/CAM i Computer Aided Geometric Design (CAGD). Możliwość zastosowania cyklografii przestrzennej na rysunku czterowymiarowym Naumowicza, jako obrazu cyklograficznego krzywej czterowymiarowej, pokazano na przykładzie powierzchni kanałowej.

4

Uogónione krzywe pościgowe

Icha A.

Modelowanie Inżynierskie

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2017

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T. 31, nr 62

29--36

PL

Praca dotyczy wybranych zagadnień krzywych przestępnych pojawiających się przy analizie problemów pościgowych. Krótko omówiono klasyczne zagadnienie pogoni oraz zaprezentowano rezultaty modelowania uogólnionych problemów pościgowych przy wykorzystaniu języka programowania MetaPost i systemu obliczeń symbolicznych Maple. Otrzymano i przedstawiono realizacje krzywych pogoni dla wybranych, zadanych krzywych ucieczki obejmujących: prostą (test), okrąg, parabolę, asteroidę, krzywą Talbota, bikorn oraz trzy krzywe autorskie.

EN

The paper deals with selected problems concerning the transcendental curves that arise when analyzing pursuit problems. The classical chase problem is described briefly, and the results of modelling the generalized pursuit problems are presented, using the MetaPost and the Maple software. The following chase curves are obtained for the prescribed escape curves, namely, the line (for the test),the circle, the parabola, the asteroid, the Talbot curve, the bicorn, and the three own curves.

5

Symplectic U7, U8 and U9 singularities

Trębska Ż.

Demonstratio Mathematica

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2015

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Vol. 48, nr 2

322--347

EN

We use the method of algebraic restrictions to classify symplectic U7, ;U8 and U9 singularities. We use discrete symplectic invariants to distinguish symplectic singularities of the curves. We also give the geometric description of symplectic classes.

6

Ocena skuteczności modeli transferu promieniowania w badaniach stanu roślinności łąk

Jarocińska A.

Teledetekcja Środowiska

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2012

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T. 48

1--52

EN

Vegetation analysis is an important problem in regional and global scale. Because of pollution of environment and changes in the ecosystems plant monitoring is very important. Remote sensing data can be easily used to plant monitoring. That kind of method is much faster and more reliable than traditional approaches. Spectrometry analyzes the interactions between radiation and object and it uses measurement of radiation intensity as a function of wavelength. Each object emits and absorbs different quantity of radiation, so it is possible to recognise the object and check its characteristics analysing the spectrum. The subject of the researches is Polish meadows. The human usage of the meadows determines its proper functioning. Grasslands, which consist of meadows and pastures, cover 10% of Poland. Meadows are most extensively use. In Poland the crops from meadows (hay and green forage) are very low. The meadows in Poland are floristically and morphologically very diverse. Many factors influence on this ecosystem and that is why the monitoring is very important. The aim of the researches is to study the possibility of use of the Radiative Transfer Models in modelling the state of the heterogeneous vegetation cover of seminatural meadows in Poland. Two approaches are used to canopy analysis: statistical and modelling. In the statistic approach, biophysical parameters calculated from the image are correlated with reflectance or transmittance from fi eld measurements. In second approach physically based model is used to represent the photon transport inside leaves and canopy. The Radiative Transfer Models are based on the laws of optics. Developing the model results in better understanding of the interaction of light in canopy and leaves. The Radiative Transfer Models are often applied to vegetation modelling. The Radiative Transfer Models are physically based models which describe the interactions of radiation in atmosphere and vegetation. Adjusted models can be used to fast and precise analysis of biophysical parameters of the canopy. The canopy can be described as homogeneous layer consisting of leaves and spaces. The Radiative Transfer Models are algorithms which vary by input and output parameters, the level of the analysis, kinds of plants and other modifications. Models are used on two levels: single leaf and whole canopy. The first model, which is used in this research, is PROSPECT, which describes the multidirectional refl ectance and diffusion on a leaf level. It is often employed with other models that describe whole canopy. Leaf has the same properties on both sides, the reflection from the leaves is Lambertian. The input parameters in the model are: chlorophyll and carotenoid content, Equivalent Water Thickness and dry matter content and also leaf structure parameter that describe the leaf structure and complexity. Second model, which is used in the study, is the canopy reflectance model SAIL (Scattering by Arbitrarily Inclined Leaves). It simulates the top of the canopy bidirectional reflectance and it describes the canopy structure in a fairly simple way. In this analysis the 4-SAIL model will be used. This version has few input parameters that describe plants and soil: spectrometric data – reflectance and transmittance from leaves (the output parameters form PROSPECT model), biophysical canopy parameters (Leaf Area Index, brown pigment content, mean leaf inclination angle), soil brightness parameter, reflectance geometry (solar zenith angle, observer zenith angle, relative azimuth angle), ratio of diffuse to total incident radiation and two hot spot size parameters. The SAIL model is often combined with the model on leaf level – the PROSAIL model. The PROSPECT and SAIL are very rarely used to meadows, this kind of ecosystem is normally rather heterogeneous and modelling is quite difficult. In this study two Radiative Transfer Models (PROSPECT-5 and 4SAIL) were used on single leaves and a whole canopy level. In order to acquire the input data to both, models model and reference spectrums the fi eld measurements were done. The input parameters were recalculated using fields measurements and put into the models: PROSPECT and PROSAIL. Only one leaf structure parameter was fitted for each polygon individually. The spectral reflectance obtained from the model was compared with field data. Based on the calculated Root Mean Square Error the simulation was verified. The RMSE values were calculated for whole range from 400 to 2500 nm and for specific ranges. The correctness of simulated spectra were analysed dependent on the type of meadows (cultivated meadows with reduced amount of biomass, cultivated meadows with high amount of biomass and not cultivated meadows) and the value of three different biophysical parameters (Leaf Area Index, fresh biomass content and water content). Better results were obtained using PROSPECT model than PROSAIL. In the visible light more accurate values were calculated using PROSAIL and in the infrared using PROSPECT. Generally bigger errors were noticed in the infrared, especially middle infrared range. The effectiveness of the reflectance simulation was not influenced by different kind of meadows. Apart from that, better results were obtained on meadows with higher biomass value, bigger Leaf Area Index and lower water content. Generally, the PROSPECT and PROSAIL radiative transfer models can be used to simulate the spectral reflectance of vegetation on heterogeneous meadows. The models can be used to estimate the biophysical parameters, but it is necessary to correct the values of input variables (especially water content). Meadows are very complex environment and some of the parameters should be adjusted.

7

Using a CAS to visualize some images of lines mapped via the harmonic cross-ratio

Korczak E., Marlewski A.

Journal Biuletyn of Polish Society for Geometry and Engineering Graphics

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2012

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Vol. 23

11--19

EN

Let K be a cubic curve in the projective space P 3 and let T1 and T2 be points determining a bisecant T1T2 of K. We fix a point A on K and a point B≠A which does not lay on K, and such that T1T2 ≠AB. We are interested in the set of points X generated by the equation (T1, T1; M, X) = –1 where M denotes the point at which AB meets the bisecant T1T2. So we consider the line congruence of order 1 and of class 3 in the aspect of the harmonic cross-ratio. We derive theoretic formulas for the set of X ‘s and we go on in the harmonic case– then the set of X ’s is a conic. We use the computer algebra system Derive 5 from Texas Instruments, Inc., USA, to produce visualizations of the images of resulting curves.

PL

Niech K będzie krzywą przestrzenną rzędu trzeciego w przestrzeni rzutowej P 3 i niech M będzie dowolnym punktem tej przestrzeni nieleżącym na K. W wiązce prostych, której wierzchołkiem jest M, znajduje się dokładnie jedna bisekanta. Punkty, w których przecina ona krzywą K, oznaczamy przez T1 i T2. Tematem pracy jest zbadanie miejsc geometrycznych punktów X i T1T2, dla których dwustosunek (T1, T2; M, X) = –1, gdy punkt M przebiega prostą, którą wyznaczają ustalone punkt krzywej K i punkt, który na K nie leży. Badanie to przeprowadzamy przy użyciu programu Derive 5 for Windows (Texas Instruments, Inc.).

8

Some properties of the curves xn +yn=1 with even exponents

Fidytek I.

Demonstratio Mathematica

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2010

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Vol. 43, nr 1

89-96

EN

We present parametric equations for the curve xn + yn = 1 with even exponents, in terms of the double area of the sector bounded by the arc between (1,0) and (x, y) and the radius vectors of these points. We determine also the area enclosed by this curve.

9

Polar invariants of plane curve singularities: intersection theoretical approach

Gwoździewicz J., Lenarcik A., Płoski A.

Demonstratio Mathematica

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2010

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Vol. 43, nr 2

303-323

EN

This article, based on the talk given by one of the authors at the Pierret-tefest in Castro Urdiales in June 2008, is an overview of a number of recent results on the polar invariants of plane curve singularities.

10

Blaschke products, (alpha)-curves and Qp spaces

Majchrowska G., Mouratides C.

Demonstratio Mathematica

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2010

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Vol. 43, nr 1

29-38

EN

We give necessary and sufficient conditions for Blaschke products with zeros on an a-curve, to belong to a Qp space, in terms only of distribution of the zeros. It comes out that the condition depends on p. As we see the 0-1 law of the non tangential case breaks. Here it is possible that a Blaschke product B belongs to Qp for some but not for all p.

11

g-chordal curves

Cieślak W., Wołowik J.

Demonstratio Mathematica

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2009

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Vol. 42, nr 2

411-422

EN

In this paper we introduce a notion of g-chordal curves which are a natural generalization of equichordal, equireciprocal and equipower curves. A Crofton-type integral formula and estimations of the area and the length of g-chordal curve are given. Moreover, a 1-parameter family of ovals with exactly four vertices in the class generated by the function g (x) = xm is constructed. A remark on the equichordal problem ends the paper.

12

A quadratic transformation of the three-dimensional space of all conics throught two points of a projective plane

Donati G.

Demonstratio Mathematica

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2009

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Vol. 42, nr 3

619-625

EN

Using the Steiner's method of projective generation of conics and its dual we define two projective mappings of a double contact pencil of conics into itself and we prove that one is the inverse of the other. We show that these projectiye mappings are induced by quadratic transformations of the three-dimensional projectiye space of all conics through two distinct points of a projective plane.

13

Some characterization of osculating curves in the Euclidean spaces

Ilarslan K., Nesovic E.

Demonstratio Mathematica

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2008

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Vol. 41, nr 4

931-939

EN

In this paper, we give some characterization for a osculating curve in 3-dimensional Euclidean space and we define a osculating curve in the Euclidean 4-space as a curve whose position vector always lies in orthogonal complement B1/1 of its first binormal vector field B1. In particular, we study the osculating curves in E4 and characterize such curves in terms of their curvature functions.

14

Krzywe stosowane w definiowaniu kształtu części w systemach CAD/CAM. Część I

Adamski W.

Mechanik

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2005

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R. 78, nr 3

196-199

15

Krzywe stosowane w definiowaniu kształtu części w systemach CAD/CAM. Część II

Adamski W.

Mechanik

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2005

|

R. 78, nr 5-6

534-537

16

Characterization of continuous functions by class C(infinity) curves

Najdecki A., Tabor J.

Demonstratio Mathematica

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2005

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Vol. 38, nr 1

31--34

EN

Let f : X -Y, where X is a Banach space and Y is a Hausdorff topological space. We prove that if f o (gamma) is continuous for every curve (gamma) : [0,1] -> X of class C(infinity), then f is continuous.

17

Interpolacja krokowa

Bożek B., Środa P.

Informatyka w Technologii Materiałów

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2004

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T. 4, Nr 1-2

65-68

PL

Zaproponowana w pracy metoda interpolacyjna polegająca na wstępnym zadaniu krzywej bądź powierzchni i takim jej przekształceniu (przy zastosowaniu prostych wzorów), aby przechodziła przez zadane punkty, umożliwia przewidzenie skutków tych przekształceń, co pozwala na jej intuicyjne stosowanie.

EN

In this paper step by step interpolation method is presented. Our method consists in that we take the shape of a curve (or a surface) and stretch it succesively (step by step) through given nodal points.

18

Development of complex aircrafturfaces using spline curves

Leonavicius K.

Research Bulletin / Warsaw University of Technology, Institute of Aeronautics and Applied Mechanics

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2001

|

nr 11

139-144

EN

The paper presents a simple designing method for complex 3D airplane models. The spline curves are employed to model both the contour and cross sections. The surface model may be shaped and easily changed using 3D isolines. This method is very efficient for the light and sport aircraft because of complexity of their geometry.