Jacob Steiner's construction of conics revisited

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F60076658%3A12410%2F19%3A43900271" target="_blank" >RIV/60076658:12410/19:43900271 - isvavai.cz</a>

Výsledek na webu

<a href="http://www.heldermann.de/JGG/JGG23/JGG232/jgg23018.htm" target="_blank" >http://www.heldermann.de/JGG/JGG23/JGG232/jgg23018.htm</a>

DOI - Digital Object Identifier

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Jazyk výsledku

angličtina

Název v původním jazyce

Jacob Steiner's construction of conics revisited

Popis výsledku v původním jazyce

We aim at presenting material on conics, which can be used to formulate, e.g., GeoGebra problems for high-school and freshmen maths courses at universities. In a (real) projective plane two pencils of lines, which are projectively related, generate, in general, a conic. This fact due to Jakob Steiner [4] allows to construct points of a cocnic given by, e.g., 5 points. Hereby the problem of transfering a given cross-ratio of four lines of the first pencil to the corresponding and uniform way we propose a method, which uses the well-known fact that a projective mapping from one line (or pencil) to another always can be decomposed into a product of perspectivities. By extending the presented graphical methods, we also construct tangents and osculating circles at points of a conic. The calculation following the graphic treatment delivers a parametrisation of conic arcs applicable also for so-called 2nd order biarcs. Even so the topic and its theoretical background is a matter of the 19th century, it is not at all well-known nowadays, as also is stated in [3]. Some of the presented constructions might also be new.

Název v anglickém jazyce

Jacob Steiner's construction of conics revisited

Popis výsledku anglicky

We aim at presenting material on conics, which can be used to formulate, e.g., GeoGebra problems for high-school and freshmen maths courses at universities. In a (real) projective plane two pencils of lines, which are projectively related, generate, in general, a conic. This fact due to Jakob Steiner [4] allows to construct points of a cocnic given by, e.g., 5 points. Hereby the problem of transfering a given cross-ratio of four lines of the first pencil to the corresponding and uniform way we propose a method, which uses the well-known fact that a projective mapping from one line (or pencil) to another always can be decomposed into a product of perspectivities. By extending the presented graphical methods, we also construct tangents and osculating circles at points of a conic. The calculation following the graphic treatment delivers a parametrisation of conic arcs applicable also for so-called 2nd order biarcs. Even so the topic and its theoretical background is a matter of the 19th century, it is not at all well-known nowadays, as also is stated in [3]. Some of the presented constructions might also be new.

Druh

J_SC - Článek v periodiku v databázi SCOPUS

CEP obor

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OECD FORD obor

10102 - Applied mathematics

Projekt

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Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2019

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Journal for Geometry and Graphics

ISSN

1433-8157

e-ISSN

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Svazek periodika

23

Číslo periodika v rámci svazku

2

Stát vydavatele periodika

DE - Spolková republika Německo

Počet stran výsledku

11

Strana od-do

189-199

Kód UT WoS článku

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EID výsledku v databázi Scopus

999