Umnožak tetiva elipse i rekurzivni nizovi polinoma

master's thesis

• Author: Lidija Sokić
• Mentor: Juraj Šiftar (mentor)
• Committee member: Juraj Šiftar (predsjednik povjerenstva)
• Committee member: Boris Širola (član povjerenstva)
• Committee member: Petar Žugec (član povjerenstva)
• Committee member: Miljenko Marušić (član povjerenstva)
• Committee member: Matko Milin (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2019-07-15, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Središnja tema ovog rada je izračunavanje umnoška \(d_n\) duljina tetiva elipse dobivene skaliranjem kružnice na kojoj je odabrano \(n\) ekvidistantnih točaka. Motivacija za ovaj problem proizlazi iz odgovarajućeg rezultata za jediničnu kružnicu gdje vrijedi \(d_n = n\). Th. E. Price izveo je poopćenje tog teorema proučavanjem stanovite familije polinoma \(P_n(z)\). Ti polinomi karakterizirani su rekurzijom koja ih dovodi u vezu s poopćenim Lucasovim i Fibonaccijevim polinomima. Za prikladni izbor parametara koji određuju promatranu elipsu dobiva se \(P_n(1) = L_n\) i \(P'_n(1) = d_n = n F_n\), pri čemu su \(L_n\) i \(F_n\) \(n\)-ti Lucasov, odnosno Fibonaccijev broj. U završnom poglavlju izložen je alternativni pristup Priceovom radu, prema nedavno objavljenom članku B. Blum - Smitha i J. Wooda (2018.). Veći dio Priceovih rezultata izveden je i protumačen na novi način, polazeći od klasičnog Cardanovog rješenja kubne jednadžbe.

Abstract (english)

The main topic of this thesis is computation of the product \(d_n\) of chord lengths for the ellipse obtained by scaling a circle on which \(n\) equidistant points were chosen. The motivation for this problem originates from the corresponding result for the unit circle, namely \(d_n = n\). Th. E. Price derived a generalization of that result by exploring a certain family of polynomials \(P_n(z)\). These polynomials are characterized by a recursion which brings them into a relation with generalized Lucas and Fibonacci polynomials. For an appropriate choice of parameters that describe the observed ellipse, one obtains \(P_n(1) = L_n\) and \(P'_n(1) = d_n = n F_n\), where \(L_n\) and \(F_n\) are the Lucas and Fibonacci numbers, respectively. In the final chapter an alternative approach to Price’s work is laid out, according to a recent paper by B. Blum-Smith and J. Wood (2018.). The main part of Price’s results is derived and explained in a new fashion, starting with the classical Cardano’s solution of the cubic equation.

• Keywords: elipsa; Lucasovi polinomi; Fibonaccijevi polinomi
• Keywords (english): ellipse; Lucas polynomials; Fibonacci polynomials
• Language: croatian
• URN:NBN: urn:nbn:hr:217:396041
• Study programme: Title: Mathematics and Physics Education; specializations in: Mathematics and Physics Education; Course: Mathematics and Physics Education; Study programme type: university; Study level: integrated undergraduate and graduate; Academic / professional title: magistar/magistra edukacije matematike i fizike (magistar/magistra edukacije matematike i fizike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2019-12-02 13:51:25