Professional paper

On the universal parabolic constant

Bojan Kovačić orcid.org/0000-0002-3893-2191 ; Tehničko veleučilište u Zagrebu, Zagreb, Hrvatska
Ivana Božić ; Tehničko veleučilište u Zagrebu, Zagreb, Hrvatska
Tihana Strmečki orcid.org/0000-0003-0574-0019 ; Tehničko veleučilište u Zagrebu, Zagreb, Hrvatska

Abstract

In this article the universal parabolic constant is defined as the analog of the number $\pi$ in relation to the circle. For this purpose, all relevant terms and results in algebraic number theory, elementary mathematics and calculus are stated, necessary to consider the subject completely and adequately. It is shown that the ratio of arc length of the parabolic segment, determined by the line through the focus parallel to the directrix, and the distance between the focus and the directrix, does not depend on the particular parabola, justifying the constant's definition. Applying the Lindemann's result that the number $\pi$ is trancedental, the proof is given that the universal parabolic constant is a transcendental and hence an irrational number. Furthermore, the property of similarity of curves is analyzed, establishing that all circles and parabolas satisfy this property, while ellipses and hyperbolas do not. Two sets of examples of the application of the parabolic constant are given. One is calculating the surface area of certain rotational solid objects, created by rotating graphs of elementary functions around the $x$-axis. The other involves analyzing problems of determining the average distance of points in a unit square to the center of the square, or to any randomly chosen but fixed vertex of the square. As to the latter, their probability theory counterparts are stated.

Keywords

the universal parabolic constant; properties; application

Hrčak ID:

105947

URI

https://hrcak.srce.hr/105947

Publication date:

30.8.2013.

Article data in other languages: croatian

Visits: 1.684 *