Version 1
: Received: 30 August 2021 / Approved: 31 August 2021 / Online: 31 August 2021 (11:04:18 CEST)
Version 2
: Received: 17 February 2023 / Approved: 21 February 2023 / Online: 21 February 2023 (03:35:09 CET)
How to cite:
Cabeza Lainez, J. An Integral Expression for Pi Obtained by Virtue of Ramanujan’s Approximation in Conoidal Areas: Applications of the Resultant Figure in Engineering. Preprints.org2021, 2021080557. https://doi.org/10.20944/preprints202108.0557.v2
Cabeza Lainez, J. An Integral Expression for Pi Obtained by Virtue of Ramanujan’s Approximation in Conoidal Areas: Applications of the Resultant Figure in Engineering. Preprints.org 2021, 2021080557. https://doi.org/10.20944/preprints202108.0557.v2
Cite as:
Cabeza Lainez, J. An Integral Expression for Pi Obtained by Virtue of Ramanujan’s Approximation in Conoidal Areas: Applications of the Resultant Figure in Engineering. Preprints.org2021, 2021080557. https://doi.org/10.20944/preprints202108.0557.v2
Cabeza Lainez, J. An Integral Expression for Pi Obtained by Virtue of Ramanujan’s Approximation in Conoidal Areas: Applications of the Resultant Figure in Engineering. Preprints.org 2021, 2021080557. https://doi.org/10.20944/preprints202108.0557.v2
Abstract
unlike the volume, the expression for the surface area of a regular conoid has not yet been obtained by means of direct integration or a differential geometry procedure. As this non-developable form is relatively used in engineering, the difficulty to determine its surface, represents a serious shortcoming for several problems which arise in radiative transfer, lighting and construction, to cite just a few. In order to solve the problem, I conceived the surface as a set of linearly dwindling ellipses which remain parallel to a circular directrix, a typical problem appears when searching the length of such ellipses. I employed a new procedure which, in principle, consists in dividing the surface into infinitesimal elliptic strips to which we have successively applied Ramanujan’s second approximation for the longitude of the ellipse. In this manner, we can obtain the perimeter of any transversal curve pertaining to the said form as a function of the radius of the directrix and the position of the ellipse’s center on the X-axis. Integrating the so-found perimeters of the differential strips for the whole span of the conoid, an unexpected solution emerges through the newly found number psi (ψ) which seems like a refined approximation to the third decimal of Pi but derived from a definite integral equation. As the strips are in truth slanted in the symmetry axis, their width is not uniform and we need to perform some adjustments in order to complete the problem with sufficient precision, but this is discussed separately in the annexes. Relevant implications for mathematical symmetry applied to multifarious architectural and engineering forms can be derived from this finding
Keywords
conoid; ellipse; Ramanujan; calculus of surface areas; number Psi; number Pi; 3D-construction of complex geometries; Engineering Design Objects; Architectural Forms
Subject
Computer Science and Mathematics, Geometry and Topology
Copyright:
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
