On the Pentacardioid : Abstract of a Dissertation Submitted to the Board of University Studies of the Johns Hopkins University in Conformity With the Requirements for the Degree of Doctor of Philosophy (Classic Reprint)

Edward C. Phillips

ISBN 10: 1332173063 / ISBN 13: 9781332173068

Published by Forgotten Books, 2015

About this Item

Zustand: Sehr gut | Sprache: Englisch | Produktart: Bücher | Keine Beschreibung verfügbar.

Seller Inventory # 26067400/2

Bibliographic Details

• Title: On the Pentacardioid : Abstract of a Dissertation Submitted to the Board of University Studies of the Johns Hopkins University in Conformity With the Requirements for the Degree of Doctor of Philosophy (Classic Reprint)
• Publisher: Forgotten Books
• Publication Date: 2015
• Language: English
• ISBN 10: 1332173063
• ISBN 13: 9781332173068
• Binding: Soft cover
• Condition: Sehr gut
• Seller catalogs:
  • Bücher

About this title

Synopsis

This book explores the Pentacardioid, the first significant curve in a series examined by Professor Morley. It begins by establishing the general symbol for an Ennacardioid as Cn, then using the corresponding symbol C5 for the Pentacardioid. The author employs a coordinate system and analysis methods established by Professor Morley to lay the groundwork for further exploration. In establishing a parameter, the author highlights the existence of three independent complex arbitrary constants which are dependent on six conditions. This allows the shape of the C5 to be determined by only two absolute conditions. The author explains construction methods for the C5, demonstrating how to build up the curve from two fundamental epicycloids which are easily constructed with simple mechanical means. The author also provides instructions on how to alter the relative sizes of the fundamental curves, and how to alter the amplitude of the determining coefficient of the resulting curve. The book delves into the singularities of the C5, which are shown to include six simple double points, three cusps, and six double tangents, for a total of 21 double points. It is also shown that the C5 is of class five and order eight. The cusps are further examined, with a self-conjugate cubic equation used to determine cusp parameters. Conditions for a node on a curve are discussed, and the author establishes the general condition which the parameters of a node must satisfy. The author provides instructions on how to calculate the twelve parameters of the six double tangents. Finally, the author examines the complete system of C5s, considering sets of C5s which are connected together by special properties or common singularities. The author restricts the determining coefficient to a third portion of the plane bounded by a pair of straight lines or rays through the origin, then considers the behavior of the three cusps when one of the cusp parameters is fixed. A graphical method is presented for determining the reality of the singularities of the C5, and a table of regions is provided which lists the number of simple singularities possessed by a given C5 based on the region in which the coefficient s1 of the given C5 happens to lie. This book provides a comprehensive examination of the Pentacardioid, exploring its equation, construction, singularities, and complete system, offering valuable insights into the mathematical properties of this fascinating curve.