Circular Cubics Research Articles

Pivot-isogonal cubics

We discuss general properties of pivotal cubics related to the isogonal transformation. We prove that circular cubics are pivotal and self-isogonal.

A classification and construction of entirely circular cubics in the hyperbolic plane

If each intersection point of a third order curve with the absolute conic of the hyperbolic plane is a tangential point, this curve will be called an entirely circular cubic. According to this definition a rough classification of such curves is given into four main types and nine sub-types. Some of them are constructed by a (1,2) or (1,1) mapping and the others are constructed by the generalized quadratic hyperbolic inversion. Thus we extend and complete Palman's paper [5] in a synthetic way.

Geometrical properties of some Euler and circular cubics. Part 2

This paper is a continuation of Part 1, previously published under the same title. The reader is asked to refer back to Part 1 for the Glossary and references.

Geometrical properties of some Euler and circular cubics. Part 1

This sequel to our earlier paper (1995) continues the investigation of the Euler cubic curves therein defined, with particular reference to perspectivities and associated conics. Study of the circular cubic in this pencil, the Neuberg cubic, brings with it some discussion of the properties of circular cubics in general.

Center-point Curves Through Six Arbitrary Points

A circular cubic curve called a center-point curve is central to kinematic synthesis of a planar 4R linkage that moves a rigid body through four specified planar positions. In this paper, we show the set of circle-point curves is a non-linear subset of the set of circular cubics. In general, seven arbitrary points define a circular cubic curve; in contrast, we find that a center-point curve is defined by six arbitrary points. Furthermore, as many as three different center-point curves may pass through these six points. Having defined the curve without identifying any positions, we then show how to determine sets of four positions that generate the given center-point curve.

Classroom Note:Horizontal Circular Curves and Cubics

Horizontal circular curves with a central angle not larger than 180 degrees or not less than 180 degrees can be completely determined when two out of seven parameters are given. A cubic can be used to solve the cases when the following are given: (1) the middle ordinate and the tangent distance or (2) the external distance and the long chord.

A Note on Two-Circuited Circular Cubics and Bicircular Quartics

A Note on Two-Circuited Circular Cubics and Bicircular Quartics

A General Construction for Circular Cubics

A General Construction for Circular Cubics

A General Construction for Circular Cubics

A General Construction for Circular Cubics

Note on the Researches of Maclaurin on Circular Cubics

In the works known to us which contain historical or bibliographical information about circular cubics no sufficient indication of the researches of Maclaurin is found. Yet many of the classic propositions connected with these curves are due to this eminent geometer, who investigated constructions for unicursal circular cubics, for certain non-unicursal circular cubics, and for the special circular cubics now known as the trisectrix of Maclaurin, the oblique cissoid, and the strophoid. His name does not appear even in the list of writings on the strophoid published by Tortolini and Günther.

Further Investigations on Circular Cubics and Bi-Circular Quartics

Last session (1910–11) I contributed two papers investigating the Focal properties of Circular Cubics and Bi-Circular Quartics by the methods of pure geometry. I find on further investigation that not only can the properties of the individual Focal Circles be directly found by the method there given, but the mutual properties of the Focal Circles can be obtained by an extension of the same method. Several geometrical results, which as far as I know are new, are obtained incidentally in the discussion.

On the locus of the foci of conics having double contact with two fixed conics

THE system of conies having double contact with two conies includes as special cases the two systems of conies passing through four fixed points and touching four fixed lines, respectively. These latter result from making the fixed conies break up into factors, in the first case, in the tangential sense, and in the second case directly. Thus the general case includes and combines together into one whole the properties of the two special cases. Now I propose to consider the locus of the foci of the general system. This must include as special cases the known loci in the case of conies through four fixed points and touching four fixed lines. The first is a curve of the sixth order, which, when the points are coneyclic, as has been shown by Professor Sylvester, breaks up into the two circular cubics having the points for foci, and the second is a circular cubic having its double focus on itself and passing through all the intersections of the four lines. Now foci, being the intersections of tangents drawn to the curve from the imaginary points at infinity, are most easily treated by tangential coordinates, which I consequently use, the equations being precisely the same as the direct ones ; viz. as given by Salmon, a conic having double contact with S, S is, if $ JcS = BF;

On Theorems relating to the Circular Cubics Which are the Inverses of a Lemniscate with respect to its Vertices

On Theorems relating to the Circular Cubics Which are the Inverses of a Lemniscate with respect to its Vertices

On an Involution System of Circular Cubics, and description of the curve by points, when the double focus is on the curve

Proceedings of the London Mathematical SocietyVolume s1-1, Issue 1 p. 96-98 Articles On an Involution System of Circular Cubics, and description of the curve by points, when the double focus is on the curve T. Cotterill M.A., T. Cotterill M.A.Search for more papers by this author T. Cotterill M.A., T. Cotterill M.A.Search for more papers by this author First published: January 1865 https://doi.org/10.1112/plms/s1-1.1.12AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat No abstract is available for this article. Volumes1-1, Issue1January 1865Pages 96-98 RelatedInformation