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On the Representation of Circles by means of Points in Space of Three Dimensions

Published online by Cambridge University Press:  03 November 2016

D. Pedoe
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Extract

The idea of the representation of a linear system of plane curves by means of a flat space is of fundamental importance in algebraic geometry. In this paper some details of the representation of circles are worked out, and an attempt is made to give a three-dimensional picture of certain parts of circle geometry The theorems given in italics are thought to be new, although in a subject with such an enormous literature it might almost be said that no theorem can be a new one.

Type
Research Article
Information
The Mathematical Gazette , Volume 21 , Issue 244 , July 1937 , pp. 210 - 215
Copyright
Copyright © Mathematical Association 1937

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References

Page no 213 note * See remark at end of paper.

Page no 214 note * This point is on the locus, for one circle of the coaxal system is orthogonal to D, and the inverse of D in this circle is D itself.

Page no 215 note * The reader may verify, that if the fixed circle be taken as

x 2+y 2+2gx+2fy+d=0,

and the coaxal system as x 2+y 2+2λx+c 2=0, the enveope is

(x 2+y 2+2λy−c 2)(x 2+y 2+2μy−c2)=0,

When λ, μ are the roots of

4t 2(d−g 2)−4ft(d−c 2)+(c 2+d)2−4c 2(f 2+g 2)=0.