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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
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.
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- Copyright © Mathematical Association 1937
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.
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