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The trisectrix and Langley's problem

Published online by Cambridge University Press:  24 February 2022

John R. Silvester*
Affiliation:
Department of Mathematics, King’s College Strand, London WC2R 2LS e-mail: john.silvester@cantab.net
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Extract

If a circle rolls without slipping around an equal fixed circle, then a point carried by the rolling circle traces out a limaçon of Pascal. (This is Etienne Pascal, father of Blaise. The word limaçon is derived from the Latin limax, a snail.) If the fixed and rolling circles have radius 1, and the point P carried by the rolling wheel is distant a from its centre, then for a > 1 the limaçon has an inner and an outer loop, joining up at a node. For a = 1 it has a cusp, and is then a cardioid, so-called because it is heart-shaped. See Figure 1, where we have plotted the cases a = $${3 \over 4}$$, a = 1 and a = $${3 \over 2}$$.

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© The Authors, 2022. Published by Cambridge University Press on behalf of The Mathematical Association

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