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Singular matrices and pairwise-tangent circles

Published online by Cambridge University Press:  15 February 2024

A. F. Beardon
A. F. Beardon*
Affiliation:
Centre for Mathematical Sciences University of Cambridge, Wilberforce Road, Cambridge CB3 0WB e-mail: afb@dpmms.cam.ac.uk
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Extract

The idea of using the generalised inverse of a singular matrix A to solve the matrix equation Ax = b has been discussed in the earlier papers [1, 2, 3, 4] in the Gazette. Here we discuss three simple geometric questions which are of interest in their own right, and which illustrate the use of the generalised inverse of a matrix. The three questions are about polygons and circles in the Euclidean plane. We need not assume that a polygon is a simple closed curve, nor that it is convex: indeed, abstractly, a polygon is just a finite sequence (v1, …, vn) of its distinct, consecutive, vertices. It is convenient to let vn + 1 = v1 and (later) Cn + 1 = C1.

Type
Articles
Information
The Mathematical Gazette , Volume 108 , Issue 571 , March 2024 , pp. 12 - 19
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Copyright
© The Authors, 2024 Published by Cambridge University Press on behalf of The Mathematical Association

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References

James, M., The generalised inverse, Math. Gaz. 62 (June 1978) pp. 109–114.CrossRefGoogle Scholar
Glencross, M. J., Calculating generalised inverse matrices, Math. Gaz. 63 (October 1979) pp. 173–175.CrossRefGoogle Scholar
Wong, E. T., Generalised inverses as linear transformations, Math. Gaz. 63 (October 1979) pp. 176–181.CrossRefGoogle Scholar
Planitz, M., Inconsistent systems of linear equations, Math. Gaz. 63 (October 1979) pp. 181–185.CrossRefGoogle Scholar
Penner, R. C., The decorated Teichmüller space of punctured surfaces, Commun. Math. Phys. 113 (1987) pp. 299–339.CrossRefGoogle Scholar
Pinelis, I., Cyclic polygons with given edge lengths: existence and uniqueness, J. Geometry 82 (2005) pp. 156–171.CrossRefGoogle Scholar
Kouǐimshá, H., Skuppin, L. and Springborn, B., A variational principle for cyclic polygons with prescribed edge lengths, Advances in discrete differential geometry, A. I. Bobenko (ed.), (2016) pp. 177–195.Google Scholar
Beardon, A. F. Pitot’s theorem, dynamic geometry and conics, Math. Gaz. 105 (March 2021) pp. 52–60 CrossRefGoogle Scholar