Articles on maths

and related tropics

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Here are articles on various topics which have taken my interest over the years.

They are all simple explanations and explorations with lots of examples. There is no original research, but the accounts are original.

A review of computer models of trees, with an introduction to basic tree botany, fractal trees, L-systems, functional-structural models based on the plant as a system of biological functions and structures, to CGI applications including Unreal Engine, Houdini, and SpeedTree.

Photogrammetry. Using matrices to calculate the positions of objects in a scene from overlapping photographs, plus other aspects of computer vision including SIFT and RANSAC algorithms.

Articles about computing eigenvalues and the statistics of random matrices

The article on perspective includes a section on matrices and quaternions for calculating rotations.

Galois theory. An informal account of the main idea with copious examples.

Algorithms for factorising large polynomials, reviewing basic background including congruences and Fermat's little theorem, and explaining with many examples how the Distinct Degree, Berlekamp and Cantor-Zassenhaus algorithms work, without going into details about computing times.

Sums of powers of integers and the Euler-Maclaurin summation formula

A revision article on Fourier series and transforms. In it I give an illustrated account of the representation of periodic functions by Fourier series, and extend this to the Fourier transform of a non-periodic continuous function. Then I illustrate sampling at discrete data points and the discrete Fourier transform. There is a brief discussion of convolution and deconvolution with relevance to recovering true signals from signals mixed with noise. An appendix describes in some detail the fast Fourier Transform algorithm.

Continued fractions . a comprehensive but non-specialist account

The shapes of musical phrases.

Electrostatic fields in rectangular and cylindrical symmetry, related to electron lens design

Peculiar oscillations of a flat-bottomed object

Sound radiation from a vibrating object

Vibrations and acoustics of violins and cellos

The `sofa constant' : the largest object which can be manoeuvred round a tight corner in a narrow corridor.

Calculating the area of an arbitrary plane figure by counting grid points inside the figure.

The astroid curve and related ellipses.

Inscribed circles and ellipses

Trigonometry of the spherical triangle.

Drawing in linear and non-linear perspective. Showing how simple shapes including an ellipse project onto the picture plane, and how series of rotations change the orientation of objects.

Group theory

Some 'teach-yourself' interactive computer programs with user notes