Articles on maths, computer graphics

and related topics

A review of computer modeling 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.

Application of CGI to re-create the 1892 Victorian railway bridge at Elvet, Durham, showing in detail probably how it was constructed. The foundations, brickwork and ironwork were modelled with Blender 4.5 and the human figures with Daz Studio. The video is on YouTube via this link.

Chess game: an amusing CGI video, modelled in Blender, of an actual game played in 1912.

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.

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.

Algorithms for computing the eigenvalues of a matrix

Statistics of the eigenvalues of random matrices

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

Galois theory. The link between symmetry groups and the solution of polynomial equations. An informal account of the main ideas 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

Continued fractions . a comprehensive but non-specialist account

The shapes of musical phrases.

Mechanics, Acoustics,

Waves &

Laplace's equation

Electrostatic fields in rectangular and cylindrical symmetry, related to electron lens design. Includes calculation of the charge distribution and capacitance of several simple shapes including the conductiing strip, ring and annulus.

Peculiar oscillations of a flat-bottomed object -- it oscillates more rapidly as it slows down.

Sound radiation from a vibrating object -- a general overview of the physics

Ultrasonic NDT: signals from rough facetted cracks. A cumputational model I developed in the 1980s

Vibrations and acoustics of violins and cellos

Review of normal modes and resonance, with application to a 'cello.

The Helmholtz resonance of violin-family instruments

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

Calculating the area of an arbitrary plane figure by counting the number of points on a square grid which lie inside the figure.

The astroid curve and related ellipses.

Inscribed circles and ellipses -- drawn within triangles and pentagons

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.

Some 'teach-yourself' interactive computer programs with user comprehensive notes. Two parallel programs which create `words' like aaab or permutations to represent group elements. The programs are a simple calculator to try creating groups for yourself. The programs and notes together form a teach-yourself course in the theory of finite groups, including their reptresentation by matrices and their character tables.

Sequence and series

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.

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

An Introduction to Mathematical Analysis’ a textbook by my friend Dr John Reade, first published in 1986 by Oxford University Press.

Puzzles & problems

Over 30 problems in maths and mechanics, and my answers.

Integral calculus

An investigation into many definite integrals involving the logarithm, ln(1+x^n), divided by a polynomial, such as int_0^inf ln(x^n+1)/(x^3+1). The ntegration uses contours within the complex plane, with limits taken around poles. It may be of interest to students learning complex integration.