## IGCSE Transformations – Rotation

Rotations in space make good Saturday morning maths at the King’s School.

## Great Circles and Spherical Triangles

In the geometry of mariners, where straight lines are Great Circles, the angle sum of a triangle is more than 180 degrees.

This method for finding the centre, and by extension the equation, of a circle given three non-colinear points, brings the ancient textbook master Euclid onto Descartes’ coordinate plane and right  into the 21 century classroom.

The sine function starts life in a triangle (year 9?) then becomes intimate with a circle at A level, as y-coord of a point rotating on a unit circle.  Later it is about infinite series or, even, differential equations.

Have got absorbed by the National Cipher Challenge.  Not sure what this animation varying the multiplier in an ‘Affine Shift’ add 5.  Functional grid shows transition from Caesar Shift 5, multiply by 1, through to multiply by 13, which is a useless cipher.  In fact any multiplier not co-prime with 26 can be seen to be not one-one, and therefore useless too.  Modular arithmetic, straight line graphs, ciphers, dotty patterns, ….