Dear Karl, Your idea is not without merit:
> Wrap the slate with a reflective strip ... > Playing around with a laser should find the > focii. This is sometimes referred to as the "Elliptical Billiard Table Problem"... If you aim at a focus, the laser path will reflect through the other focus and so on. Eventually it will settle down into running backwards and forwards along the major axis BUT... If you aim the laser so that its path passes OUTSIDE the line joining the two foci, the path traced will, after an indefinite number of reflections, leave a dead area in the centre which is ITSELF an ellipse. If you aim the laser so that its path passes BETWEEN the two foci, the path traced will, after an indefinite number of bounces, leave two dead areas around each end of the major axis. The inner boundaries of these areas are the turning points of a hyperbola. The real excitement comes if you aim the laser so that you get a return to the starting point after a finite number of reflections. You then get a nice pretty pattern. I have knocked up the attached example where there are 46 reflections. You can prove all this using Projective Geometry. This is a delightful subject which includes splendid concepts such as "The Circular Points at Infinity". In the 1950's, Projective Geometry was in the UK A-level Mathematics syllabus and taught to 17- and 18-year olds. As far as I know, these days, this subject isn't taught ANYWHERE in the UK even in Universities. Geometry is deemed a useless subject because "you don't really need it". End of rant. Frank
Ellipse.pdf
Description: Ellipse.pdf
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