Reflections

By Terence Harris, The University of New South Wales

In everyday life, images can be seen reflecting off the surface of a lake, a tinted window, or more commonly, through a mirror. Most mirrors are flat, making it easier for us to see ourselves; we know that to achieve this we need to stand perpendicular to the face of the mirror. In mathematics, points can be reflected over a line in a similar way, but they can also be reflected over more general curves. For example, every point on the inside of a circle can be a reflected onto the outside,  with the centre of the circle being the only point reflected “to infinity”.

It turns out there are certain restrictions on a curve which characterise how unpredictable a reflection over it can be, and how much the reflection can distort shapes. A reflection over a parabola, for example, has no limit on how much it can distort the ratio of side lengths in a rectangle. This is not the case for an ordinary mirror, which doesn’t change the shape of a rectangle at all. Unlike a parabola, a reflection over a hyperbola can’t distort shapes too much either.

Another problem a curve may have is a sharp corner. A reflection over a curve with a sharp corner does not behave in a predictable way at points close to the corner. If there are no corners however, the reflection will just behave like a flat mirror at points very close to the curve. This is analogous to the fact that, even if a surface is curved, it appears flat if you “zoom in” far enough. A good example of this is the apparent flatness of the Earth’s surface.

My AMSI project involved characterising those curves which admit reflections that behave predictably at points close to the curve. It turned out that a slightly stronger assumption than the absence of sharp corners was needed.

 

Terrence Harris was one of the recipients of a 2014/15 AMSI Vacation Research Scholarship.

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