Circular Inversion – Reflecting in a Circle
This topic is a great introduction to the idea of mapping – where one point is mapped to another. This is a really useful geometrical tool as it allows complex shapes to be transformed into isomorphic (equivalent) shapes which can sometimes be easier to understand and work with mathematically.
One example of a mapping is a circular inversion. The inversion rule maps a point P onto a point P’ according to the rule:
OP x OP’ = r2
To understand this, we start with a circle radius r centred on O. The inversion therefore means that the distance from O to P multiplied by the distance from O to P’ will always give the constant value r2
This is an example of the circular inversion of the point A to the point A’.
We have the distance of OA as √2 and the radius of the circle as 2. Therefore using the formula we can find OA’ by:
OA’ = r2 / OA = 4/√2
OA’ = 21.5. This means that the point A’ is a distance of 21.5 away from O on the same line as OA.
We can check that the Geogebra plot is correct – because this point A’ is plotted at (2,2) – which is indeed (using Pythagoras) a distance of 21.5 from O.
A point near to the edge of the circle will have an inversion also close to the circle
A point near to the centre will have an inversion a long way from the circle. The point (0,0) will be undefined as no point outside the circle will satisfy the inversion equation.
So, that is the basic idea behind circular inversion – though it gets a lot more interesting when we start inverting shapes rather than just points.
Circles through the origin map onto straight lines to infinity (see above).
Circles centred on the origin map to other circles centred on the origin (above).
Ellipses create these shapes (above).
The straight line through A B maps to a circle through the origin (above).
The solid triangle ABC maps to the pink region (above).
The solid square ABCD maps to the pink region (above).
These shapes can all be explored using the reflect object in circle button on Geogebra.
It is possible to extend the formula to 3 dimensions to give spherical inversion:
The above image is a 3D human head inverted in a sphere (from the Space Symmetry Structures website. There’s lots to explore on this topic – it’s a good example of how art can be mathematically generated, as well as introducing isomorphic structures.
If you enjoyed this post you might also like:
The Riemann Sphere : Another form of mapping using spheres is the Riemann Sphere – which is a way of mapping the entire complex plane to the surface of a sphere.
Fractals, Mandelbrot and the Koch Snowflake. Using maths to model infinite patterns.