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Circular inversions II
There are some other interesting properties of circular inversions. One of which is that they preserve the “angle” between intersecting circles. Firstly, how can circles have an angle between them? Well, we draw 2 tangents to both the circles at the point of intersection, and then measure the angle between the 2 tangents:
Therefore we can see that the “angle” between these 2 circles is 59.85 degrees. If we then carry out a circular inversion we see the following:
The inversion has been done with regards to the black circle centred around the origin. The red and blue circles are mapped from outside the the black circle onto circles inside the black circle. Now if we do the same as before – by finding the 2 tangents at the point of intersection, we find that the angle has remained the same – it is still 59.85 degrees.
It is also possible to find circles which remain unchanged under the inversion. This happens when a circle is orthogonal (at a 90 degree angle) to the circle with which the inversion is being carried out.
The small circle has an angle of 90 degrees with the large circle, and therefore when we invert with respect to the large circle, we map the small circle onto itself.
The question is, why is all this useful? Well, an entire branch of mathematics (non-Euclidean geometry) is concerned with being able to map points in our traditional Euclidean worldview (the geometry of high school triangles, parallel lines and circle theorems) to different geometrical systems entirely. Circular inversion is a good introduction to this concept.
Also, circular inversion can sometimes make studying mathematical shapes easier to understand and explain. For example, (from Wolfram):
It would be very difficult to explain mathematically how the shape above is generated – whilst there are patterns, it is not obvious how to explain them. However, if we invert this shape through a circular inversion (with the circle at centre of the image) then we get the following:
This is the image inside the circle – and now we can clearly see the pattern behind the generated image. So, inversion has a lot of potential for simplifying geometrical problems.
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.
