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.