## Inversion in a single circle

You might think that discussing the inversion in a circle is somewhat underwhelming. But, as I am using multiple inversion in many circles to create fractal images, I found that there are some important ideas you will not find so easily. I will present them briefly here and use them in the next posts.

Inversion in a circle is nearly the same as a mirror image at a straight line. Actually, in a stereographic projection of a sphere and in the Poincaré disc representation of hyperbolic space, most geodesic lines are represented by circles, see my earlier post “Straight lines in elliptic and hyperbolic space“. A mirror image in these geodesic lines thus becomes an inversion in the corresponding circle. You can use this to create Kaleidoscopes for elliptic and hyperbolic space.

Any circle can be transformed to a straight line by an inversion in a suitably chosen second circle. The inversion in the first circle then becomes a mirror image at the straight line. This is useful to get a systematic overview of the images resulting from multiple inversions in several circles.

I will show you some illustrations of the above. But first I am briefly repeating some well-known details. For a circle with radius R and centre at position c, it is easy to calculate the inverted image q of a point p using

This is as easy to calculate as a mirror image at a straight line and it has similar properties. Points on the circle are fixed points. Points inside the circle are mapped to the outside and inversely. However, inversion maps the centre of the circle to infinity. We get an intuitive image of inversion in a circle if the circle is the stereographic projection of a great circle of the Riemann sphere. Then the inversion is simply a mirror image at the plane containing the great circle and exchanges the corresponding halves of the sphere.

I do not want to discuss more details – I’d rather prefer to show how inversion acts on an image. This is an example:

The inverting circle is shown as a blue line. The photo of a butterfly is directly mapped into its inside. The outside shows an inversion of the inside. It results from inverting the position of each pixel in the circle and looking up the colour at the inverted position. This image results from the browser app at http://geometricolor.ch/singleCircleApp.html. Use it to learn more about inversions in circles making your own experiments with other images. You can see we have indeed an approximate mirror symmetry close to the circle. Note the strong distortions far away from the circle.

In spite of the strong distortions the inversion in a circle is very much the same as a mirror image at a straight line. You can see this in the next image:

Here, the centre of the red circle lies on the blue circle. Thus an inversion in the red circle maps the blue circle into a straight line, which is the orange line. The inversion in the blue circle, as seen in the first image, now becomes a mirror image at the orange line. Here, you can see that details of the flower and the head of the butterfly are mirrored in the same way as in the previous image. You cannot see how one of the wings intersects the orange line and gets mirrored because the image is too small. Again, this is a result of http://geometricolor.ch/singleCircleApp.html and you can do your own experiments.

For the following posts, the inversion in a circle is both a building block for creating images and an isometry of non-Euclidean planes. It simplifies systematic searches similarly as scaling, rotation, translation and mirror symmetries.

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