There is ongoing research into the number of distinct ways a certain number of circles can intersect. In this problem there are a number of criteria. Two intersections are considered distinct if the circles cannot be smoothly transformed into each other without crossing other circles. Also circles that intersect must intersect at two distinct points.

I intend to take multiple approaches to attempt an answer to this question. One approach is to rigorously describe the posible orientations of circles at each count and use the formalization to categorize the arrangments.

We will start with a single circle. A circle has three parameters: the x coordinate of its center, the y coordinate of its center and its radius. The coordinates belong to the entire real line will the radius must be positive. This space of circles is equivalent to half of the 3d hyperplane. We will, however, not represent a single circle by these three parameters.

Consider an arrangement of three circles. Notice that translating, rotating and scaling all the circles does not affect the intersections. We can treat all arrangements that can be transformed in these ways into each other as equivalent. We can, therefore, choose one of them to represent the whole set. We will choose one of the circles and designate it $C_0$. We can then translate the circles so that the center of $C_0$ is at the origin. We will also scale the circles so that $C_0$ has a radius of 1. We will choose the largest circle as our $C_0$.

If we apply that operation to the case of a single circle the parameters are no longer free and our space of circles goes from 3 dimensions to 0 dimensions.

moving on to two circles, we gain three more parameters for the center and radius of the next circle. Since our arrangement has a unit circle at the origin it is mor natural to specify coordinates in polar form.

Let us again consider the case of three circles. What we want is a canonical transformation the represent the whole class of equivalent arangements but currently the rotation of the arrangement is free. We will therefore designate the second largest circle as $C_1$ and enforce that $C_1$ must have its center on the positive x axis.

This constrains the orientation of most arrangements, however the case of $C_1$ having its center at the origin makes the rotation arbitrary. We can fix this by first indexing all circles by their radius: the largest circle becomes $C_0$, the next largest becomes $C_1$, and the $C_2$, etc. Then we can change the restriction to specify that the circle with the lowest index whose center is not at the origin must have its center on the positive x axis. We will denote the center coordinates of the circle $C_i$ as $c_i = ($x_i, y_i) = ($\rho_i, \theta_i)$ and its radius as $r_i$

Returning to two circles, with our additional constraint the angle coordinate of the second circle's center is constrained leaving only the radial component, $\rho_1$, and the radius, $r_1$, two degrees of freedom.

Now we can partition the space into distinct arrangements. There are three categories: the second circle is outside, inside or intersecting the first. The second circle is inside if $\rho_1 + r_1 < 1$, outside if $\rho_1 - r_1 > 1$ and intersects otherwise ($\rho_1 + r_1 > 1 \vee \rho_1 - r_1 < 1$).

For increasing numbers of circles we will parameterize the sub spaces belonging to the different categories. This will create a tree of categories. We will see, however, that different leaves in the tree will be equivalent so we will need tocheck if subspaces are equal or if they intersect.