2 Two circles
2.1 Configuration
We will first focus on a special case of the theorem: two circles. This later will turn out to be a good model for the majority of general cases. To make it even simpler, we will fix Inner to be the unit circle, and restrict the center of Outer on the x-axis
where \(u\) and \(r\) are the only two free parameters we are allowed to change. As mentioned before, all variables are in a field \(K\) with characteristic 0. We will also assume the existence of a constant \(k\) satisfying a degree-2 equation
which will be used in the next chapter. We can always consider the algebraic closure of \(K\) to ensure the existence of \(k\). However, even if we restrict ourselves in \(\mathbb {R}\), we still can find such \(k\) for all useful cases: depending on the sign we choose for \(r\), \(u + r\) represents either the left-most or the right-most point of Outer, so to say \(k\) exists, we need to ensure at least one of the two points is not inside Inner. Indeed, if both points are inside Inner, then Outer is completely enclosed by Inner in the real plane, making the polygon impossible to form.
Finally, we don’t allow \(u = 0\) (concentric circles) or \(r = 0\) (degenerate Outer). We will discuss these cases in a later chapter
We encode these in the following definition
A two-circle configuration in field \(K\) with characteristic 0 is a tuple \((u, r, k)\) such that \(u \ne 0\), \(r \ne 0\), and \(k^2 = (u + r)^2 - 1\)
We call a configuration singular if the two circles are tangent to each other. That is, \(u + r = \pm 1\) or \(u - r = \pm 1\). We further categorize \(u + r = \pm 1\) as positive-singular, and \(u - r = \pm 1\) as negative-singular. This distinction is inconsequential geometrically, as we can change the sign of \(r\) while preserving the geometry, but this will show algebraic significance later.
Moving on to vertices and edges, we consider the pair
where \([x : y : z]\) is the projective coordinates of a point, and \([a : b : c]\) is the projective coordinates of a line \(ax+by=cz\). We define a subset \(Dom\) of \(\mathbf{P}(2, K) \times \mathbf{P}(2, K)\) that represents legal vertex-edge pairs for the configuration
The subset domain (\(\mathrm{Dom}\)) of \(\mathbf{P}(2, K) \times \mathbf{P}(2, K)\) consists of pairs \(([x : y : z], [a : b : c])\) such that
\((x-uz)^2 + y^2 = r^2z^2\)
\(a^2 + b^2 = c^2\)
\(ax+by=cz\)
This means that the vertex is on Outer, the edge is tangent to Inner, and the vertex is on the line of the edge. We also effectively defined directed edges: while each edge has two associated vertices, we specify one of them as the "starting" vertices. Dually, this also assign a direction to vertices: we specify the "forward" edge among the two edges associated to the vertex.
As an immediate observation, we characterize the cardinality of \(\mathrm{Dom}\) when fixing one of the two variables
There are at most two possible values for \(E\) for a fixed \(V\) such that \((V, E)\in \mathrm{Dom}\)
Once \(V\) is fixed, one can reduce the system of equations to a quadratic equation of one of the affine coordinate of \(E\), which has at most two distinct roots, and other coordinates of \(E\) are linear to the root. Several edge cases for potentially including point at infinity needs to be considered, but it doesn’t add addition possible values beyond two.
There are at most two possible values for \(V\) for a fixed \(E\) such that \((V, E)\in \mathrm{Dom}\)
Same proof as the previous one
These are effectively stating that one can draw at most two tangents from a point to a circle, and there are at most two intersections between a line and a cicle.
2.2 Operation
We can think of the process of forming a polygon as iteratively generating new elements in \(\mathrm{Dom}\) from previous ones, where the new starting vertex is the ending vertex of the previous edge, and the new forward edge is the edge associated with the new starting vertex, different from the previous edge. We use the function \(\mathrm{next}: \mathrm{Dom} \to \mathrm{Dom}\) to represent a such iteration.
We first recognize that \(\mathrm{next}\) is a composition of two functions \(\mathrm{rPoint}\) (reflect point) and \(\mathrm{rChord}\) (reflect chord).
\(\mathrm{rPoint}: \mathrm{Dom} \to \mathrm{Dom}\) keeps the edge component, and sends the vertex to the other one associated with the edge:
where
\(\mathrm{rChord}: \mathrm{Dom} \to \mathrm{Dom}\) keeps the vertex component, and sends the edge to the other one associated with the vertex:
where
Despite the case split, these are in fact rational functions on the \(Outer \times Inner\) as an algebraic curve. The special casing is to patch up \(0/0\). This will be a common theme in more functions to be defined.
We verify that these are legitimate by proving their characterizing lemma
\(\mathrm{rPoint}\) is an involution.
Expand the function composition and verify the equality for all cases
\(\mathrm{rPoint}\) is a bijection from \(\mathrm{Dom}\) to itself.
Trivial from being an involution after verifying the domain and the range
\(\mathrm{rPoint}\) maps a vertex to itself if and only if the input edge is a shared tangent to Inner and Outer.
The formula of \(\mathrm{rPoint}\) is constructed by calculating a root of a quadratic equation when given another root. Two roots coincide if and only if the discriminant vanishes. The rest is verifying the equations.
\(\mathrm{rChord}\) is an involution.
Expand the function composition and verify the equality for all cases
\(\mathrm{rChord}\) is a bijection from \(\mathrm{Dom}\) to itself.
Trivial from being an involution after verifying the domain and the range
\(\mathrm{rChord}\) maps an edge to itself if and only if the input vertex is on Inner. In other words, it is an intersection of Inner and Outer.
The formula of \(\mathrm{rChord}\) is constructed by calculating a root of a quadratic equation when given another root. Two roots coincide if and only if the discriminant vanishes. The rest is verifying the equations.
Using these, we define the function \(\mathrm{next}\)
\(\mathrm{next}: \mathrm{Dom} \to \mathrm{Dom}\) sends a pair of vertex and edge to the next one on the polygon.
It is obvious that
\(\mathrm{next}\) is a bijection from \(\mathrm{Dom}\) to itself.
\(\mathrm{next}\) is a composition of two bijections
We call \((V, E)\) singular if \(\mathrm{next}(V, E) = (V, E)\). This is important because applying \(\mathrm{next}\) on a singular point repeatedly will result in a degenerate polygon where all vertices coincide. Obviously this doesn’t tell us how other vertices behaves, so we will need to exclude it from the theorem. We characterize singular points with the following lemma
\(\mathrm{next}(V, E) = (V, E)\) if and only if the \(V\) is an intersection and \(E\) is a shared tangent to Inner and Outer. Consequently, Inner and Outer must be tangent to each other at \((V, E) = ([1 : 0 : 1], [1 : 0 : 1])\) or at \((V, E) = ([-1 : 0 : 1], [-1 : 0 : 1])\).
The first part is directly from the property of 9 and ??. One can then solve the system of equations for \(V\) and \(E\) to get the coordinates.
This tells us that the singular point only exists for a singular configuration, is unique if exists, and is precisely the tangent point. This allows us to define the domain of non-singular points:
The subset \(\mathrm{Dom_0} \subset \mathbf{P}(2, K) \times \mathbf{P}(2, K)\) is the set \(\mathrm{Dom}\) with \(([1 : 0 : 1], [1 : 0 : 1])\) and \(([-1 : 0 : 1], [-1 : 0 : 1])\) excluded.
We now give the statement of a version of Poncelet’s closure theorem
Given a two-circle configuration in field \(K\), if for a natural number \(n\), and some \((V, E) \in \mathrm{Dom_0}\) it holds that \((\mathrm{next})^n(V, E) = (V, E)\), then the same holds for the same \(n\) and all \((V, E) \in \mathrm{Dom_0}\).