Dependencies

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 \((V, E) \in \mathrm{Dom_0}\).

The curve \(E\) is singular if and only if the two circles are tangent to each other

If \(k = 0\), \(f\) becomes a constant function

When \(k \ne 0\), \(f\) is a bijection between \(E_0\) and \(\mathrm{Dom_0}\)

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 \(k^2 = (u + r)^2 - 1\)

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

• \((x-uz)^2 + y^2 = r^2z^2\)

• \(a^2 + b^2 = c^2\)

• \(ax+by=cz\)

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.

For \(k \ne 0\), define

When \(k \ne 0\), for all \((X, Y) \in E_0\),

\(E\) is an affine cubic curve defined by the following equation, plus a point at infinity as the group identity

\(E_0\) is the set of non-singular points of \(E\) including the point at infinity, which forms an abelian group.

There are at most two possible values for \(E\) for a fixed \(V\) such that \((V, E)\in \mathrm{Dom}\)

There are at most two possible values for \(V\) for a fixed \(E\) such that \((V, E)\in \mathrm{Dom}\)

For all \(p \in E_0\),

For all \(p \in E_0\), \(n \in \mathbb {N}\),

When \(k \ne 0\), for all \((V, E) \in \mathrm{Dom_0}\),

\(f\) is injective when \(k \ne 0\).

For all \(p \in E_0\),

For all \(p \in E_0\),

A finite circular sequence of points, or vertices is called a proper polygon if the following conditions hold

• Two adjacent vertices do not coincide.

• Two adjacent edges (the affine span of two adjacent vertices) do not lie on the same line

Given a two-circle configuration in field \(K\) with \(k = 0\), 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 \((V, E) \in \mathrm{Dom_0}\).

When \(k = 0\), \((\mathrm{next})^n(V, E) = (V, E)\) for \((V, E) \in \mathrm{Dom_0}\) if and only if \(n \cdot g = 0\).

If \(k \ne 0\), \(f(X, Y) \in \mathrm{dom_0}\)

\(\mathrm{next}: \mathrm{Dom} \to \mathrm{Dom}\) sends a pair of vertex and edge to the next one on the polygon.

\(\mathrm{next}\) is a bijection from \(\mathrm{Dom}\) to itself.

\(\mathrm{next}(V, E) = (V, E)\) 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 \((V, E) = ([-1 : 0 : 1], [-1 : 0 : 1])\).

where the subtraction follows the group law of \(E\)

Given two concentric circles \(I\) and \(O\) in Euclidean plane where \(I\) is inside \(O\), if there exists a proper \(n\)-sided polygon simultaneously inscribed in \(O\) and circumscribed around \(I\), then from any poing \(P\) on \(O\), one can draw another proper \(n\)-sided polygon with the same property.

\(\mathrm{rChord}: \mathrm{Dom} \to \mathrm{Dom}\) keeps the vertex component, and sends the edge to the other one associated with the vertex:

where

\(\mathrm{rChord}\) is a bijection from \(\mathrm{Dom}\) to itself.

\(\mathrm{rChord}\) 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.

\(\mathrm{rChord}\) is an involution.

\(\mathrm{rPoint}: \mathrm{Dom} \to \mathrm{Dom}\) keeps the edge component, and sends the vertex to the other one associated with the edge:

where

\(\mathrm{rPoint}\) is a bijection from \(\mathrm{Dom}\) to itself.

\(\mathrm{rPoint}\) maps a vertex to itself if and only if the input edge is a shared tangent to Inner and Outer.

\(\mathrm{rPoint}\) is an involution.

The curve \(E\) contains a singular point only in the following 4 cases

• \(u + r = \pm 1\) (positive singular), and the singular point is \((-u / r, 0)\)

• \(u - r = \pm 1\) (negative singular), and the singular point is \((u / r, 0)\)

The two-circle configuration over \(\mathbb {R}\) corresponding to a Euclidean configuration has the following parameters

• \(u = \lVert O_{center} I_{center}\rVert / I_{radius}\)

• \(r = O_{radius}/ I_{radius}\)

• \(k = \sqrt{(u + r)^2 - 1}\)