4 Euclidean plane
In this chapter, we prove a version of Poncelet’s closure theorem in real Euclidean plane. We still restrict ourselves to only two circles, but now they no longer have fixed coordinates and radius in the affine plane. We will only discuss the case where the circle Inner is completely incide Outer, as in other cases, the it is possible to get polygon with degenerate vertices or edges, creating difficulty in Euclidean geometry. After ruling out degenerate cases, we represents polygons as a finite circular sequence of points, and require them to be proper
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
The whole proof amounts to translating points and edges between Euclidean plane and the affine portion of the two-circle configuration. Before doing so, we first recognize that we includes a special case in the Euclidean version that two-circle configuration doesn’t handle, which is when two circles are concentric:
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.
This case is not handled by two-circle configuration, but because the two circles are concentric, one can freely rotate all vertices around the center, creating the desired polygon once a vertex lands on \(P\)
For all other cases, we create the corresponding two-circle configuration
The two-circle configuration over \(\mathbb {R}\) corresponding to a Euclidean configuration has the following parameters
\(u = \lVert O_{center} I_{center}\rVert / I_{radius}\)
\(r = O_{radius}/ I_{radius}\)
\(k = \sqrt{(u + r)^2 - 1}\)
which effectively declare \(I_{center}\) as the origin, \(I_{radius}\) as length 1, and \(O_{center} I_{center}\) as x-axis. We then setup correspondence between the affine part of \(\mathbf{P}(2, \mathbb {R})\) and the Euclidean plane.