Unified Approach to the Fuss Relations in Poncelet’s Porism

In the present paper, we introduce two important modifications to the recurrence relation for the Fuss relations derived in Cieślak (Comput Aided Geom Des 66:19–30, 2018). These modifications allow for a simpler and unified method for determining the relations, and additionally explain why their derivation is extremely difficult. We introduce two convenient notions of unified and reduced Fuss relations (u.F.r. and r.F.r.) and, using the resulting theory, derive such relations for the cases of \(n=3, \ldots , 10\) $n=3,\dots ,10$ . The introduced modifications of the recursion allow us to formulate two theorems about the nature of polynomials, by means of which Fuss relations are determined. We show that for n even we get one formula u.F.r., while for n odd we get two r.F.r., corresponding to odd and even indices of rotation of the closed Poncelet transversals. The introduced method of notation allows a significant reduction in the number of terms in the Fuss relations, which we have shown in the considered examples and which can be easily verified on all Fuss relations known so far.