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Given a cubic K. Then for each point P there is a conic $$C_P$$ associated to P. The conic $$C_P$$ is called the polar conic of K with respect to the pole P. We investigate the situation when two conics $$C_0$$ and $$C_1$$ are polar conics of K with respect to some poles $$P_0$$ and $$P_1$$ , respectively. First we show that for any point Q on the line $$P_0P_1$$ , the polar conic $$C_Q$$ of K with respect to Q belongs to the linear pencil of $$C_0$$ and $$C_1$$ , and vice versa. Then we show that two given conics $$C_0$$ and $$C_1$$ can always be considered as polar conics of some cubic K with respect to some poles $$P_0$$ and $$P_1$$ . Moreover, we show that $$P_1$$ is determined by $$P_0$$ , but neither the cubic nor the point $$P_0$$ is determined by the conics $$C_0$$ and $$C_1$$ .

The exponential pencil Gλ:=G1(G−10G1)λ−1, generated by two conics G0,G1, carries a rich geometric structure: It is closed under conjugation, it is compatible with duality and projective mappings, it is convergent for λ→±∞ or periodic, and it is connected in various ways with the linear pencil gλ=λG1+(1−λ)G0. The structure of the exponential pencil can be used to characterize the position of G0 and G1 relative to each other., Beiträge zur Algebra und Geometrie = Contributions to algebra and geometry, 59 (3), ISSN:0138-4821, ISSN:2191-0383

We investigate closed chains of conics which carry Poncelet triangles. In particular, we show that every chain of conics which carries Poncelet triangles can be closed. Furthermore, for k=3 and k=4 we show that there are closed chains of pairwise conjugate conics which carry Poncelet k-gons such that the contact points of each k-gon are the vertices of the next k-gon—such miraculous chains of conics do not exist for 5≤k≤23., Beiträge zur Algebra und Geometrie = Contributions to algebra and geometry, 58 (2), ISSN:0138-4821, ISSN:2191-0383