Given three non-parallel segments OP1, OP2, OP3 in a plane ω, we consider the ellipses EP1,P2 , EP2,P3 , EP3,P1 having as conjugate semi-diameters the pairs (OP1,OP2), (OP2,OP3) and (OP3,OP1), respectively. We find the necessary and sufficient conditions for (i) the existence of a common point P ∈ EP1,P2 ∩ EP2,P3 ∩ EP3,P1 and (ii) the existence of a pair of parallel and distinct lines tangent to the three ellipses. In this later case, we solve the problem by introducing the definition of cylindrical Pohlke’s projection.
ON POLHKE’S TYPE PROJECTIONS IN THE CYLINDRICAL CASE
RENATO MANFRIN
2024-01-01
Abstract
Given three non-parallel segments OP1, OP2, OP3 in a plane ω, we consider the ellipses EP1,P2 , EP2,P3 , EP3,P1 having as conjugate semi-diameters the pairs (OP1,OP2), (OP2,OP3) and (OP3,OP1), respectively. We find the necessary and sufficient conditions for (i) the existence of a common point P ∈ EP1,P2 ∩ EP2,P3 ∩ EP3,P1 and (ii) the existence of a pair of parallel and distinct lines tangent to the three ellipses. In this later case, we solve the problem by introducing the definition of cylindrical Pohlke’s projection.
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