A note on a secondary Pohlke’s projection

Given three segments OP1,OP2,OP3 in a plane ω, which are not contained in a line, we find a simple necessary and sufficient condition for the existence of two distinct ellipses centered at O and circumscribing the three ellipses having as conjugate semi-diameters the pairs (OP1,OP2), (OP2,OP3) and (OP3,OP1). We prove this result by showing that it is equivalent to the existence of a secondary Pohlke’s projection closely related to the (always existing) projection given by Pohlke’s fundamental theorem of oblique axonometry.