QG-2Cu1: Perspective Squares Double Cubic

Eckart Schmidt discovered that the locus of perspectors of the Reference Quadrigon being point perspective with a Square comprises a double cubic: QG-2Cu1a/b.

Explanation:
It is not quite simple to construct a Square perspective with a random Quadrigon.
First of all we have to distinguish 2 types of perspectivity in the Quadri-environment:
• Point perspectivity . This concerns 2 quadrigons from which the lines through the corresponding vertices coincide in one point called the perspector.
• Line perspectivity . This concerns 2 quadrigons from which the intersection points of the corresponding sidelines are collinear on a line called the perspective axis (or perspectrix).

Quadrigons that are point perspective are not automatically line perspective. However when two quadrigons are line perspective they are also point perspective.
When a Reference Quadrigon is line perspective with a Square there is a very limited set of points that can function as a perspector. These points are:
• The intersection points QG-2P4a/b of the QA-DT-Thales Circle (QG-Ci1) and the QL-DT-Thales Circle (QG-Ci2). For the construction of the corresponding line perspective squares see Ref-34, attachment QFG-message #1240.
• All points on the sidelines of the Reference Quadrigon, especially the vertices of the Reference Quadrigon.
• Certain infinity points. Equations:
CT-coordinates Equation QG-2Cu1a in 1st QA-Quadrigon:
(r S + q SA + q SB) x2 y + (r S - p SA - p SB) x y2
+ (2 q SA + r SA + r SC) x2 z + (-p SA - p SC - 2 q SC) x z2
+ (-p S + r SB + r SC) y2 z + (-p S - q SB - q SC) y z2
+ (-p S + r S - 2 p SA + 2 r SC) x y z = 0
CT-coordinates Equation QG-2Cu1b in 1st QA-Quadrigon:
(r S - q SA - q SB) x2 y + (r S + p SA + p SB) x y2
+ (-2 q SA - r SA - r SC) x2 z + (p SA + p SC + 2 q SC) x z2
+ (-p S - r SB - r SC) y2 z + (-p S + q SB + q SC) y z2
+ (-p S + r S + 2 p SA - 2 r SC) x y z = 0

DT-coordinates Equation QG-2Cu1b in 1st QA-Quadrigon:
p r y³(c²p²-a²r²) + 2 p r y²(p² SA z-r² SC x)
-p r y((c²q²+b²r²)x²-(b²p²+a²q²)z²)
-2 p q²r x z(SA x-SC z) +q² S(x+y+z)(r²x²-p²z²) = 0
Changing the sign of S will give the other cubic. See Ref-34, Eckart Schmidt, QFG#2931.

Properties:
• These points lie on QG-2Cu1a/b:
– P1, P2, P3, P4: the vertices of the Reference Quadrigon lie on both cubics.
– QG-2P4a and QG-2P4b: intersection points QG-Ci1 and QG-Ci2
(each point lying on one of both cubics).
– QG-2P5a and QG-2P5b: intersection points QG-Ci2 and QG-Diagonals
(both points lying on both cubics).
– The infinity point of the Newton Line (lying on both cubics).
– The two circular points at infinity (each point lying on one of both cubics).

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