**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 1*

^{st}QA-Quadrigon: (r S + q SA + q SB) x

^{2}y + (r S - p SA - p SB) x y^{2}+ (2 q SA + r SA + r SC) x

^{2}z + (-p SA - p SC - 2 q SC) x z^{2}+ (-p S + r SB + r SC) y

^{2}z + (-p S - q SB - q SC) y z^{2}+ (-p S + r S - 2 p SA + 2 r SC) x y z

*CT-coordinates Equation QG-2Cu1b in 1*

^{st}QA-Quadrigon: (r S - q SA - q SB) x

^{2}y + (r S + p SA + p SB) x y^{2}+ (-2 q SA - r SA - r SC) x

^{2}z + (p SA + p SC + 2 q SC) x z^{2}+ (-p S - r SB - r SC) y

^{2}z + (-p S + q SB + q SC) y z^{2}+ (-p S + r S + 2 p SA - 2 r SC) x y z

**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).

– 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).