QA-Tf18: 2nd QA-Hung’s Transformation


QA-Tf18 maps two random lines into a point.
Let P1, P2, P3, P4 be the defining Quadrangle Points.
r is a random line.
r meets lines P1P2, P1P3, P1P4, P2P3, P2P4, P3P4 at the points R12, R13, R14, R23, R24, R34.
Let T1, T2, T3 be the midpoints of segments R12R34, R13R24, R14R23.
d is another random line.
Line d1, d2,d3 pass through T1, T2, T3 and are parallel to d.
d1 meets lines P1P2, P3P4 at points M12, M34.
Let R12' be the reflection of R12 in M12.
Let R34' be the reflection of R34 in M34.
Line d1' connects points R12' and R34'.
Define similarly the lines d2' and d3'.
Then d1', d2' and d3' are concurrent at a point P.
This points is a transformation of lines r,d to point P in QA.
This transformation is found by Tran Quang Hung. See Ref-34, QFG#3716.

QA Tf18 2nd QA Hung's Transformation 01
CT-Coordinates of QA-Tf18(r,d), where r=(l:m:n), d=(x:y:z)
          ( (z - y) ( l q r (z - y)2 + p r (x - z) ( n x + l y - n y - l z) + p q (x - y) (m x - l y + l z - m z)) :
          (x - z) (m p r (x - z)2 + q r (y - z) (m x - n x + n y - m z) + p q (x - y) (m x - l y + l z - m z)) :
          (y - x) (n p q (y - x)2 + q r (y - z) (m x - n x + n y - m z) + p r (x - z) ( n x + l y - n y - l z)) )

CT-Coordinates of QA-Tf18(r,ip), where r=(l:m:n), ip=(x:y:z) = infinitypoint line-d
          (x ( l (p + q) r x y + l q (p + r) x z + p (m q + n r) y z) :
          y (m (p + q) r x y + q (l p + n r) x z + m p (q + r) y z) :
          z ((l p + m q) r x y + n q (p + r) x z + n p (q + r) y z) )
(Note that in this expression of the coordinates x + y + z = 0)

Properties:
Denote Pij = QA-Tf18(QA-Li,QA-Lj), where QA-L0 is denoted as the line at infinity. For more information about next properties see Ref-34, QFG#3723:
• For a given QA-Lj, all Pij-points are collinear on the 5th Point Tangent QA-Tf9 (InfinityPoint(QA-Lj)). See construction at QA-Tf9.
• all P0i-points and Pii-points are Infinity Points
• all Pi0-points are not defined
• all other Pij-points are Finite Points
• P0j, P1j, P2j, P3j, P4j, P5j, P6j, P7j, P8j, P9j are collinear for j=1,2,...,9
• Pi1, Pi4, Pi9 are collinear for i=0,1,2,...,9
• P0i = Pii, they are the infinity points of QA – Li
• P14 = QA-P3
• P17 = P04 = P44 = QA-Tf2(QA-P3) = InfinityPoint
• P19 = P29 = QA-P2
• P24 = 5P-s-P4 (P1,P2,P3,P4,QA-P4)
• P12 lies on QA-P2.QA-P33
• P41 lies on QA-P1.QA-Tf2(QA-P2)
• P42 lies on QA-P23.QA-P33
• P69 lies on QA-P14.QA-P24
• P92 lies on QA-P1.QA-P6.QA-P23
For more properties (from Eckart Schmidt) see Ref-34, QFG#3726-3729.