QA-Tf10: QA-QuadriPole


The QA-QuadriPole is an equivalent of the Tripole (also named Trilinear Pole) in a triangle. It transforms in a Quadrangle “harmonically” a line into a point.
QA-Tf10(L) = QA-Tf2(DT-TP(L)), where DT-TP = Trilinear Pole wrt the QA-Diagonal Triangle QA-DT (=QA-Tr1).
The combination QA-Tf10/QA-Tf11 in a Quadrangle is the equivalent of the combination QL-Tf10/QL-Tf11 in a Quadrilateral.
In particular QA-Tf10 is the dual of QL-Tf11 and has the same coordinates as QL-Tf11 when substituting (l:m:n) > (p:q:r).
 
QA Tf10 QuadriPole 01
 
One of the advantages of QA-Tf10 is that enveloping lines eLi can be transformed by QA-Tf10 into points ePi, producing a point driven locus, whereafter the tangents at ePi can be obtained, which can be transferred back by QA-Tf10, delivering the points of tangency at eLi, which produce a point driven locus tangent to the initial enveloping lines. So an envelope of lines can be transferred into a point driven locus. See QA-8.
An example of transfering back tangents to points of tangency at an original curve can be found at Ref-34, QFG#2212.
 
CT-coordinates
Let L = (x:y:z), then QA-Tf10(L) =
(p (2 p x + q y + r z) : q (p x + 2 q y + r z) : r (p x + q y + 2 r z))
 
Properties
QA-Tf10(QA-Tf11(P)) = P and QA-Tf11(QA-Tf10(L)) = L.
QA-Tf10(L) also can be obtained as DT-Tripole(QL-Tf2*(L)), where QL-Tf2* = QL-Tf2-transformation wrt the dual QL with defining lines Li = DT-Tripolar(Pi) (i=1,2,3,4). Therefore it is also QL-Tf10(L) wrt the dual QL. See QA-8.