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At this site you will find mathematical subjects I investigated. All pages are in English. Some pages are also available in Dutch. When a change of language is possible it is shown at the top of the page.
Chris van Tienhoven, the Netherlands
Op deze site vindt u enige wiskundige onderwerpen waar ik onderzoek naar gedaan heb. Alle pagina's zijn in het Engels. Sommige pagina's zijn ook in het Nederlands. Wanneer er een taalkeuze is, dan staat dit bovenaan de pagina d.m.v. een (NL-)vlag.
Chris van Tienhoven, Nederland
QL-Tf10: QL-QuadriPole

The QL-QuadriPole is an equivalent of the Tripole (also named Trilinear Polar) in a trilateral. It transforms in a Quadrilateral “harmonically” a line into a point.
QL-Tf10(L) = DT-TP(QL-Tf2(L)), where DT-TP = Trilinear Pole wrt the QL-Diagonal Triangle QL-DT (=QL-Tr1).
The combination QL-Tf10/QL-Tf11 in a Quadrilateral is the equivalent of the combination QA-Tf10/QA-Tf11 in a Quadrangle.
In particular QL-Tf10 is the dual of QA-Tf11 and has the same coordinates as QA-Tf11 when substituting (p:q:r) >(l:m:n).
QL Tf10   QL QuadriPole 01
One of the advantages of QL-Tf10 is that enveloping lines eLi can be transformed by QL-Tf10 into points ePi, producing a point driven locus, whereafter the tangents at ePi can be obtained, which can be transferred back by QL-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 picture below and QL-8.
QL Tf10 QuadriPolar 11
Let L = (x:y:z), then QL-Tf10(L) =
(m n (3 m n x - l n y - l m z) : l n (-m n x + 3 l n y - l m z) : l m (-m n x - l n y + 3 l m z))
Let L = (x:y:z), then QL-Tf10(L) =
(x/l2 : y/m2 : z/n2)

QL-Tf10(QL-Tf11(P)) = P and QL-Tf11(QL-Tf10(L)) = L.
QL-Tf10(L) also can be obtained as QA-Tf2*(DT-TP(L)), where QA-Tf2* = QA-Tf2-transformation wrt the dual QA with vertices Pi=DT-TP(Li) (i=1,2,3,4). Therefore it is also QA-Tf10(L) wrt the dual QA. See QL-8. 


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