QL-Tf8: QL-Involutary Centerline
QL-Tf8 is a transformation mapping a point into a special line through this point.
This line has involutary properties. In fact it is the counterpart of the Involution Center of a Line Involution in a Quadrangle.
Involutary Centerline
In a Quadrangle every line indicates an Involution Center (see QA-Tf1).
In the same way in a Quadrilateral every point indicates an Involutary Centerline.
In a Quadrangle every line indicates an Involution Center (see QA-Tf1).
In the same way in a Quadrilateral every point indicates an Involutary Centerline.
Construction 1
1. Let P be some random point
2. L = QL-Tf11(P) = QL-Tf2(DT-Tripolar(P)),
3. Construct the dual Quadrangle P1.P2.P3.P4 (Pi=DT-Tripole(Li), see QL-8).
4. Let IC = QA-Tf1(L) wrt P1.P2.P3.P4.
5. Finally QA-Tf8(P) = QL-Tf11(P) = QL-Tf2(DT-Tripolar(IC)).
where DT-Tripolar=Trilinear Polar wrt QL-DT
and DT-Tripole =Trilinear Pole wrt QL-DT.
1. Let P be some random point
2. L = QL-Tf11(P) = QL-Tf2(DT-Tripolar(P)),
3. Construct the dual Quadrangle P1.P2.P3.P4 (Pi=DT-Tripole(Li), see QL-8).
4. Let IC = QA-Tf1(L) wrt P1.P2.P3.P4.
5. Finally QA-Tf8(P) = QL-Tf11(P) = QL-Tf2(DT-Tripolar(IC)).
where DT-Tripolar=Trilinear Polar wrt QL-DT
and DT-Tripole =Trilinear Pole wrt QL-DT.
Construction 2
From Eckart Schmidt. See Ref-34, QFG#2186.
1. Let P be some random point
2. Let Co0 = inscribed QL-conic tangent to P.QL-P13
3. QA-Tf8(P) = 2nd tangent from P to Co0.
From Eckart Schmidt. See Ref-34, QFG#2186.
1. Let P be some random point
2. Let Co0 = inscribed QL-conic tangent to P.QL-P13
3. QA-Tf8(P) = 2nd tangent from P to Co0.
Doublelines
In a Quadrangle every line indicates not only an Involution Center but also two Doublepoints (see QA-Tf1).
In the same way in a Quadrilateral every point indicates two Doublelines.
Note that a Doublepoint is defined as one point where originally two points coincide and a Doubleline is one line where originally two lines coincide. However there are normally two Doublepoints created by a Line Involution as well as there are two Doublelines created by a Point Involution.
In a Quadrangle every line indicates not only an Involution Center but also two Doublepoints (see QA-Tf1).
In the same way in a Quadrilateral every point indicates two Doublelines.
Note that a Doublepoint is defined as one point where originally two points coincide and a Doubleline is one line where originally two lines coincide. However there are normally two Doublepoints created by a Line Involution as well as there are two Doublelines created by a Point Involution.
Construction 1 of the related Doublelines
1. Let P be some random point
2. L = QL-Tf11(P) = QL-Tf2(DT-Tripolar(P)),
3. Construct the dual Quadrangle P1.P2.P3.P4 (Pi=DT-Tripole(Li), see QL-8).
4. Let DP1 and DP1 be the two Doublepoints of the QA-Line involution on L wrt P1.P2.P3.P4. See QA-Tf1.
5. Then the Doublelines DL1 and Dl2 can be obtained by:
DL1 = QL-Tf11(DP1) = QL-Tf2(DT-Tripolar(DP1)),
DL2 = QL-Tf11(DP2) = QL-Tf2(DT-Tripolar(DP2)).
where DT-Tripolar=Trilinear Polar wrt QL-DT
and DT-Tripole =Trilinear Pole wrt QL-DT.
1. Let P be some random point
2. L = QL-Tf11(P) = QL-Tf2(DT-Tripolar(P)),
3. Construct the dual Quadrangle P1.P2.P3.P4 (Pi=DT-Tripole(Li), see QL-8).
4. Let DP1 and DP1 be the two Doublepoints of the QA-Line involution on L wrt P1.P2.P3.P4. See QA-Tf1.
5. Then the Doublelines DL1 and Dl2 can be obtained by:
DL1 = QL-Tf11(DP1) = QL-Tf2(DT-Tripolar(DP1)),
DL2 = QL-Tf11(DP2) = QL-Tf2(DT-Tripolar(DP2)).
where DT-Tripolar=Trilinear Polar wrt QL-DT
and DT-Tripole =Trilinear Pole wrt QL-DT.
Construction 2 of the related Doublelines
From Eckart Schmidt. See Ref-34, QFG#2188.
The Doublelines are the equivalent of the Doublepoints in QA-Tf1.
1. There are two QL-inscribed conics Co1 and Co2 through P.
2. The tangents in P to these conics are the Doublelines.
From Eckart Schmidt. See Ref-34, QFG#2188.
The Doublelines are the equivalent of the Doublepoints in QA-Tf1.
1. There are two QL-inscribed conics Co1 and Co2 through P.
2. The tangents in P to these conics are the Doublelines.
Construction 3 of the related Doublelines
There is an relatively easy way of constructing the related Doublelines.
Let L1,L2,L3,L4 be the basic lines of the Reference Quadrilateral and let L0 be some random line.
Let Sij = Li.Lj and let Tij = L0^P.Sij, where (i,j) are different numbers from (1,2,3,4).
Now there will be a line involution (see QA-Tf1) on L created by any two pairs of points of (Tij,Tkl), where (i,j,k,l) each are different numbers from (1,2,3,4). These are actually the pairs (T12,T34), (T13,T24) and (T14,T23). Any two pairs of them will create the same line involution.
Given the line involution there will be two Doublepoints DP1 and DP2 (see QA-Tf1) on this line. These points will lie on the Doublelines and so by connecting P with DP1 and DP2 will deliver the Doublelines.
Even more easier: let L0 coincide with one of the basic lines e.g. L1.
Then two suitable pairs of points are
(L1^L4, L1^(P.(L2^L3))),
(L1^L2, L1^(P.(L3^L4))).
These points are easy to draw and so line L1 with involutary scale is present in order to construct the Doublepoints like described in QA-Tf1. Connect these Doublepoints with P in order to obtain the Doublelines.
There is an relatively easy way of constructing the related Doublelines.
Let L1,L2,L3,L4 be the basic lines of the Reference Quadrilateral and let L0 be some random line.
Let Sij = Li.Lj and let Tij = L0^P.Sij, where (i,j) are different numbers from (1,2,3,4).
Now there will be a line involution (see QA-Tf1) on L created by any two pairs of points of (Tij,Tkl), where (i,j,k,l) each are different numbers from (1,2,3,4). These are actually the pairs (T12,T34), (T13,T24) and (T14,T23). Any two pairs of them will create the same line involution.
Given the line involution there will be two Doublepoints DP1 and DP2 (see QA-Tf1) on this line. These points will lie on the Doublelines and so by connecting P with DP1 and DP2 will deliver the Doublelines.
Even more easier: let L0 coincide with one of the basic lines e.g. L1.
Then two suitable pairs of points are
(L1^L4, L1^(P.(L2^L3))),
(L1^L2, L1^(P.(L3^L4))).
These points are easy to draw and so line L1 with involutary scale is present in order to construct the Doublepoints like described in QA-Tf1. Connect these Doublepoints with P in order to obtain the Doublelines.
(y z (2 l2 m x + 2 l2 n x - 2 l m n x + l m2 y + l m n y - 3 m2 n y + l m n z + l n2 z - 3 m n2 z) :
x z (l2 m x - 3 l2 n x + l m n x + 2 l m2 y - 2 l m n y + 2 m2 n y + l m n z - 3 l n2 z + m n2 z) :
-x y (3 l2 m x - l2 n x - l m n x + 3 l m2 y - l m n y - m2 n y + 2 l m n z - 2 l n2 z - 2 m n2 z))
x z (l2 m x - 3 l2 n x + l m n x + 2 l m2 y - 2 l m n y + 2 m2 n y + l m n z - 3 l n2 z + m n2 z) :
-x y (3 l2 m x - l2 n x - l m n x + 3 l m2 y - l m n y - m2 n y + 2 l m n z - 2 l n2 z - 2 m n2 z))
Properties
• QL-Tf8(QL-P1) = QL-P1.QL-P26
• QL-Tf8(QL-P8) = QL-P8.QL-P23
• The Involutary Centerline is the 4th harmonic line of the Doublelines and P.QL-P13. See Ref-34, Eckart Schmidt, QFG#2187.
• QL-Tf8(QL-P1) = QL-P1.QL-P26
• QL-Tf8(QL-P8) = QL-P8.QL-P23
• The Involutary Centerline is the 4th harmonic line of the Doublelines and P.QL-P13. See Ref-34, Eckart Schmidt, QFG#2187.