nL-n-Luc5a nL-Ref/Par1 construction
nL-n-Luc5a is a Level-up construction which uses the Reference n-Line Ref and the 1st generation parallel n-Line Par1 wrt some lower-level center to construct a Perspective / Homothetic Center. See nL-n-Luc5.
Examples:
|
3L-point
|
4L-point
|
5L-point
|
6L-point
|
|
|
3L-n-P8
= X(2)
|
4L-n-P8
= QL-P12
(Ref/Par1-HC, Ref=Par2)
|
5L-n-P8
(Ref/Par1/Par2-HC)
|
6L-n-P8
(Ref/Par1/Par2-HC)
|
Etc.
|
|
3L-n-P3/P9
= X(3)
|
4L-n-Px
= QL-P5
(Ref/Par1-HC, Ref=Par2)
|
No Ref/Par1-relationship.
No Ref/Par2-relationship.
No Par1/Par2-relationship.
|
||
|
3L-n-P4/P10
= X(4)
|
4L-n-Px
= QL-P20
(Ref/Par1-HC, Ref=Par2)
|
5L-s-P7 (Par1/Par2-HC)
No Ref/Par1-relationship.
No Ref/Par2-relationship.
|
No Ref/Par1-relationship.
No Ref/Par2-relationship.
No Par1/Par2-relationship.
|
|
|
3L-n-P5/P11
= X(5)
|
4L-n-P2
= QL-P22
(Ref/Par1-HC, Ref=Par2)
|
No Ref/Par1-relationship.
No Ref/Par2-relationship.
No Par1/Par2-relationship.
|
||
|
Points on line 4L-n-P8.4L-n-P5
4L-n-P8 = QL-P12
4L-n-P5 = QL-P30
4L-n-P5. 4L-n-P8 (2:-1)= QL-P27
|
Points on line 5L-n-P8. 5L-n-P2
preserving distance ratio’s
(Ref/Par1/Par2-HC)
5L-n-P8
5L-n-P2
5L-n-P2. 5L-n-P8 (2:-1)
|
No Ref/Par1-relationship.
No Ref/Par2-relationship.
No Par1/Par2-relationship.
|
Note that the mutual distance ratios of points X(2), X(3), X(4), X(5) (lying on the 3L-Eulerline) and resp. of QL-P12, QL-P5, QL-P20, QL-P22 (lying on the 4L-Newton Line) are identical.
Summary
General cases
• nL-n-Luc5a (nL-n-P8) = pL-n-P8, where p=(n+1)
• nL-n-Luc5a (nL-n-P5) = pL-n-P2, where p=(n+1)
• nL-n-Luc5a (nL-n-Px) = pL-n-Py, where p=(n+1) and
nL-n-Px = Ratiopoint nL-n-P8.nL-n-P5 (s:t)
pL-n-Py = Ratiopoint pL-n-P8.pL-n-P2 (s:t),
meaning that the line nL-n-P8.nL-n-P5 is leveled-up into the line pL-n-P8.pL-n-P2
and that distance ratios are preserved.
Specific cases
• nL-n-Luc5a (X(3)) = QL-P5, there is no nL-n-Luc5a (QL-P5)
• nL-n-Luc5a (X(4)) = QL-P20, there is no nL-n-Luc5a (QL-P20)
• nL-n-Luc5a (X(5)) = QL-P22, there is no nL-n-Luc5a (QL-P22)
• nL-n-Luc5a (X(3)) = QL-P5, there is no nL-n-Luc5a (QL-P5)
• nL-n-Luc5a (X(4)) = QL-P20, there is no nL-n-Luc5a (QL-P20)
• nL-n-Luc5a (X(5)) = QL-P22, there is no nL-n-Luc5a (QL-P22)