nL-n-Luc5 is called a Level-up construction because circumstantially it transforms a Central Point of an n-Line into a Central point of an (n+1)-Line.
nL-n-Luc5 is a class of constructions which will be subdivided later. See below.
nL-n-Luc5 is a class of constructions which will be subdivided later. See below.
nL-n-Luc5 transforms an n-Line into another n-Line by drawing lines through the n versions of some Central Point (n-1)-Px perpendicular or parallel to the omitted line.
• The reference n-Line is called Ref.
• When drawing parallel lines through the n versions of (n-1)-Px the result will be an n-Line called Par. When drawing more than one generations the resulting n-Lines will be called Par1, Par2, etc.
• When drawing perpendicular lines through the n versions of (n-1)-Px the result will be an n-Line called Per. When drawing more than one generations the resulting n-Lines will be called Per1, Per2, etc.
• When a pair of the occurrences of Ref, Par1, Par2, Per1, Per2 are perspective there will be a Perspective Center XXX/YYY-PC(Px), where XXX and YYY are different names taken from the group Ref, Par1, Par2, Per1, Per2, etc.
• When the corresponding lines of XXX and YYY are parallel and XXX and YYY are perspective, then this Perspective Center will be called Homothetic Center XXX/YYY-HC(Px), where XXX and YYY are different names taken from the group Ref, Par1, Par2, Per1, Per2, etc.
• The reference n-Line is called Ref.
• When drawing parallel lines through the n versions of (n-1)-Px the result will be an n-Line called Par. When drawing more than one generations the resulting n-Lines will be called Par1, Par2, etc.
• When drawing perpendicular lines through the n versions of (n-1)-Px the result will be an n-Line called Per. When drawing more than one generations the resulting n-Lines will be called Per1, Per2, etc.
• When a pair of the occurrences of Ref, Par1, Par2, Per1, Per2 are perspective there will be a Perspective Center XXX/YYY-PC(Px), where XXX and YYY are different names taken from the group Ref, Par1, Par2, Per1, Per2, etc.
• When the corresponding lines of XXX and YYY are parallel and XXX and YYY are perspective, then this Perspective Center will be called Homothetic Center XXX/YYY-HC(Px), where XXX and YYY are different names taken from the group Ref, Par1, Par2, Per1, Per2, etc.
More specific:
1. Every n-Line has n Component (n-1)-Lines, each (n-1)-Line constructed by omitting one line of the n-Line.
2. Through the n (n-1)L-versions of some central point parallels are drawn to the omitted line, thus producing a new n-Line called Par1.
3. When this construction is repeated by using Par1 as Reference n-Line the outcome will be a 2nd generation n-Line called Par2.
4. Through the (n-1)L-versions of some central point perpendiculars are drawn to the omitted line, thus producing a new n-Line called Per1.
5. When this last construction is repeated by using nL-Per1 as Reference n-Line the outcome will be a 2nd generation n-Line called Per2.
It appears that all kind of combinations of nL-Ref, Par1, Par2, Per1, Per2 can be homothetic or perspective, where they give rise to a Homothetic Center(HC) / Perspective Center (PC).
Examples
Although most of the times there will no perspectivity there are plenty of positive examples:
• nL-n-P5 applied in n (n-1)-Lines gives homothetic Ref / Par1, creating nL-n-P2.
• X(4) applied in 4 3-Lines gives a Ref/Par1-HC, being QL-P20.
• X(4) applied in 4 3-Lines gives a Par1/Per1-PC, being QL-P21.
• 5L-s-P1 applied in 6 5-Lines gives a Ref/Par2-HC, being 6L-s-P2.
• 5L-s-P1 applied in 6 5-Lines gives a Ref/Per2-HC, being 6L-s-P3.
• 5L-s-P1 applied in 6 5-Lines gives a Par2/Per2-HC, being 6L-s-P4.
• etc.
Not always a Perspective Axis
Note that although there is a Perspective Center/Homothetic Center of two n-Lines for n>3 there not always is a Perspective Axis. Actually there mostly is no Perspective Axis. There is a Perspective Axis when the intersection points of corresponding lines are collinear on a Perspective Axis.
A nice example is the Perspective Axis of Par1/Per1-Perspective Center QL-P21, being the Steiner Line QL-L2.
Note that although there is a Perspective Center/Homothetic Center of two n-Lines for n>3 there not always is a Perspective Axis. Actually there mostly is no Perspective Axis. There is a Perspective Axis when the intersection points of corresponding lines are collinear on a Perspective Axis.
A nice example is the Perspective Axis of Par1/Per1-Perspective Center QL-P21, being the Steiner Line QL-L2.
Present state of research
There is a huge differentiation in perspective pairs of n-Lines coming from (Ref, Per1, Per2, Per3, Per4, Par1, Par2, Par3, Par4).
Most common are the perspectivities of these pairs of n-Lines:
• Ref/Par1 (consequently also Par1/Par2, etc.)
• Par1/Par2 (without perspectivity of Ref/Par1)
• Ref/Per2
• Par1/Per2
But it has to be said that most of the times there will be no homothetic / perspective pair of n-Lines.
So each occurrence of a Ref-Per-Par-perspectivity for some Px is special.
There is a huge differentiation in perspective pairs of n-Lines coming from (Ref, Per1, Per2, Per3, Per4, Par1, Par2, Par3, Par4).
Most common are the perspectivities of these pairs of n-Lines:
• Ref/Par1 (consequently also Par1/Par2, etc.)
• Par1/Par2 (without perspectivity of Ref/Par1)
• Ref/Per2
• Par1/Per2
But it has to be said that most of the times there will be no homothetic / perspective pair of n-Lines.
So each occurrence of a Ref-Per-Par-perspectivity for some Px is special.
Examples ETC-points applied in a 4-Line
X(2) in a 4-Line
Perspective/Homothetic Centers:
• Ref/Par1 = QL-P12
• Ref/Per2 = Ref/Per4 = Per1/Per3 = Per2/Per4 = QL-Px =
Midpoint QL - P5.QL - P29 = Midpoint QL - P2.QL - P20 = Reflection of QL - P6 in QL - P22
• Par1/Per2=QL-P12.QL-Px (4:1)
• Par1/Per4=QL-P12.QL-Px (40:1)
These 4 Perspective/Homothetic Centers are collinear.
X(4) in a 4-Line
Note: Ref=Par2=Par4, Par1=Par3
Perspective/Homothetic Centers:
• Ref/Par1 = QL-P20
• Ref/Per2 = InfinityPoint (QL-P2.QL-P20)
• Par1/Per1 = QL-P21
• Par1/Per2 = QL-P2.QL-P20 (-1:2)
Click at next links for extra properties of all possible Ref/Par/Per-Constructions when Par and Per are constructed until the 2nd generation:
| nL-n-Luc5a nL-Ref/Par1-Construction |
| nL-n-Luc5b nL-Ref/Par2-Construction |
| nL-n-Luc5c nL-Ref/Per1-Construction |
| nL-n-Luc5d nL-Ref/Per2-Construction |
| nL-n-Luc5e nL-Par1/Par2-Construction |
| nL-n-Luc5f nL-Par1/Per1-Construction |
| nL-n-Luc5g nL-Par1/Per2-Construction |
| nL-n-Luc5h nL-Par2/Per1-Construction |
| nL-n-Luc5i nL-Par2/Per2-Construction |
| nL-n-Luc5j nL-Per1/Per2-Construction |