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

• When drawing parallel lines through the n versions of (n-1)-Px the result will be an n-Line called

• When drawing perpendicular lines through the n versions of (n-1)-Px the result will be an n-Line called

• When a pair of the occurrences of Ref, Par1, Par2, Per1, Per2 are perspective there will be a Perspective Center

• 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

• 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.

**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.

**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

*of all possible Ref/Par/Per-Constructions when Par and Per are constructed until the 2***extra properties**^{nd}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 |