nL-n-Tf4 2^nd Generation nL-Orthopolar

Let L be a random line.

Let nLL be the 2^nd generation n-Line made up from the n versions of (n-1)L-n-Tf2(L).

The 3^rd generation n-Line constructed in a similar way upon nLL are n coinciding Lines, being nL-n-Tf4(L). The lines L, nL-n-Tf2(L), nL-n-Tf4(L) coincide in one point being nL-n-Tf3(L).

5-Line Example

The 2^nd Generation nL-Orthopolar best can be understood by the example in a 5-Line.

The 5 versions of 4L-n-Tf2(L) form the 2^nd generation Orthopolar 5-Line, being called here 5LL.

The 5 versions of 4L-n-Tf2(L) wrt 5LL delivers 5 times the same line, which is 5L-n-Tf4(L).

The lines L, 5L-n-Tf2(L), 5L-n-Tf4(L) coincide in one point being 5L-n-Tf3(L).

Notes

1. Note that for all L the sides of 5LL have a fixed direction regardless the position of L.
2. Note that for all L in the same direction corresponding 5LL’s are congruent.

Properties

• Let nLL-n-Tf2(L) be nL-n-Tf2(L) wrt nLL.
• nLL-n-Tf2 has some very special properties:
  • nLL-n-Tf2(La) // nLL-n-Tf2(Lb), where La and Lb are two different random lines.
  • nLL-n-Tf2(L) will be a line passing through the intersection point of L and nL-n-Tf2(L).
  • the n versions of (n-1)LL-n-Tf2(LLi) coincide with a line // nLL-n-Tf2(L), where LLi is the omitted line of nLL and i=1, …, n.
  • the n versions of (n-1)LL-n-Tf2(L) coincide with nLL-n-Tf2(L).

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