Results in Mathematics

, Volume 47, Issue 3–4, pp 305–326 | Cite as

Relations between the K-loop and the defect of an absolute plane

  • Helmut Karzel
  • Mario Marchi


Let \({\cal P}\) be the point set of an absolute plane, let \({\cal {\tilde P}}\) be the set of all point reflections, let ℳ, resp. ℳ+, be the group of all, resp. of all proper, motions and let
$$^\sim:{\cal P\times P\rightarrow \tilde P};\ \ \ (a,\ b)\mapsto\ \widetilde {a,\ b}$$
be the map where \({\widetilde {a,\ b}}\) denotes the uniquely determined point-reflection interchanging a and b.
$$\delta\:\ {\cal P}^{3}\rightarrow {\cal M}^{+};\ \ \ (a,b,c)\mapsto \delta_{a;b,c}\:=\ {\tilde a}\ {\rm o}\ \widetilde {a,\ b}\ {\rm o}\ \widetilde {b,\ c}\ {\rm o}\ \widetilde {c,\ a}$$
is called the defect function, or shortly the defect.

We show that δa;b,c is a rotation around the point a where the angle of δa;b,c is exactly the angle defect of the triangle (a, b, c) (cfr. 3.5).

After fixing a point \(o\ \in {\cal P}\) and setting \(a+b\:=\widetilde {o,\ a}\ {\rm o}\ {\tilde o}\ (b),\ ({\cal P},+)\) becomes a K-loop and the so called precession function
$$\delta_{a,b}\:=\ \big((a+b)^{+}\big)^{-1}\ {\rm o}\ a^{+}\ {\rm o}\ b^+$$
of the loop (\({\cal P}, +)\) coincides with the defect of the triangle (o, a, −b) (cfr. (4.4.1)), hence δa,b = δo;a,−b for all \(a, b \in {\cal P}\).
With the order relation of the absolute plane we associate an orientation function
$$\Omega\:\ \Delta\ \times \Delta \rightarrow \lbrace -1,+1\rbrace$$
defined on the pairs of triangles (cfr. (2.8)). If (a, b, c) ∈ Δ is a triangle and d a point of the line \(\overline {b,\ c}\) 1, then (cfr. (3.9.2)):
$$\delta_{a;b,c}\ {\rm o}\ \delta_{a;c,d}=\delta_{a;b,d}$$
and moreover, if d is even a point of the open segment ]b, c[ then (cfr. (2.8.5)):
$$\Omega(a,\ b,\ c;\ a,\ b,\ d)=\Omega(a,\ b,\ d;\ a,\ d,\ c)=+1.$$

Thus the angle defect of the triangle (a, b, c) is the sum of the angle defects of the triangles (a, b, d) and (a, d, c).


Rotation Angle Proper Motion Elliptic Case Ordinary Case Hyperbolic Case 
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Copyright information

© Birkhäuser Verlag, Basel 2005

Authors and Affiliations

  • Helmut Karzel
    • 1
  • Mario Marchi
    • 2
  1. 1.Techn. Universität München Fakultät für MathematikMünchen
  2. 2.BresciaItaly

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