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

Abstract

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.

Then

$$\delta\:\ {\cal P}^{3}\rightarrow {\cal M}^{+};\ \ \ (a,b,c)\mapsto \delta_{a;b,c}\:=\ {\tilde a}\ {\rm o}\ \widetilde {a,\ b}\ {\rm o}\ \widetilde {b,\ c}\ {\rm o}\ \widetilde {c,\ a}$$

is called the defect function, or shortly the defect.

We show that δ_a;b,c is a rotation around the point a where the angle of δ_a;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 δ_a,b = δ_o;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).