12.6: Axiom III
Note that the first part of Axiom III follows directly from the definition of the h-angle measure defined on page . It remains to show that \(\measuredangle_h\) satisfies the conditions Axiom IIIa , Axiom IIIb , and Axiom IIIc .
The following two claims say that \(\measuredangle_h\) satisfies IIIa and IIIb .
Given an h-half-line \([O P)_h\) and \(\alpha\in(-\pi,\pi]\) , there is a unique h-half-line \([O Q)_h\) such that \(\measuredangle_h P O Q= \alpha\) .
For any h-points \(P\) , \(Q\) , and \(R\) distinct from an h-point \(O\) , we have
\(\measuredangle_h P O Q+\measuredangle_h Q O R \equiv\measuredangle_h P O R.\)
- Proof of \(\PageIndex{1}\) and \(\PageIndex{2}\)
-
Applying the main observation, we may assume that \(O\) is the center of the absolute. In this case, for any h-point \(P \ne O\) , the h-half-line \([OP)_h\) is the intersection of the Euclidean half-line \([OP)\) with h-plane. Hence the claim \(\PageIndex{1}\) and Claim \(\PageIndex{2}\) follow from the axioms Axiom IIIa and Axiom IIIb of the Euclidean plane.
The function
is continuous at any triple of points \((P,Q,R)\) such that \(Q\ne P\) , \(Q\ne R\) , and \(\measuredangle_h P Q R\ne\pi\) .
- Proof
-
Suppose that \(O\) denotes the center of the absolute. We can assume that \(Q\) is distinct from \(O\) .
Suppose that \(Z\) denotes the inverse of \(Q\) in the absolute; suppose that \(\Gamma\) denotes the circle perpendicular to the absolute and centered at \(Z\) . According to Lemma 12.3.1 , the point \(O\) is the inverse of \(Q\) in \(\Gamma\) .
Let \(P'\) and \(R'\) denote the inversions in \(\Gamma\) of the points \(P\) and \(R\) respectively. Note that the point \(P'\) is completely determined by the points \(Q\) and \(P\) . Moreover, the map \((Q,P)\mapsto P'\) is continuous at any pair of points \((Q,P)\) such that \(Q\ne O\) . The same is true for the map \((Q,R)\mapsto R'\)
According to the main observation
\(\measuredangle_h P Q R\equiv -\measuredangle_h P' O R'.\)
Since \(\measuredangle_h P' O R'=\measuredangle P' O R'\) and the maps \((Q,P)\mapsto P'\) , \((Q,R)\mapsto R'\) are continuous, the claim follows from the corresponding axiom of the Euclidean plane.