# Twistor Geometry and Non-Linear Systems by H.D. Doebner, T.D. Palev

By H.D. Doebner, T.D. Palev

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A2 ≺ A1 . Conversely, if either A1 ≺ A2 ≺ . . ≺ An−1 ≺ An or An ≺ An−1 ≺ . . ≺ A2 ≺ A1 , then the points A1 , A2 , . . , An are in order [A1 A2 . . An ]. Proof. 14). ✷ The following simple corollary may come in handy, for example, in discussing properties of vectors on a line. 2. If points A, B both precede a point C (in some order, direct or inverse, defined on a line a), they lie on the same side of C. Proof. We know that A ≺ C & B ≺ C ⇒ A = C & B = C. 5. Thus, from the definition of CA we see that B ∈ CA , as required.

That is, [ABC] ⇔ (A ≺ B & B ≺ C) ∨ (C ≺ B & B ≺ A). Proof. Suppose [ABC]. 3. If A, B ∈ OP , C = O then [ABO] ⇒ (B ≺ A)OP ⇒ (A ≺ B)a ; also B ≺ C in this case from definition of order on line. 2 If A, B ∈ OP , C ∈ OQ then [ABC] & [BOC] =⇒ [ABO] ⇒ (A ≺ B)a and B ∈ OP & C ∈ OQ ⇒ (B ≺ C)a . For A ∈ OP , B = O, C ∈ OQ see definition of order on line. 2 For A ∈ OP , B, C ∈ OQ we have [AOB] & [ABC] =⇒ [OBC] ⇒ B ≺ C. If A = O and B, C ∈ OQ , we have [OBC] ⇒ B ≺ C. Conversely, suppose A ≺ B and B ≺ C in the given direct order on a.

2 [ADB] ∨ [BDC], [BAC] & [ADC] =⇒ [BDC], [ACB] & [ADC] =⇒ [ADB], which contradicts ABa & BCa. If ¬∃b (A ∈ b & B ∈ b & C ∈ b) (see Fig. 4 a & [ADC]) =⇒ ∃E (E ∈ a & [AEB]) ∨ ∃F (F ∈ a & [BF C]), which contradicts ABa & BCa. e. 52 The line a is called the edge of the half-plane aA . The edge a of a half-plane χ will also sometimes be denoted by ∂χ. 50 In particular, if an open interval (CD) is included in the open interval (AB), the points C, D both lie on the segment [AB]. this theorem can be formulated as follows: Consider a ray OA , a point B ∈ OA , and a convex set A.