By A. Heyting, N. G. De Bruijn, J. De Groot, A. C. Zaanen
Bibliotheca Mathematica: a sequence of Monographs on natural and utilized arithmetic, quantity V: Axiomatic Projective Geometry, moment version specializes in the rules, operations, and theorems in axiomatic projective geometry, together with set idea, prevalence propositions, collineations, axioms, and coordinates. The booklet first elaborates at the axiomatic procedure, notions from set idea and algebra, analytic projective geometry, and occurrence propositions and coordinates within the airplane. Discussions concentrate on ternary fields hooked up to a given projective airplane, homogeneous coordinates, ternary box and axiom procedure, projectivities among traces, Desargues' proposition, and collineations. The e-book takes a glance at occurrence propositions and coordinates in house. themes comprise coordinates of some degree, equation of a airplane, geometry over a given department ring, trivial axioms and propositions, 16 issues proposition, and homogeneous coordinates. The textual content examines the basic proposition of projective geometry and order, together with cyclic order of the projective line, order and coordinates, geometry over an ordered ternary box, cyclically ordered units, and basic proposition. The manuscript is a useful resource of knowledge for mathematicians and researchers drawn to axiomatic projective geometry.
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4. In ^(Dl0): The 772's on I with given coinciding invariant points from a group. PftooF as for the preceding theorem. 5. I n 5$: B y every collineation a Πη between two lines is transformed into a Πη between their image lines. P R O O F . This is immediate from the definition of a Πη and the fact t h a t a collineation transforms lines into lines and preserves the incidence relations.
PROOF. See the proof of Th. 2. 15. I n 5β(Ζ)9), the dual of D99 denoted by dD99 is a theorem. dD9 can be formulated as follows. Let the trilaterale a1a2a3 and b±b2b3 be given, such t h a t cor responding sides as well as corresponding vertices are different. Let AiB{ be denoted by ci9 a{ n b{ by C{. If Cl9 C 2 , C 3 are on a line l9 while ^4X € bl9 A2eb29 then c l 5 c 2 , c 3 pass through a point 0. Let us denote cx n c2 by 0 ; we prove t h a t 0 € c 3 . By the remark following Th. 2 we m a y suppose t h a t Βζφ09 04α29 02Φ C 3 , bx^l9 Α2Φ C1.
A centre of a collineation (£ is an invariant point of (£. II. A collineation with two different centres is the identity. III. If in a central collineation © with centre C the line Z, not through C, is invariant, then every point of I is invariant. IV. A collineation with two different lines of invariant points is the identity (dual of II). 2. In $ : Every central collineation which is not the identity has one and only one line of invariant points. PROOF. Let K be a central collineation with centre C.