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In a Hilbert plane II, show: (a) Any rigid motion with at least three noncollinear fixed points must be the identity. (b) Any rigid motion is equal to the product of at most three reflections.

In a Hilbert plane II, show: (a) Any rigid motion with at least three noncollinear fixed points must be the identity. (b) Any rigid motion is equal to the product of at most three reflections.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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17.3 In a Hilbert plane II, show: (a) Any rigid motion with at least three noncollinear fixed points must be the identity. (b) Any rigid motion is equal to the product of at most three reflections.
Transcribed Image Text:17.3 In a Hilbert plane II, show: (a) Any rigid motion with at least three noncollinear fixed points must be the identity. (b) Any rigid motion is equal to the product of at most three reflections.
Definition If II is a geometry consisting of the undefined notions of point, line, between- ness, and congruence of line segments and angles, which may or may not sat- isfy various of Hilbert's axioms, we define a rigid motion of II to be a mapping 9: II → II defined on all points, such that: (1) p is a 1-to-1 mapping of the points of II onto itself. (2) sends lines into lines. (3) 9 preserves betweenness of collinear points. (4) For any two points A, B, we have AB (A)q(B). (5) For any angle α, we have Lα = Ly(a). In other words, o preserves the structures determined by the undefined notions in our geometry.
Transcribed Image Text:Definition If II is a geometry consisting of the undefined notions of point, line, between- ness, and congruence of line segments and angles, which may or may not sat- isfy various of Hilbert's axioms, we define a rigid motion of II to be a mapping 9: II → II defined on all points, such that: (1) p is a 1-to-1 mapping of the points of II onto itself. (2) sends lines into lines. (3) 9 preserves betweenness of collinear points. (4) For any two points A, B, we have AB (A)q(B). (5) For any angle α, we have Lα = Ly(a). In other words, o preserves the structures determined by the undefined notions in our geometry.
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