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Rigid Motion Consider a Hilbert plane II, and 6: II → II is a rigid motion. We say that a point x in the plane II is a fixed point of o, if (x) = x. (a) show that any rigid motion with at least three non-collinear fixed points must be the identity. (b) Show that a rigid motion having exactly one fixed point must be a rotation. (c) Show that any rotation can be written as the composition of two reflections. (d) Show that any rigid motion can be written as the product of at most three reflections.

Rigid Motion Consider a Hilbert plane II, and 6: II → II is a rigid motion. We say that a point x in the plane II is a fixed point of o, if (x) = x. (a) show that any rigid motion with at least three non-collinear fixed points must be the identity. (b) Show that a rigid motion having exactly one fixed point must be a rotation. (c) Show that any rotation can be written as the composition of two reflections. (d) Show that any rigid motion can be written as the product of at most three reflections.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter11: Topics From Analytic Geometry
Section11.5: Polar Coordinates
Problem 97E
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3. Rigid Motion Consider a Hilbert plane II, and 6: II → II is a rigid motion. We say that a point x in the plane II is a fixed point of o, if (x) = = x. (a) show that any rigid motion with at least three non-collinear fixed points must be the identity. (b) Show that a rigid motion having exactly one fixed point must be a rotation. (c) Show that any rotation can be written as the composition of two reflections. (d) Show that any rigid motion can be written as the product of at most three reflections.
Transcribed Image Text:3. Rigid Motion Consider a Hilbert plane II, and 6: II → II is a rigid motion. We say that a point x in the plane II is a fixed point of o, if (x) = = x. (a) show that any rigid motion with at least three non-collinear fixed points must be the identity. (b) Show that a rigid motion having exactly one fixed point must be a rotation. (c) Show that any rotation can be written as the composition of two reflections. (d) Show that any rigid motion can be written as the product of at most three reflections.
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