Prove that every proper affine plane isometry F[𝐱]=Q 𝐱+𝐛 of ℝ^2, where det Q=1, is either (i) a

Step 1: Understand the form of the affine plane isometry.u000AGiven $F[u005Cmathbf{x}] u003D Q u005Cmathbf{x} +

Prove that every proper affine plane isometry F[𝐱]=Q 𝐱+𝐛 of...

Prove that every proper affine plane isometry $F[\mathbf{x}]=Q \mathbf{x}+\mathbf{b}$ of $\mathbb{R}^2$, where $\operatorname{det} Q=1$, is either (i) a translation, or (ii) a rotation (7.43) centered at some point $\mathbf{c} \in \mathbb{R}^2$.
Hint: Use Exercise 1.5.7.

Key Concepts

Affine Transformation

An affine transformation is a function on the plane of the form F(x) = Q x + b, where Q is a linear transformation and b is a fixed translation vector. Affine transformations preserve collinearity and ratios of distances along parallel lines, forming a fundamental class of maps in geometry that combine linear effects like rotation, scaling, or shear with translations.

Euclidean Isometry

A Euclidean isometry is a distance-preserving map, meaning it maintains the length of vectors and the angles between them. In the plane, isometries include translations, rotations, reflections, and glide reflections. These maps are fundamental in studying the symmetry and structure of geometric figures in Euclidean space.

Proper Isometry

A proper isometry is an isometry that preserves the orientation of the plane. For affine maps represented as F(x) = Q x + b, the condition that det Q = 1 indicates that Q is an element of the special orthogonal group SO(2), meaning no reflection is present. Proper isometries in the plane thus include translations and rotations only.

Translation

A translation is an affine transformation where the linear part Q is the identity matrix, so every point is shifted by a constant vector b. Translations preserve distances and directions identically across the plane and are characterized by having no fixed rotation or reflection component.

Rotation

A rotation is an affine isometry with a nontrivial orthogonal matrix Q that represents a rotation about some center point c. In such cases, the transformation can be rewritten in the form F(x) = Q (x - c) + c, indicating that the point c remains fixed. Rotations are distinguished from translations by having a fixed point and a non-identity linear part.

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