Rotate, reflect and interpret fixed points and area

Anticlockwise rotation about the origin

The unit basis points rotate to (cosθ,sinθ) and (−sinθ,cosθ). Put these images in columns. Clockwise rotation through θ uses R₋θ=Rθᵀ. Angles may be specified exactly by both sine and cosine

Rotate (2,1) anticlockwise through π/2 about the origin. Find the image x

For cosθ=3/5 and sinθ=4/5, find entry (1,2) of Rθ

A different centre needs subtraction and addition

Subtract the centre before rotating and add it afterwards. The middle expression gives the affine translation part; the last expression reverses the rotation

Rotate P=(4,1) through π/2 anticlockwise about C=(1,−1). Find the image x.

Reflect in a line through the origin

Rotate the mirror to the x-axis, reflect there, then rotate back. For mirror direction θ this gives Sθ=RθFR₋θ. The double angle matters: θ+π describes the same line. For an offset mirror through C, use C+Sθ(p−C)

For the mirror with cosθ=3/5 and sinθ=4/5, find entry (1,1) of Sθ

Mirror through the origin

\(\displaystyle \cos\theta=\frac35,\quad\sin\theta=\frac45\)

Find entry (1,2) of the same reflection matrix

Fixed points and invariant sets differ

Solve the displayed equation to find fixed points. A set is invariant if its image equals itself; its points may move. A reflection fixes every point on its mirror, but also maps the perpendicular line onto itself while reversing its nonzero points. A nontrivial origin rotation fixes only the origin

For reflection in the line of direction (3/5,4/5), write the fixed-point set as y=mx. Find m.

Under (x,y)↦(x,−y), what is true of the y-axis?

Determinants control signed area

A linear map scales planar area by |det A|. The determinant sign records preservation or reversal of orientation when nonzero. A singular map collapses the plane to at most a line and has zero area factor. Translation does not change area. Rotations have determinant 1 and reflections −1.

Matrix

\(\displaystyle A=\begin{bmatrix}2&3\\0&-2\end{bmatrix}\)

A region of area 7 has image under the displayed matrix A. Find the image area

Matrix

\(\displaystyle S=\begin{bmatrix}1&2\\0&1\end{bmatrix}\)

A shear has determinant 1. Which conclusion follows?

Derive and justify the geometric laws, ungraded

Derive the rotation and reflection matrices from basis images and composition. Prove rotations preserve lengths, reflections square to I, and a rotation with θ not a multiple of 2π has only the origin fixed. Explain the determinant area factor using a transformed unit square

Model: verify algebra against geometry

RθᵀRθ=I by sin²θ+cos²θ=1, so squared lengths are unchanged. For F=diag(1,−1), S=RθFR₋θ gives S²=RθF²R₋θ=I. The fixed-point matrix Rθ−I has determinant 2−2cosθ, positive unless θ is a multiple of 2π; thus only p=0 solves it. The unit square maps to the parallelogram of the two matrix columns; its area is |ad−bc|. Triangulation and linear scaling extend the factor to planar regions, while translations preserve all differences

Sketch and identify invariants, ungraded

Sketch P=(2,1), its π/2 rotation Q and the two radial segments, using equal axis scales. Compare with reflection in y=x. Identify the fixed points and give an invariant set that is not pointwise fixed for each map

Model: separate point and set claims

The rotation gives Q=(−1,2), preserves radius √5 and has only the origin fixed. Every circle centred at the origin is invariant as a set. Reflection gives (1,2), fixes the entire line y=x, and leaves y=−x invariant as a set while reversing its nonzero points