The Lumenary · Early undergraduate

Transformation geometry and the Erlangen programme

Name: ______________________

Date: ______________________

You will learn to

Prove that every isometry of the plane is a translation, a rotation, a reflection or a glide reflection, and that each is a product of at most three reflections
Represent isometries, similarities and affine maps by complex numbers and by matrices in homogeneous coordinates, and compose them
Prove that affine maps preserve parallelism and ratios along lines but in general not lengths or angles
Prove that projective transformations of a line preserve the cross-ratio, and use it in perspective drawings
Explain Klein's Erlangen programme and compare the invariants of Euclidean, similarity, affine and projective geometry

1.
What kind of isometry is $f(z) = -z + 4$ ?
- A.The rotation by $180^\circ$ about the point $2$
- B.A translation by $4$
- C.A reflection
- D.The rotation by $180^\circ$ about $0$
2.
Which of these isometries of the plane move every point? (Choose all that apply.)
- A.A translation by a non-zero vector
- B.A glide reflection
- C.A rotation by $90^\circ$
- D.A reflection
3.
Two mirror lines cross at the point $P$ at an angle of $40^\circ$ . What is the composite of the two reflections?
- A.A rotation about $P$ by $80^\circ$ (clockwise or anticlockwise, depending on the order)
- B.A rotation about $P$ by $40^\circ$
- C.A reflection in the line bisecting the two mirrors
- D.A translation
4.
The points $0, 1, 3, -3$ have cross-ratio $(0, 1;\, 3, -3) = \dfrac{(3 - 0)(-3 - 1)}{(3 - 1)(-3 - 0)} = 2$ . The projective map $m(x) = \dfrac{2x + 1}{x + 1}$ sends them to $1, \tfrac32, \tfrac74, \tfrac52$ . What is the cross-ratio $(1, \tfrac32;\, \tfrac74, \tfrac52)$ ?
5.
Four equally spaced fence posts stand along a straight road. In a photograph, measured along the image of the road, the first three appear at $0$ , $10$ and $15$ mm. At how many millimetres does the fourth appear? (Use the cross-ratio: equally spaced points $0, 1, 2, 3$ have $(0, 1; 2, 3) = \tfrac43$ .)
6.
Which property is preserved by every affine map $x \mapsto Ax + b$ of the plane?
- A.Two lines are parallel
- B.Two lines are perpendicular
- C.A curve is a circle
- D.Two segments have equal length
7.
Claim: an isometry of the plane is determined by the images of two distinct points $p$ and $q$ . The claim is false. Which counterexample shows it?
- A.The identity and the reflection in the line through $p$ and $q$
- B.Two rotations about $p$ by different angles
- C.Two different translations
- D.A rotation about $p$ and a non-zero translation
8.
Which Lean statement correctly formalises the glide reflection $z \mapsto \bar z + t$ (with $t$ real and non-zero) has no fixed point?
- A.theorem _ {t : ℝ} (ht : t ≠ 0) (z : ℂ) : conj z + t ≠ z
- B.theorem _ {t : ℝ} (ht : t ≠ 0) : ∃ z : ℂ, conj z + t ≠ z
- C.theorem _ {t : ℝ} (z : ℂ) : conj z + t ≠ z
- D.theorem _ {t : ℝ} (ht : t ≠ 0) (z : ℂ) : conj (conj z + t) + t = z + 2 * t
9.
An isometry is stored as (a, b, opposite) meaning $z \mapsto az + b$ (if opposite is False) or $z \mapsto a\bar z + b$ (if True), with $|a| = 1$ . Write classify(a, b, opposite) returning 'identity', 'translation', 'rotation', 'reflection' or 'glide', using the criteria from the proof. The tests use Gaussian integers, so exact comparisons with == are safe.
Computational — Practice

Answer key — Transformation geometry and the Erlangen programme

1.
Direct, with $a = -1$ and $b = 4$ . Since $a \neq 1$ it is a rotation about $c = b/(1 - a) = 4/2 = 2$ , by the angle $\arg(-1) = 180^\circ$ : a half-turn about $2$ .
2.
Translations (by $b \neq 0$ ) and glide reflections have no fixed points; rotations fix their centre and reflections fix their mirror line. This is why 'every isometry fixes a point' is false.
3.
Both reflections fix $P$ , and two reflections give a direct isometry, so the composite is a rotation about $P$ . With $P = 0$ and mirrors at angles $\beta$ and $\alpha = \beta + 40^\circ$ : $z \mapsto e^{2i\alpha}\overline{e^{2i\beta}\bar z} = e^{2i(\alpha - \beta)}z$ , a rotation by $80^\circ$ . Reversing the order gives $-80^\circ$ .
4.
By the invariance theorem the answer is $2$ . Directly: $\frac{(3/4)(1)}{(1/4)(3/2)} = \frac{3/4}{3/8} = 2$ . Note that ordinary ratios are not preserved: the gaps $1, 2, 6$ between $0, 1, 3$ and $-3$ become $\tfrac12, \tfrac14, \tfrac34$ .
5.
$\frac{3(x - 10)}{x} = \frac43$ gives $9(x - 10) = 4x$ , so $x = 18$ mm. The gaps $10, 5, 3$ shrink towards the horizon, which (for the map $t \mapsto 30t/(t + 2)$ ) is at $30$ mm.
6.
Parallelism: the line $p + tv$ goes to $f(p) + tAv$ , so lines with a common direction keep a common direction. Perpendicularity, circles and equal lengths are destroyed by shears or stretches; they belong to Euclidean (or similarity) geometry, not affine geometry.
7.
The reflection in the line $pq$ fixes $p$ and $q$ but is not the identity. Two points determine an isometry only up to this reflection; three non-collinear points determine it completely (the lemma on three distances).
8.
'No fixed point' is a universal statement, ∀ z, conj z + t ≠ z, and it needs t ≠ 0. The existential version is far too weak, and the version without the hypothesis is false for t = 0.

def classify(a, b, opposite):
    a, b = complex(a), complex(b)
    if not opposite:
        if a == 1:
            return 'identity' if b == 0 else 'translation'
        return 'rotation'
    s = a * b.conjugate() + b
    return 'reflection' if s == 0 else 'glide'

For example $z \mapsto -\bar z + 2i$ has $s = -\overline{2i} + 2i = 4i \neq 0$ : it is a glide reflection, along the imaginary axis (reflect in $\operatorname{Re} z = 0$ , then slide by $2i$ ).