The Lumenary · Advanced undergraduate

Projective geometry

Name: ______________________

Date: ______________________

You will learn to

Use homogeneous coordinates to describe points and lines of the real projective plane, including points at infinity, and compute joins and meets with cross products
Prove that two distinct points lie on exactly one line and two distinct lines meet in exactly one point, and explain the principle of duality
State and prove the theorems of Desargues and Pappus
Prove that projective transformations preserve the cross-ratio, and that any four points in general position can be mapped to any other four by a unique projective transformation
Show that all non-degenerate real conics are projectively equivalent, and explain ellipse, parabola and hyperbola as one conic meeting the line at infinity in 0, 1 or 2 points

1.
In $\mathbb{R}P^2$ , where do the lines $y = 3x + 1$ and $y = 3x - 2$ (in the chart $z = 1$ ) meet? (Choose all correct descriptions of the meeting point.)
- A. $[1 : 3 : 0]$
- B. $[-2 : -6 : 0]$
- C. $[3 : 1 : 0]$
- D. $[0 : 0 : 1]$
- E.They do not meet.
2.
How many points does the projective plane $\mathbb{F}_3P^2$ over the field with three elements have?
3.
Four distinct collinear points have cross-ratio $(a, b; c, d) = 3$ . Over all $24$ orderings of the four points, how many different values does the cross-ratio take?
4.
Which properties are preserved by every projective transformation of $\mathbb{R}P^2$ ? (Choose all that apply.)
- A.Three points being collinear
- B.The cross-ratio of four collinear points
- C.One point being the midpoint of two others
- D.The angle between two lines
- E.A line being tangent to a conic
- F.A conic being a circle
5.
In the proof of Theorem 2 (four points), we write $v_4 = \lambda_1 v_1 + \lambda_2 v_2 + \lambda_3 v_3$ and claim every $\lambda_i \neq 0$ . Why?
- A.If, say, $\lambda_3 = 0$ , then $v_4$ lies in the span of $v_1, v_2$ , so $p_1, p_2, p_4$ would be collinear — excluded by general position.
- B.Because $v_1, v_2, v_3$ form a basis, so all coordinates of any vector are non-zero.
- C.Because we may rescale $v_4$ .
- D.Because the transformation must be invertible.
6.
The non-degenerate conic $x^2 - 4xy + 4y^2 + x = 0$ meets the line at infinity in how many points, and what kind of affine conic is it?
- A.One (double) point, $[2 : 1 : 0]$ : a parabola
- B.None: an ellipse
- C.Two: a hyperbola
- D.It is a pair of lines
7.
The Lean theorem line_unique has hypotheses (hpq : p ⨯₃ q ≠ 0) (hp : ℓ ⬝ᵥ p = 0) (hq : ℓ ⬝ᵥ q = 0) and concludes ∃ c : K, c • (p ⨯₃ q) = ℓ. What goes wrong if hpq is dropped?
- A.If p and q represent the same point, p ⨯₃ q = 0, so the conclusion would force ℓ = 0 — false for any genuine line through p.
- B.Nothing: the theorem stays true without it.
- C.The theorem would need K to be algebraically closed.
- D.The dual statement about lines would fail but the statement about points would hold.
8.
Which statements about the theorems of Desargues and Pappus are true? (Choose all that apply.)
- A.In the projective plane over the quaternions, Desargues holds but Pappus fails.
- B.Pappus' theorem implies Desargues' theorem (in any projective plane).
- C.In a finite projective plane, Desargues implies Pappus.
- D.Every projective plane (satisfying the incidence axioms) satisfies Desargues.
- E.Pappus fails in $\mathbb{C}P^2$ because $\mathbb{C}$ is not ordered.
9.
Write join(p, q) returning the cross product of two triples (the line through two points, or the point on two lines) and collinear(p, q, r) returning True when the three points of $\mathbb{R}P^2$ lie on a line (use the triple product $p \cdot (q \times r)$ , with exact integer or fraction arithmetic). Then confirm Pappus' theorem on an example.
Computational — Practice

Answer key — Projective geometry

1.
$(3, -1, 1) \times (3, -1, -2) = ((-1)(-2) - 1 \cdot (-1),\ 1 \cdot 3 - 3 \cdot (-2),\ 3 \cdot (-1) - (-1) \cdot 3) = (3, 9, 0)$ , so the meet is $[3 : 9 : 0] = [1 : 3 : 0] = [-2 : -6 : 0]$ : the point at infinity of slope $3$ , the same in both directions.
2.
$(3^3 - 1)/(3 - 1) = 26/2 = 13 = 3^2 + 3 + 1$ : the $9$ points of the affine chart plus $4$ points at infinity (one for each of the $4$ directions, i.e. slopes $0, 1, 2, \infty$ ).
3.
With $\lambda = 3$ : $3, \tfrac13, -2, -\tfrac12, \tfrac23, \tfrac32$ — six distinct values. Each value is taken by $4$ orderings (the Klein four-group of double swaps acts trivially), and $24/4 = 6$ . The first experiment lists them.
4.
Preserved: collinearity, cross-ratio and tangency. Not preserved: midpoints, angles, being a circle. Klein's view: these last notions belong to smaller groups (affine, Euclidean) that fix extra structure — a line at infinity, and the circular points on it.
5.
If $\lambda_3 = 0$ then $v_4 = \lambda_1 v_1 + \lambda_2 v_2$ lies in the plane spanned by $v_1, v_2$ , i.e. $p_4$ is on the line $p_1p_2$ — three collinear points, contradicting general position. With all $\lambda_i \neq 0$ , the columns $\lambda_i v_i$ are still a basis and their sum is $v_4$ .
6.
Homogenise: $x^2 - 4xy + 4y^2 + xz = 0$ . Setting $z = 0$ gives $(x - 2y)^2 = 0$ , a double root: the single point $[2 : 1 : 0]$ , where the line at infinity is tangent. So it is a parabola with axis direction $(2, 1)$ . Its matrix $\begin{pmatrix} 1 & -2 & 1/2 \\ -2 & 4 & 0 \\ 1/2 & 0 & 0 \end{pmatrix}$ has determinant $-1$ , so it is non-degenerate.
7.
p ⨯₃ q ≠ 0 says p, q are linearly independent, i.e. distinct points (cross_ne_zero_iff). Without it, p = q is allowed; then p ⨯₃ q = 0, and the conclusion c • 0 = ℓ fails for every non-zero ℓ through p. Geometrically: one point does not determine a line.
8.
True: quaternions (Desargues yes, Pappus no), Hessenberg (Pappus ⇒ Desargues), and finite planes (Desargues ⇒ field coordinates ⇒ Pappus). False: non-Desarguesian planes exist, and Pappus holds over every field, including $\mathbb{C}$ .
9.
```
def join(p, q):
    return (p[1] * q[2] - p[2] * q[1],
            p[2] * q[0] - p[0] * q[2],
            p[0] * q[1] - p[1] * q[0])


def collinear(p, q, r):
    l = join(q, r)
    return p[0] * l[0] + p[1] * l[1] + p[2] * l[2] == 0
```
The same function computes joins of points and meets of lines — duality in code. With integer coordinates everything is exact, so the Pappus check is an exact verification of one configuration; the coordinate proof in the lesson covers all of them.