Math Problems Vault

Problem 53.1

(a)(b)

Find some projective transformation of the plane taking given lines to given lines:

?

(a)

$x=0 \mapsto x=0, \quad y=0 \mapsto x=1$ ;

(b)

$x+y=1 \mapsto x=1, \quad x+y=0 \mapsto y=0$ .

Problem 53.2

(a)(b)

Find some projective transformation of the plane taking given curves to given curves:

?

(a)

$x^{2}+y^{2}=1 \mapsto y=x^{2}$ ;

(b)

$x^{2}-y^{2}=1 \mapsto x^{2}+y^{2}=1$ .

Problem 53.3

(a)(b)

Find some projective transformation of the plane taking the circle $x^{2}+y^{2}=1$ to itself and:

?

(a)

taking the point $(0,0)$ to the point $(1 / 2,0)$ ;

(b)

taking the line $x=2$ to the line at infinity.

Problem 53.4

(a)(b)(c)

Find some projective transformation of space taking a given quadric to a given quadric:

?

(a)

$x^{2}+y^{2}+z^{2}=1 \mapsto x y-z^{2}=1$ ;

(b)

$x y=z \mapsto x^{2}+y^{2}-z^{2}=1$ ;

(c)

$x y=z^{2} \mapsto y=x^{2}$ .

Problem 53.5

(a)(b)

Find the maximum dimension of planes contained in the quadric:

?

(a)

$x_{1}^{2}+\ldots +x_{k}^{2}-x_{k+1}^{2}-\ldots -x_{n}^{2}=1$ ;

(b)

$x_{1}^{2}+\ldots +x_{k}^{2}-x_{k+1}^{2}-\ldots -x_{n}^{2}=x_{n}$ .

Problem 53.6

Prove that over the field of complex numbers, any projective transformation has at least one fixed point.

?

Problem 53.7

Prove that in a real projective space of even dimension, any projective transformation has a fixed point.

?

Problem 53.8

Prove that if a projective transformation of an $n$ -dimensional projective space over an infinite field has finitely many fixed points, then this number does not exceed $n+1$ .

?

Problem 53.9

Prove that for every finite set of points $A$ in a projective space over an infinite field there exists an affine chart $A$ containing it.

?

Problem 53.10

Prove that any $(k-1)$ -dimensional subspace in $\mathbb {P}^{n}$ can be covered by $k$ affine charts and cannot be covered by fewer affine charts.

?

Problem 53.11

Find the number of points of an $n$ -dimensional projective space over a field of $q$ elements.

?

Problem 53.12

Find the number of $k$ -dimensional subspaces of an $n$ -dimensional projective space over a field of $q$ elements.

?

Problem 53.13

Find the number of projective transformations of an $n$ -dimensional projective space over a field of $q$ elements.

?

Problem 53.14

Let $M_{1}$ and $M_{2}$ be disjoint planes in $\mathbb {P}^{n}, L_{1}$ and $L_{2}$ disjoint planes having the same dimensions. Prove that there exists a projective transformation which takes $M_{1}$ to $L_{1}$ and $M_{2}$ to $L_{2}$ .

?

Problem 53.15

Prove that if a projective transformation takes some affine chart to itself, then it induces an affine transformation on this chart.

?

Problem 53.16

Prove that every bijective transformation of a two-dimensional projective plane that takes lines to lines and preserves the cross-ratio of points on each line is projective.

?

Problem 53.17

Prove that by means of a suitable projective transformation, any four lines in the projective plane, no three of which meet at a single point, can be taken to any four lines possessing the same property.

?

Problem 53.18

Prove that there exists a projective transformation of the plane that preserves a given triangle and takes a given point inside this triangle to any other given point inside it.

?

Problem 53.19

Prove that there exists a transformation of the plane that preserves a circle and takes a given point inside this circle to any other given point inside it.

?

Problem 53.20

Prove that using a straightedge alone, one cannot construct the center of a given circle.

?

Problem 53.21

Prove, using projective transformations, that the segments joining the vertices of a triangle to points on the opposite sides meet at a single point if and only if these points are the points of tangency of some ellipse inscribed in the triangle.

?

Problem 53.22

(a)(b)

A painting depicts an avenue of trees. Denote by $l$ the distance from the first tree to the horizon line along the line of the avenue, and by $a_{k}$ the distance between the $k$ -th and $(k+1)$ -th trees. Express:

?

(a)

$a_{3}$ in terms of $a_{1}$ and $a_{2}$ ;

(b)

$a_{2}$ in terms of $l$ and $a_{1}$ .

Problem 53.23

(a)(b)

A projective transformation of the plane is called a homology if it fixes every point lying on some line (the axis of the homology) and every line passing through some point (the center of the homology). Prove that:

?

(a)

there exists a unique homology with a given axis $l$ and a given center $O$ that takes a given point $A \neq O, A \notin l$ , to a given point $A^{\prime } \neq O, A^{\prime } \notin l$ , lying on the line $O A$ ;

(b)

every projective transformation of the plane is a product of two homologies.

Problem 53.24

Prove that there exists a unique projective transformation of the plane that preserves the circle $x^{2}+y^{2}=1$ and takes three given points on this circle to three given points also lying on this circle.

?

Problem 53.25

Prove that if the lines $A A^{\prime }, B B^{\prime }$ , $C C^{\prime }$ meet at a single point, then the points of intersection of the lines $A B$ and $A^{\prime } B^{\prime }, B C$ and $B^{\prime } C^{\prime }, A C$ and $A^{\prime } C^{\prime }$ lie on a single line.

?

Problem 53.26

Prove that the points of intersection of opposite sides of a hexagon inscribed in a circle lie on a single line.

?

Problem 53.27

Prove that the points of intersection of opposite sides of a hexagon whose vertices lie alternately on two given lines lie on a single line.

?

Problem 53.28

Let $a_{1}, a_{2}, a_{3}, a_{4}$ be lines in the plane passing through a point $O, l$ a line not passing through $O$ . Prove that the cross-ratio of the points of intersection of the lines $a_{1}, a_{2}, a_{3}, a_{4}$ with the line $l$ does not depend on $l$ (the cross-ratio of the lines $a_{1}, a_{2}, a_{3}, a_{4}$ ).

?

Problem 53.29

(a)(b)(c)(d)

Let $f$ be a nonsingular bilinear function on an $(n+1)$ -dimensional vector space $V$ . To each $(k+1)$ -dimensional subspace $U \subset V$ associate the $(n-k)$ -dimensional subspace

U^{\perp }=\left\{ y \in V \mid f(x, y)=0 \text{ for every } x \in U\right\} .

This correspondence defines a map $K_{f}$ on the projective space $\mathbb {P}\left(V\right)$ , which assigns to each $k$ -dimensional plane an $(n-k-1)$ -dimensional plane (a correlation with respect to the function $f$ ).

Prove that:

?

(a)

the correlation preserves incidence, i.e.,

U_{1} \subset U_{2} \leftrightarrow K_{f}\left(U_{1}\right) \supset K_{f}\left(U_{2}\right) ;

(b)

if the function $f$ is symmetric or skew-symmetric, then the correlation $K_{f}$ is involutive, i.e.,

K_{f}\left(K_{f}(U)\right)=U

(c)

the composition of a correlation and a projective transformation is a correlation;

(d)

every correlation is the composition of a fixed correlation and some projective transformation.

Problem 53.30

Prove that every correlation of a projective line acts on points in the same way as some projective transformation.

?

Problem 53.31

Prove that a correlation on the projective plane preserves the cross-ratio.

?

Problem 53.32

Prove that the correlation on the projective plane with respect to a symmetric bilinear function $f$ takes each point of the curve $f(x, x)=0$ to the tangent to this curve passing through the given point.

?

Problem 53.33

State the theorem (Brianchon's theorem) obtained from Pascal's theorem (see 53.26) by applying a correlation.

?

Problem 53.34

Using the notion of correlation, prove that the points of intersection of the tangents to a given circle, drawn through the endpoints of all possible chords passing through a given point, lie on a single line.

?