Projective Geometry

On the affine-projective line, find the coordinates of the midpoint of the segment $E_{1} E_{2}$ in the projective coordinate system $E_{1} E_{2} E$, if $E$ is the point at infinity of the line.

Let $E$ be the point at infinity of an affine-projective line. Find the projective coordinates of the point dividing the segment $E_{1} E_{2}$ in the ratio $\lambda$, in the projective coordinate system $E_{1} E_{2} E$.

Taking the points $E_{1}(-4), E_{2}(2)$ as the fundamental points of a projective system, and the point $E(3)$ as the unit point, find the relation between the projective coordinates $(x_{1}: x_{2})$ of a point $M$ and its affine coordinate $x$.

An affine coordinate $x$ is introduced on a line. Taking the point with coordinate 0 as the fundamental point $E_{1}$, the point with coordinate 1 as the fundamental point $E_{2}$, and the point at infinity as the unit point $E$ of the projective coordinate system, find the relation between the coordinate $x$ of an arbitrary point and its projective coordinates $\left(x_{1}: x_{2}\right)$.

Given three points $A(-2), B(3)$ and $C(-1)$. Taking these points respectively as the points $E_{1}, E_{2}$ and $E$, find the projective coordinates of the points $D(5), F(-7), G(4), H(2), O(0), P(1)$ in the projective coordinate system $E_{1} E_{2} E$.

With the affine coordinate $x$ on the affine-projective plane, the standard projective coordinate system $E_{1} E_{2} E$ is associated. It is known that the cross ratio $(E_{1} E_{2} E M)$ equals $a$. What is the affine coordinate of the point $M$?

Find the cross ratio $(A B C D)$ of four points $A, B, C, D$ of a projective line in each of the following cases:

On a projective line, three points $A(1: 2), B(-1: 1)$, $C(3: 5)$ are given. It is known that $(A B C D) = 1 / 2$. Find the projective coordinates of the point $D$.

The cross ratio $(A B C D)$ of four points $A, B, C$, $D$ equals $\lambda$. What is the cross ratio:

In the triangle $P Q R$, the median $P M$ is drawn. Considering the pencil of lines passing through the point $P$ as a projective line, find in it the fourth harmonic line to the lines $P Q$, $P R, P M$.

In the triangle $P Q R$, the bisector $P M$ is drawn. Considering the pencil of lines passing through the point $P$ as a projective line, find in it the fourth harmonic line to the lines $P Q, P R, P M$.

In the plane, a triangle $P Q R$ is given whose angle $P$ is right. In the triangle, the altitude $P M$ is drawn. Considering the pencil of lines passing through the point $P$ as a projective line, find in it the fourth harmonic line to the lines $P Q, P R, P M$.

In the plane, a line $\ell$ is given, on which three points $A, B, C$ are marked. Show how, using a straightedge, one can construct a fourth point $D$ harmonic to $A, B, C$, i.e., such that $(A B C D) = -1$.

In each of the following cases, find the projective transformation carrying the points $A, B, C$ into the points $A^{\prime }, B^{\prime }, C^{\prime }$ respectively:

(Types of projective transformations of a line). Under what necessary and sufficient condition does the projective transformation of the projective line

Prove that the projective transformation

is elliptic.

Can a projective transformation of a line be improper if it is:

Find the general form of the transformation leaving fixed the fundamental points $E_{1}$ and $E_{2}$.

Find the fixed points of the projective transformation

Find all projective transformations under which two distinct points $(a: b)$ and $(c: d)$ are fixed.

(Reflection with respect to a pair of points). Find all projective transformations of a line whose square equals the identity transformation, and determine their type.

Three distinct points $A, B, C$ are given on a projective line. Prove that the projective transformation carrying these points respectively into the points $B, C, A$ is elliptic.

A projective transformation of a line carries the points $A$, $B, C$ into the points $A, C, B$ respectively. Using a straightedge, construct the second fixed point of this transformation.

How many parabolic projective transformations of a projective line are there under which the given points $A$ and $B$ go to the given points $A^{\prime }$ and $B^{\prime }$, provided that at least three of the four points $A, B, A^{\prime }, B^{\prime }$ are distinct?

Let $f$ be an elliptic projective transformation of a line, and let $f^{k} = \underbrace{f \circ \ldots \circ f}_{k}$ be its $k$-th power. Prove that if the transformation $f^{k}$ has a fixed point, then it is the identity.

On the affine-projective plane, a projective coordinate system $E_{1} E_{2} E_{3} E$ and an affine coordinate system $O x y$ are introduced so that in the system $O x y$ the lines $E_{2} E_{3}, E_{3} E_{1}, E_{1} E_{2}$ are given by the equations $x-4 = 0, y-3 = 0,3 x+4 y-12 = 0$ respectively, and the point $E$ has coordinates $(3,2)$. Find:

In the affine coordinate system $O x y$, the lines $E_{1} E_{2}, E_{2} E_{3}$, $E_{3} E_{1}$ are given by the equations $y = 2, x = 0$ and $y = 0$ respectively, and the point $E$ has coordinates $(1,1)$. Find, in the projective coordinate system $E_{1} E_{2} E_{3} E$, the center of the pencil of lines parallel to the axis $O y$.

The fundamental points and the unit point of the projective coordinate system $E_{1} E_{2} E_{3} E$ have the following Cartesian coordinates with respect to the affine coordinate system $O x y: E_{1}(1,1), E_{2}(-1,1)$, $E_{3}(0,0), E(0,1 / 2)$. Find, in the coordinate system $E_{1} E_{2} E_{3} E$, the equations of the coordinate axes $O x, O y$ and of the line at infinity.

In the plane with the affine coordinate system $O x y$, a point $S(1 / 2,1 / 3)$ is given. Find the relation between the homogeneous coordinates $(x: y: z)$ of a point $M$ in the standard projective system associated with $O x y$, and its projective coordinates $(x_{1}: x_{2}: x_{3})$ with respect to the system $E_{1} E_{2} E_{3} E$, if the points $E_{1}, E_{2}, E_{3}, E$ are taken to be respectively the points $O, Q, R, S$, where $Q$ and $R$ are the points at infinity of the axes $O x$ and $O y$ respectively.

In the plane, points $A(1,2), B(-1,2), C(2,3), D(-3,4)$ are given. Taking them respectively as the fundamental points $E_{1}, E_{2}, E_{3}, E$ of a projective coordinate system, find in it the coordinates of the points $F(-2,0)$ and $G(5,3)$.

With respect to the projective coordinate system $E_{1} E_{2} E_{3} E$, the fundamental points and unit point of another system are given: $E_{1}^{\prime }(4: 1: 1)$, $E_{2}^{\prime }(4: 4: 1), E_{3}^{\prime }(0: 4: 1), E^{\prime }(2: 1: 1)$. Find the formulas expressing the coordinates $(x_{1}: x_{2}: x_{3})$ of an arbitrary point with respect to the coordinate system $E_{1} E_{2} E_{3} E$ in terms of its coordinates $(x_{1}^{\prime }: x_{2}^{\prime }: x_{3}^{\prime })$ in the system $E_{1}^{\prime } E_{2}^{\prime } E_{3}^{\prime } E$.

On the affine-projective plane, a triangle $E_{1} E_{2} E_{3}$ is given, with $E$ the point of intersection of its medians. In the projective coordinate system $E_{1} E_{2} E_{3} E$, find the coordinates of the points at infinity of the sides of the triangle and the equations of its medians.

Find the equation of the line at infinity of the affine-projective plane in the projective coordinate system $E_{1} E_{2} E_{3} E$, if $E$ is taken to be the point of intersection of the bisectors of the interior angles of the triangle $E_{1} E_{2} E_{3}$, whose sides $E_{2} E_{3}, E_{3} E_{1}, E_{1} E_{2}$ have lengths $a, b, c$ respectively.

Find the equation of the line at infinity of the affine-projective plane in the projective coordinate system $E_{1} E_{2} E_{3} E$, if $E$ is taken to be the point of intersection of the altitudes of the triangle $E_{1} E_{2} E_{3}$, whose sides $E_{2} E_{3}, E_{3} E_{1}, E_{1} E_{2}$ have lengths $a, b, c$, and whose opposite angles are $\alpha , \beta , \gamma$ respectively.

In a projective coordinate system, write the equation of the line passing through the points $(1: 2:-1)$ and $(3: 5:-2)$.

In some arithmetic model of the projective plane, four points $A(1: 2: 3), B(-1: 2: 2), C(3: 1: 5), D(2: 0: 1)$ are given. Find the point of intersection of the lines $A B$ and $C D$.

(Desargues' theorem). Points $A, B, C, P, Q, R$ are given such that the points of intersection of the pairs of lines $P Q$ and $A B$, $Q R$ and $B C$, $P R$ and $A C$ lie on a single line. Prove that the lines $A P, B Q, C R$ belong to a single pencil.

On the line $E_{1} E_{3}$, two arbitrary distinct points $A$ and $B$ are taken, different from the points $E_{1}$ and $E_{3}$; on the line $E_{1} E_{2}$, two arbitrary distinct points $C$ and $D$ are taken, different from the points $E_{1}$ and $E_{2}$. Prove that the points of intersection of the pairs of lines $E_{2} E_{3}$ and $B C$, $E_{2} A$ and $D B$, $E_{3} D$ and $C A$ lie on a single line.

Through two distinct points $Q$ and $R$ of the projective plane, lines are drawn: through the point $Q$ the lines $\ell_{1}, \ell_{3}, \ell_{5}$; through the point $Q$ the lines $\ell_{2}, \ell_{4}, \ell_{6}$. Denote by $(i k)$ the point of intersection of the lines $\ell_{i}$ and $\ell_{k}$. Prove that the lines passing through the pairs of points (12) and (45), (23) and (56), (34) and (61) belong to a single pencil.

State and prove the proposition dual to the proposition of the previous problem.

Prove that if $A_{1}, A_{2}, A_{3}, A_{4}$ are four points of the projective plane, no three of which lie on a single line, then the three points of intersection of the lines $A_{1} A_{2}$ and $A_{3} A_{4}, A_{1} A_{3}$ and $A_{2} A_{4}, A_{1} A_{4}$ and $A_{2} A_{3}$ also do not lie on a single line.

With respect to a projective coordinate system, four points $A(1: 1: 2), B(3:-1: 2), C(11:-1: 10), D(3: 7: 10)$ are given. Prove that they lie on a single line. Find the cross ratio $(A B C D)$.

Three points $A(1: 2: 3), B(-3: 2: 4), C(-2: 4: 7)$ are given. Prove that they lie on a single line, and find the fourth harmonic point $D$ to them.

In the projective plane, a triangle $A B C$ is given and a point $S$ not lying on any of its sides. Let $P, Q$, and $R$ be the points of intersection of the lines $A S, B S$, and $C S$ respectively with the lines $B C, C A$, and $A B$, and let $L, M$, and $N$ be the points that are the fourth harmonic points to the triples $B C P$, $C A Q$, and $A B R$ respectively. Prove that the points $L, M$, and $N$ lie on a single line.

For the proposition of the previous problem, state and prove the proposition:

To what points do the points $A(1: 2: 3), B(2:-1: 4)$, $C(-1: 0: 1)$ go under the projective transformation $\left(x_{1}^{\prime }: x_{2}^{\prime }: x_{3}^{\prime }\right) = \left(\left(2 x_{1}-3 x_{2}+x_{3}\right):\left(3 x_{1}-x_{2}+4 x_{3}\right):\left(3 x_{1}-2 x_{2}+5 x_{3}\right)\right)$?

To what points do the fundamental points and the unit point go under the projective transformation

On the projective plane, four points are given, no three of which lie on a single line. Prove that there exists a projective transformation carrying these points into the vertices of a parallelogram.

Find the projective transformation that leaves fixed the fundamental points of a given projective coordinate system, and carries the unit point $E$ into the point $E^{\prime }\left(c_{1}: c_{2}: c_{3}\right)$.

Find the projective transformation carrying the fundamental point $E_{1}$ of a given projective system into $E_{2}$, the point $E_{2}$ into $E_{3}$, the point $E_{3}$ into $E_{1}$, and leaving the unit point fixed.

Find the projective transformation of the affine-projective plane that leaves fixed the origin and the unit point, and carries the points at infinity of the axes $O x$ and $O y$ into the points $(1,0)$ and $(0,1)$ respectively.

To what lines do the lines $\ell_{1}: x_{1}+3 x_{2}+2 x_{3} = 0, \quad \ell_{2}:-x_{1}+2 x_{2}+5 x_{3} = 0, \quad \ell_{3}: 4 x_{2}-3 x_{3} = 0$ go under the projective transformation

Find the general form of the projective transformations which:

Find the projective transformation that leaves fixed the point $(0,0)$ and carries the lines

respectively into the lines

Determine the projective class of the following curves, using Lagrange's method:

Find the projective transformation that carries:

Find the equation of the circle circumscribed about a square, taking three of its vertices as the fundamental points, and the fourth as the unit point of the projective coordinate system.

How does the equation of the parabola $y^{2} = 4 x$ transform if one passes from the Cartesian coordinates $(x, y)$ to the projective coordinates $(x_{1}: x_{2}: x_{3})$, taking $E_{1}(1,0), E_{2}(-1,2), E_{3}(-1,-2)$ as the fundamental points, and the origin as the unit point of the projective coordinate system?

How does the equation of the circle $x^{2}+y^{2} = 1$ transform if one passes from the Cartesian coordinates $(x, y)$ to the projective coordinates $(x_{1}: x_{2}: x_{3})$, taking as the sides $E_{2} E_{3}, E_{3} E_{1}, E_{1} E_{2}$ of the fundamental triangle the lines $1+y = 0,1-y = 0, x = 0$, and as the unit point the point $E(1,0)$?

Find the projective transformation that carries the circle $x^{2}+y^{2} = 25$ into itself, and the points $A(3,-4), B(3,4)$ and the midpoint $M$ of the chord $A B$ into the points $A^{\prime }(-4,3), B^{\prime }(-4,-3)$ and the midpoint $M^{\prime }$ of the chord $A^{\prime } B^{\prime }$.

Find the projective transformation that carries the ellipse

into the same ellipse, and the points of the ellipse $A(0,5), B(-10,0), C(8,3)$ respectively into the points $A^{\prime }(0,-5), B^{\prime }(0,5), C^{\prime }(10,0)$.

Find the general form of the projective transformations under which the hyperbola

goes into the same hyperbola, and the tangents at the vertices of the hyperbola go into its asymptotes.

Find the projective transformation under which the parabola $y^{2} = 4 x$ goes into the same parabola, and the points of the parabola $A(1,2)$, $B(16,-8), C(1,-2)$ go respectively into the points $A^{\prime }(1,2)$, $B^{\prime }(1,-2), C^{\prime }(0,0)$.

Find the projective transformation of the parabola $y^{2} = 4 x$ into the circle $x^{2}+y^{2} = 25$, under which the points of the parabola $A(4,-4), B(0,0)$, $C(4,4)$ go into the points of the circle $A^{\prime }(0,-5), B^{\prime }(5,0), C^{\prime }(0,5)$.

In the projective coordinate system $E_{1} E_{2} E_{3} E$, find the general equation of an oval tangent to the two sides $E_{3} E_{1}$ and $E_{3} E_{2}$ of the triangle $E_{1} E_{2} E_{3}$ at the vertices $E_{1}$ and $E_{2}$.

Prove that any oval on the affine-projective plane can be written parametrically in the form

where the matrix $(c_{i j})$ is nonsingular. Under what conditions does this equation define:

Write the parametric equations of the circle $x^{2}+y^{2} = 1$, taking as the parameter $t$ of a point $M$ the slope of the line passing through the point $M$ and a fixed point $(\cos \varphi , \sin \varphi )$ of the circle.

Find parametric equations of the form (9.2) for the hyperbola

such that for the parameter values $t = -1, t = 1, t = 0$ one obtains the points $(-a, 0),(a, 0)$ and the point at infinity of the asymptote $\frac{x}{a}+\frac{y}{b} = 0$ respectively.

Find the general equation of the second-order curve given by the parametric equations:

Let two parametric curves $(x(t), y(t))$ and $(u(t), v(t))$ of the form (9.2) define the same oval on the affine-projective plane, i.e., there exists a function $s(t)$ such that $u(s(t)) = x(t)$, $v(s(t)) = y(t)$. Prove that the function $s(t)$ is then fractional-linear:

where $\left(\begin{smallmatrix} a & b \\ c & d\end{smallmatrix}\right)$ is a nonsingular matrix.

Let $A_{1}, A_{2}, A_{3}$ be three arbitrary points lying on an oval, and let $\ell_{1}, \ell_{2}, \ell_{3}$ be the tangents to the oval at these points. Prove that the three points of intersection of the lines $\ell_{1}$ and $A_{2} A_{3}, \ell_{2}$ and $A_{3} A_{1}, \ell_{3}$ and $A_{1} A_{2}$ lie on a single line.

Prove that if the triangle $A_{1} A_{2} A_{3}$ is circumscribed about a real oval, and $A_{4}, A_{5}, A_{6}$ are the points of tangency of the sides $A_{1} A_{2}, A_{2} A_{3}$, $A_{3} A_{1}$ with the given curve, then the lines $A_{1} A_{5}, A_{2} A_{6}, A_{3} A_{4}$ pass through a single point.

Prove that if the equation

defines a real non-degenerate second-order curve, then the condition

is necessary and sufficient for the point $M\left(x_{1}: x_{2}: x_{3}\right)$ to be an interior point.

Let $A B C$ and $A^{\prime } B^{\prime } C^{\prime }$ be two triangles lying on the projective plane such that the lines $A A^{\prime }, B B^{\prime }, C C^{\prime }$ pass through a single point. Prove that the six points of intersection of the lines $A B$ and $A^{\prime } C^{\prime }, A B$ and $B^{\prime } C^{\prime }, A C$ and $A^{\prime } B^{\prime }, A C$ and $B^{\prime } C^{\prime }, B C$ and $C^{\prime } A^{\prime }, B C$ and $A^{\prime } B^{\prime }$ lie on one and the same second-order curve.

Let $A_{1}, A_{2}, A_{3}, A_{4}, A_{5}$ be five arbitrary points lying on a real oval. Prove that if $\ell$ is the tangent to this oval at the point $A_{1}$, then the three points of intersection of the lines $\ell$ and $A_{3} A_{4}$, $A_{1} A_{2}$ and $A_{4} A_{5}, A_{2} A_{3}$ and $A_{1} A_{5}$ lie on a single line.

(Brianchon's theorem). A hexagon $A_{1} A_{2} A_{3} A_{4} A_{5} A_{6}$ is circumscribed about a real non-degenerate second-order curve. Prove that the lines $A_{1} A_{4}, A_{2} A_{5}, A_{3} A_{6}$ pass through a single point.

Prove that if the sides of two triangles are tangent to a second-order curve, then a second-order curve can be drawn through the six vertices of these triangles.

A real non-degenerate second-order curve is given by the general second-degree equation in projective coordinates. Find the necessary and sufficient condition for the line $A x_{1}+B x_{2}+C x_{3} = 0$ to intersect the given curve.

Find the polar of the point $A$ with respect to the second-order curve given by the equation $F(x, y) = 0$ in each of the following cases:

Prove that the directrices of a second-order curve are the polars of the corresponding foci with respect to the given curve.

Find the pole of the line $\ell$ with respect to the second-order curve $F$ in each of the following cases:

Find the equation of the line passing through the point $(2,1)$ and polar-conjugate to the line $4 x-y+30 = 0$ with respect to the second-order curve $x^{2}-6 x y+9 y^{2}-12 x+14 y-7 = 0$.