Second-Order Curves in Projective Coordinates

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.