Affine and Isometric Transformations

Given two affine transformations

Find the transformation $f^{-1}$ inverse to $f$, and the compositions $f^{-1} \circ g$ and $g \circ f^{-1}$.

With respect to the coordinate system $Oxy$, an affine transformation $x^{\prime } = 2 x+3 y+1, y^{\prime } = x+2 y-2$ is given. Find the coordinates of the preimages of the points $(0,1),(1,0),(1,1)$, given in the coordinate system $O^{\prime } u v$, if the new coordinates are related to the old ones as follows:

Find the formulas of the affine transformation carrying the points $(2,0),(0,3),(1,2)$ into the points $(1,3),(-2,1),(0,0)$ respectively.

Find the affine transformation carrying the point $(-1,1)$ into the point $(1,2)$, and the vectors $(1,1)$ and $(0,1)$ into the vectors $(0,2)$ and $(2,-1)$ respectively.

Find the affine transformation under which all points of the line $2 x+3 y-5 = 0$ are fixed, and the point $(0,1)$ goes to the point $(2,-1)$.

Under an affine transformation, the point $(0,0)$ goes to the point symmetric to it with respect to the line $\frac{x}{2}+\frac{y}{3} = 1$, and the points $(2,0)$ and $(0,3)$ are interchanged. Find the formulas for this transformation. The coordinate system is rectangular.

Find the affine transformation under which the lines $10 x-4 y-1 = 0$ and $3 x-3 y+7 = 0$ go to the lines $x-2 y+3 = 0$ and $x-y-6 = 0$ respectively, and the point $(1,-2)$ goes to $(0,10)$.

Find the affine classification of triples of distinct lines.

With respect to the coordinate system Oxy, an affine transformation is given:

Write this transformation in the coordinate system $O^{\prime } u v$, if the new coordinates are related to the old ones as follows:

Given the affine transformation

On the line $x+y+2 = 0$, find the point which under this transformation goes to a point also lying on this line.

Given the affine transformation

and the point $A = (1,4)$. Find the line passing through the point $A$ which goes to a line also passing through $A$.

Given the affine transformation

Find the vector which under this transformation goes to a vector orthogonal to it. The coordinate system is rectangular.

Find the fixed point of the affine transformation carrying the vertices of the triangle $A, B, C$ into the points $B, C, A$ respectively.

It is known that an affine transformation carries the vertices of the triangle $A B C$ into the midpoints $K, L, M$ of the opposite sides. Find the images of the points $K, L, M$ and of the point $O$ of intersection of the medians of the triangle $A B C$ under this transformation. Determine the geometric meaning of the transformation.

Find the fixed points and invariant lines of the affine transformation

Do there exist affine transformations:

How will the affine transformation $x^{\prime } = x+4 y, y^{\prime } = 2 x+3 y$ be written if the invariant lines are taken as the axes of the new coordinate system $O^{\prime } u v$?

Prove that if an affine transformation has a unique fixed point, then every invariant line passes through this point. How many invariant lines can there be in this case?

In a rectangular coordinate system, write the formulas for the rotation of the plane by an angle $\varphi$ about the point $\left(x_{0}, y_{0}\right)$.

Write the formulas for the homothety transformation with coefficient $k$ and center at the point $(x_{0}, y_{0})$.

Find the similarity transformation that is the composition of a rotation by an angle $\pi / 2$ about the point $(-1,2$) and a homothety centered at this point with coefficient 2. The coordinate system is rectangular.

Find the similarity transformation with fixed point $(1,3)$ carrying the point $(4,-1)$ into the point $(-9,3)$. The coordinate system is rectangular.

Find the affine transformation that is the composition of a reflection with respect to the line $x+2 y+3 = 0$ and a homothety centered at the point $(1,-2)$ on this line with coefficient 5. The coordinate system is rectangular.

Find the affine transformation of compression toward the line $x+y-2 = 0$ along the direction perpendicular to it with compression coefficient equal to 3. The coordinate system is rectangular.

Determine the geometric meaning of the affine transformation:

With respect to a rectangular coordinate system, an affine transformation $x^{\prime } = k_{1} x, y^{\prime } = k_{2} y, k_{1} > 1 > k_{2} > 0$ is given. Find the lines passing through the origin which are mapped isometrically onto their images by this transformation.

With respect to a rectangular coordinate system, an affine transformation $x^{\prime } = 2 x+2 y, y^{\prime } = x+2 y$ is given. Find a pair of mutually perpendicular vectors which, under this transformation, goes to a pair of mutually perpendicular vectors.

Prove that the transformation

is a similarity transformation reversing orientation. Represent this transformation as the composition of a reflection with respect to a line and a homothety centered on this line. The coordinate system is rectangular.

Find the affine transformation carrying the vertices of the tetrahedron $A(0,0,0), B(1,0,0), C(0,1,0), D(0,0,1)$:

Find the set swept out by the lines that pass through the origin and are mapped isometrically onto their images by the affine transformation $x^{\prime } = k_{1} x, y^{\prime } = k_{2} y, z^{\prime } = k_{3} z$, where not all $k_{i}$ equal one.

Determine the geometric meaning of the following affine transformation (the coordinate system is rectangular, $a^{2}+b^{2} \neq 0$):

Prove that if an affine transformation has a unique fixed point, then all invariant lines and planes pass through this point.

Find the fixed points, invariant lines, and invariant planes of the affine transformation:

Find the affine transformation under which the coordinate planes $x = 0, y = 0$ and $z = 0$ go to the planes $x-y+z-1 = 0$, $x+y+1 = 0$ and $2 x-z+1 = 0$ respectively, and the point $(1,1,1)$ goes to the point $(1,1,0)$.

Write the formulas for the orthogonal projection onto the plane $x+y+z = 1$. The coordinate system is rectangular.

Does there exist an affine transformation having no invariant lines and no fixed points, but having an invariant plane?

Find the affine transformation of compression along the normal to the plane $x+2 y+2 z+2 = 0$ with compression coefficient 4. The coordinate system is rectangular.

Find the formulas for the affine transformation that is the composition of a reflection with respect to the line $x = 1+2 t, y = 2-t$, $z = 5 t$ and a subsequent compression of space by a factor of two toward this line. The coordinate system is rectangular.

Let $\ell_{1}, \ell_{2}, \ell_{3}$ be three pairwise skew lines not parallel to a single plane, and let $\ell_{1}^{\prime }, \ell_{2}^{\prime }, \ell_{3}^{\prime }$ be another three pairwise skew lines, also not parallel to a single plane. Prove that there exists a unique affine transformation of space carrying the first triple of lines into the second.

Let $\ell_{1}, \ell_{2}, \ell_{3}$ be three pairwise skew lines parallel to a single plane, and let $\ell_{1}^{\prime }, \ell_{2}^{\prime }, \ell_{3}^{\prime }$ be another three pairwise skew lines, also parallel to some plane. Does there exist an affine transformation of space carrying the first triple of lines into the second?

Prove that any plane passing through the midpoints of skew edges of an arbitrary tetrahedron divides it into two parts of equal volume.

Find the equations of the preimages of the curves

Under the conditions of the previous problem, find the equations of the images of the given curves under this affine transformation.

Problem 853: given the curves

Prove that if $a$ and $b(a \geqslant b)$ are the semi-axes of the image of the unit circle under an affine transformation, $\mathbf{v}$ is an arbitrary vector, and $\mathbf{v}^{\prime }$ is its image under the same transformation, then

Find the locus of points of intersection of tangents to an ellipse passing through the endpoints of two of its conjugate diameters.

Show that if the second-order curve is the ellipse

then the locus of points from which tangents having conjugate directions can be drawn to this ellipse is the ellipse

If, however, the curve is a hyperbola or a parabola, then no tangents having conjugate directions can be drawn to it.

Find the locus of midpoints of the chords of an ellipse joining the endpoints of conjugate diameters.

Prove that the centroid of a triangle inscribed in one ellipse and circumscribed about another ellipse homothetic to the first with the same center coincides with the center of the ellipses.

Determine the locus of midpoints of the chords of the ellipse

cutting off segments of constant area $S$ from it.

Prove that the lines joining the endpoints of conjugate diameters of an ellipse are tangent to an ellipse homothetic to the given one. Find the homothety coefficient.

Prove that the segments $M M_{1}$ and $M M_{2}$ of the tangents drawn from a point $M$ to an ellipse ($M_{1}$ and $M_{2}$ being the points of tangency) are to each other as the diameters to which they are parallel.

Prove that the vertices of any rhombus circumscribed about an ellipse lie on the axes of symmetry of the same ellipse.

Prove that the ellipse of largest area inscribed in a given parallelogram is tangent to its sides at their midpoints.

Prove that the lengths of the chords subtending the arcs enclosed between two parallel secants to an ellipse are to each other as the diameters of the ellipse parallel to these chords.

A quadrilateral is circumscribed about an ellipse. Prove that the sum of the areas of the two triangles having a common vertex at the center of the ellipse and having as bases two opposite sides of the quadrilateral is equal to the sum of the areas of the other two such triangles.

Compute the area of the figure bounded by an ellipse with semi-axes $a$ and $b$.

Through a point $A$ of the plane, a secant to the ellipse is drawn, intersecting this ellipse at points $P$ and $Q$. Prove that $\frac{\left|A P\right| \cdot \left|A Q\right|}{d^{2}} =$ const, where $d$ is the diameter of the ellipse parallel to the secant.

Prove that two conjugate diameters of an ellipse divide it into four parts of equal area.

Find the ellipse of smallest area passing through three points $A, B, C$ of the plane, and compute its area.

Find all affine transformations that carry into itself the ellipse

and, moreover:

Find all affine transformations under which the hyperbola $x y = c$ goes into itself.

Find all affine transformations that carry into itself the hyperbola

and, moreover:

Using the hyperbolic functions $\operatorname {sh} t, \operatorname {ch} t$, determine the general affine transformation that carries into itself the hyperbola

Find a proper affine transformation under which the hyperbola $x^{2}-y^{2} = 4$ goes into itself, and the point $(2,0)$ goes to the point $(\sqrt{5}, 1)$. To what points does this transformation carry the foci of the hyperbola?

Consider two central radius vectors of a hyperbola such that the area of the hyperbolic sector formed by these radii and the arc of the hyperbola has a given value. Prove that:

Find the affine transformations carrying the parabola $y^{2} = 2 p x$ into itself. Among them, find the unimodular transformations, i.e., the transformations whose determinant equals 1.

Find the affine transformation carrying the parabola $y^{2} = 2 x$ into itself, under which the points $(0,0)$ and $(2,2)$ go to the points $(2,2)$ and $(2,-2)$ respectively. Into what lines do the directrix and the axis of the parabola go under this transformation?

Determine the affine transformation carrying the parabola

into the parabola $y^{2} = 2 p x$.

Determine the locus of points corresponding to the focus of the parabola $y^{2} = 2 p x$ under all its unimodular transformations into the same parabola.

Prove that there exists exactly one affine transformation carrying a parabola into itself, under which two given points of the parabola go to two other given points of the parabola.

Prove that the area of the segment of a parabola cut off by a chord $A B$ is $2 / 3$ of the area of the triangle whose sides are the chord $A B$ and the tangents to the parabola at the endpoints of the chord.

Find the locus of midpoints of the chords of the parabola $y^{2} = 2 p x$ cutting off segments of the parabola of constant area $S$.

Find the locus of centers of the ellipses obtained in the intersection of the ellipsoid $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}} = 1$ with planes passing through the point $\left(x_{0}, y_{0}, z_{0}\right)$.

Prove that the planes passing through the endpoints of three mutually conjugate diameters of the ellipsoid $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}} = 1$ are tangent to the ellipsoid $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}} = \frac{1}{3}$.

Find the fixed point of the isometric transformation $x^{\prime } = x \cos \varphi -y \sin \varphi +x_{0}, \quad y^{\prime } = x \sin \varphi +y \cos \varphi +y_{0}, \quad 0 < \varphi < 2 \pi$.

Write the formulas for the isometric transformation that is the composition of a reflection with respect to the line passing through the point $\left(x_{0}, y_{0}\right)$ with direction vector $(\cos \varphi , \sin \varphi )$, and a translation along this line by a vector having the direction of the given vector and length $d$.

Given the isometric transformation $x^{\prime } = x+x_{0}, y^{\prime } = -y+y_{0}$, which reverses the orientation of the plane. Find the axis of symmetry and the translation vector along the axis of symmetry.

Given the isometric transformation

Find the axis of symmetry and the translation vector along the axis of symmetry.

Find the isometric transformation that is a reflection with respect to the line $A x+B y+C = 0$.

Find the isometric transformation preserving orientation and carrying the point $(0,2)$ into the point $(1, \sqrt{3})$, and the point $(1, \sqrt{3})$ into the point $(2,2)$. Find the angle of rotation and the fixed point of this transformation.

Find the isometric transformation reversing orientation and carrying the point $(2,0)$ into the point $(1, \sqrt{3})$, and the point $(1, \sqrt{3})$ into the point $(3, \sqrt{3}$). Find the axis of symmetry and the translation vector along the axis of symmetry.

Prove that any rotation of the plane can be represented as the composition of two reflections.

Let an isometric transformation be given:

Prove that if its matrix $A$ is different from the identity matrix and $\det A = 1$, then this transformation is the composition of a rotation about some axis and a translation along this axis. Find the angle $\varphi$ of this rotation and the direction vector $\mathbf{a}$ of the axis of rotation in two cases: when $\mathbf{a}$ is not collinear with the unit vector $\mathbf{e}_{1}$, and when they are collinear. Find the vector $\mathbf{c}$ of translation along the axis of rotation, the equations of the axis of rotation, and the canonical form of the transformation.

With the same notation, prove that if $\det A = -1$ and $a_{11} + a_{22}+a_{33} \neq 1$, then the transformation (8.1) is the composition of a reflection with respect to some plane and a rotation about some axis perpendicular to this plane. Find the angle $\varphi$ of this rotation and the direction vector $\mathbf{a}$ of the axis of rotation in two cases: when $\mathbf{a}$ is not collinear with the unit vector $\mathbf{e}_{1}$ and $\varphi \neq \pi$, and when they are collinear. Find the (unique) fixed point, the plane of symmetry, the axis of rotation, and the canonical form of the transformation.

Prove that if $\det A = -1$ and $a_{11}+a_{22}+a_{33} = 1$ (with the same notation), then the transformation (8.1) is the composition of a reflection with respect to some plane and a translation by a vector $\mathbf{d}$, coplanar with this plane. Find the plane of symmetry, the vector $\mathbf{d}$, and the canonical form of the transformation.

Using the results of the previous problems, prove that any isometric transformation of space can be represented as the composition of a finite number of reflections.

Find the isometric transformation that is the composition of a rotation by an angle $\pi / 2$ and a translation along the axis of rotation by a vector with coordinate 9 with respect to the axis of rotation, given that the axis of rotation passes through the point $(1,0,0)$ and has direction vector $(-1,2,2)$.

Find a non-identity isometric transformation leaving fixed the three points $(1,0,0),(0,1,0)$ and $(0,0,1)$.

In an oriented space, two mutually perpendicular vectors $\mathbf{a}$ and $\mathbf{b}$ are given. Find the vector $\mathbf{a}^{\prime }$ obtained from the vector $\mathbf{a}$ under a rotation of space by an angle $\varphi$ about the axis with direction vector $\mathbf{b}$.

Determine the geometric meaning and find the canonical form of the following isometric transformations of space: