Movements in the Plane
At night, the lights along the Malecón in Santo Domingo look "upside down" in the water. Where exactly does each reflected light end up if you know its coordinates? Reflecting or sliding a figure are movements of the plane, and all of them can be computed with simple operations. Also, all of them preserve distances: the figure that moves is not deformed.
Translation
If you know the starting point $A$ and the ending point $B$, the vector of the translation is $\vec{AB}=B-A$ (end minus start).
Two translations in a row are equivalent to a single one, with the sum vector: $P + \vec u + \vec v = P + (\vec u + \vec v)$.
A delivery drone is at position $P(3, -2)$, expressed in kilometers relative to its base. The controller sends it an instruction to make a straight-line displacement…
A delivery drone is at position $P(3, -2)$, expressed in kilometers relative to its base. The controller sends it an instruction to make a straight-line displacement defined by the translation vector $\vec{v} = (-5, 4)$. Determine the coordinates of the drone's new position.
A laser cutting machine moves in a straight line on a metal sheet from the initial point $A(-1, 5)$ to the final point $B(4, 2)$. Calculate the translation vector that…
A laser cutting machine moves in a straight line on a metal sheet from the initial point $A(-1, 5)$ to the final point $B(4, 2)$. Calculate the translation vector that describes this movement exactly.
A fishing boat undergoes two consecutive movements on the open sea. First, it translates according to the vector $\vec{u} = (2, 6)$ and then according to the vector…
A fishing boat undergoes two consecutive movements on the open sea. First, it translates according to the vector $\vec{u} = (2, 6)$ and then according to the vector $\vec{v} = (3, -1)$. If it departed from a port located at coordinates $P(-4, -5)$, determine its final position after both displacements.
Point reflection
Why? If $M=\dfrac{A+A'}{2}$, solving for $A'$ gives $A'=2M-A$.
Point reflection of a line. The image of a line is another line parallel to it. To get its equation, replace $x$ with $2a-x$ and $y$ with $2b-y$, where $C(a, b)$ is the center. Another option is to transform two of its points.
In the design of a symmetrical garden, a fountain is to be placed at point $F'$, which is the reflection of a statue located at $E(4, 7)$ with respect to a central post…
In the design of a symmetrical garden, a fountain is to be placed at point $F'$, which is the reflection of a statue located at $E(4, 7)$ with respect to a central post located at the origin $O(0, 0)$. Find the coordinates where the fountain should be installed.
Two communications satellites, $S_1$ and $S_2$, orbit symmetrically with respect to a central space station. If $S_1$ is at position $(8, -3)$ and the space station is…
Two communications satellites, $S_1$ and $S_2$, orbit symmetrically with respect to a central space station. If $S_1$ is at position $(8, -3)$ and the space station is at $(2, 1)$, calculate the coordinates of the position of satellite $S_2$.
A railway track runs along the line $r \equiv 3x + 4y - 12 = 0$. A town planner wants to layout a new avenue $s$ that is perfectly symmetrical to the railway track with…
A railway track runs along the line $r \equiv 3x + 4y - 12 = 0$. A town planner wants to layout a new avenue $s$ that is perfectly symmetrical to the railway track with respect to a central monument located at $C(1, -1)$. Find the general equation of the avenue $s$.
Reflection over a line
Cases you can do in your head
| Axis | Reflection of $(x, y)$ |
|---|---|
| $x$-axis ($y=0$) | $(x, -y)$ |
| $y$-axis ($x=0$) | $(-x, y)$ |
| line $y=x$ | $(y, x)$ |
| vertical line $x=k$ | $(2k-x,\ y)$ |
| horizontal line $y=k$ | $(x,\ 2k-y)$ |
Any axis $Ax+By+C=0$
Substitute the point into the equation of the line and use
$$P'=P-\frac{2\,(Ax_0+By_0+C)}{A^2+B^2}\,(A, B)$$Why? The vector $(A, B)$ is perpendicular to the line. To get from $P$ to the axis you have to move in that direction; the factor gives exactly the distance, and it is doubled to reach the other side.
An architect designs a bridge whose arch has a critical anchoring point at $A(-3, 5)$. To maintain visual symmetry in a 3D rendering with respect to the water surface,…
An architect designs a bridge whose arch has a critical anchoring point at $A(-3, 5)$. To maintain visual symmetry in a 3D rendering with respect to the water surface, which coincides with the x-axis ($y=0$), they need to locate the virtual reflection of this point under the water. Determine the coordinates of this symmetrical point.
In a graphic design program, a polygon with a key vertex at point $V(5, -2)$ has been drawn. An axial symmetry (reflection) across the angle bisector of the first and…
In a graphic design program, a polygon with a key vertex at point $V(5, -2)$ has been drawn. An axial symmetry (reflection) across the angle bisector of the first and third quadrants (line $y=x$) is applied. Analytically determine the coordinates of the reflected vertex $V'$.
A laser beam is emitted from a projector situated at point $P(-2, 4)$. In front of it, there is a plane mirror placed exactly along the line with equation…
A laser beam is emitted from a projector situated at point $P(-2, 4)$. In front of it, there is a plane mirror placed exactly along the line with equation $r \equiv x - 2y + 5 = 0$. Calculate the coordinates of the virtual point $P'$ (the reflection of $P$ across the mirror) from which the reflected light appears to originate.
Axis of symmetry and shortest paths
The axis joining two points. If $A$ and $B$ are reflections of each other over a line, that line is the perpendicular bisector of segment $AB$: it passes through its midpoint and is perpendicular to it.
Shortest path with a bounce. A point $P$ on a line $r$ minimizes $AP+PB$ (with $A$ and $B$ on the same side of $r$) like this: you reflect $B$ over $r$ and get $B'$. Since $PB=PB'$, the sum $AP+PB'$ is minimal when $A$, $P$, $B'$ are collinear. It is the same idea as a bank shot in pool: angle of incidence equals angle of reflection.
Two forest watchtowers are located at points $T_1(-3, 1)$ and $T_2(5, 5)$. A straight firebreak is to be laid out as an axis of symmetry between both towers, such that…
Two forest watchtowers are located at points $T_1(-3, 1)$ and $T_2(5, 5)$. A straight firebreak is to be laid out as an axis of symmetry between both towers, such that any point on the firebreak is equidistant from $T_1$ and $T_2$. Calculate the general equation of the line defining this firebreak.
Two villages, $A(1, 5)$ and $B(7, 2)$, need to connect to a main water pipe running in a straight line along the x-axis ($y=0$). A pumping station $P$ is to be built on…
Two villages, $A(1, 5)$ and $B(7, 2)$, need to connect to a main water pipe running in a straight line along the x-axis ($y=0$). A pumping station $P$ is to be built on the main pipe such that the total length of the secondary pipes ($AP + PB$) is minimised. Using the concept of reflection, calculate the exact coordinates of point $P$ and the minimum total length of the pipe.
On a billiard table, a cue ball is located at point $B(2, 3)$ and a red ball at $R(8, 6)$. A player wishes to hit the red ball by having the cue ball bounce first off…
On a billiard table, a cue ball is located at point $B(2, 3)$ and a red ball at $R(8, 6)$. A player wishes to hit the red ball by having the cue ball bounce first off the top cushion of the table, which lies along the line $y = 10$. Assuming the angle of incidence is equal to the angle of reflection, calculate the coordinates of the exact point on the cushion where the cue ball must strike to achieve its objective.
Translating figures and equations
Why? A point $(x, y)$ is on the new figure if, when you undo the translation, $(x-h,\ y-k)$ was on the original figure.
Moving the center (or the vertex) to the origin. It is a translation by $\vec v=-(h, k)$, where $(h, k)$ is the center or the vertex.
The trajectory of a conveyor belt follows the line with equation $y = 2x - 1$. Due to space requirements in the factory, the entire belt is translated $3$ units to the…
The trajectory of a conveyor belt follows the line with equation $y = 2x - 1$. Due to space requirements in the factory, the entire belt is translated $3$ units to the left and $2$ units upwards. Write the general equation of the new line that defines the translated trajectory.
A cell phone tower provides coverage in a circular area whose Cartesian equation is $x^2 + y^2 - 4x + 6y - 12 = 0$. Due to interference issues with a nearby building,…
A cell phone tower provides coverage in a circular area whose Cartesian equation is $x^2 + y^2 - 4x + 6y - 12 = 0$. Due to interference issues with a nearby building, the company decides to translate the tower (the center of the circle) according to the vector $\vec{v} = (-3, 5)$. Determine the general equation of the new coverage area.
The orbit of a comet is projected onto an astronomical plane according to the equation of the ellipse $4x^2 + 9y^2 - 24x + 36y + 36 = 0$. By translating the axes, it is…
The orbit of a comet is projected onto an astronomical plane according to the equation of the ellipse $4x^2 + 9y^2 - 24x + 36y + 36 = 0$. By translating the axes, it is possible to express this equation in its canonical form centered at the origin. Calculate the translation vector that moves the geometric center of the conic to the origin, and write the equation of the ellipse after this translation.
Reflecting figures
To reflect a figure, reflect its key points:
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Polygon: reflect the vertices.
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Circle: reflect the center; the radius doesn't change.
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Line: if it crosses the axis, the intersection point stays fixed; reflect another point and draw the line through both. If it is parallel to the axis, it stays parallel.
An agricultural plot is in the shape of a triangle with vertices at $A(0, 0)$, $B(4, 0)$ and $C(2, 5)$. The municipality expropriates the plot and, in exchange, gives…
An agricultural plot is in the shape of a triangle with vertices at $A(0, 0)$, $B(4, 0)$ and $C(2, 5)$. The municipality expropriates the plot and, in exchange, gives the owner another plot obtained by applying a reflection across the vertical line $x = 6$ to the original triangle. Calculate the vertices of the new plot and prove analytically that the area of the plot has not changed.
The outline of a circular gear has the equation $x^2 + y^2 - 2x - 4y - 4 = 0$. This gear is reflected in an inclined mirror represented by the line…
The outline of a circular gear has the equation $x^2 + y^2 - 2x - 4y - 4 = 0$. This gear is reflected in an inclined mirror represented by the line $r \equiv 3x - 4y + 10 = 0$. Determine the general equation of the circle representing the reflected image of the gear.
A ray of light travels along the line $r \equiv 2x - y + 4 = 0$ and hits a plane mirror located on the line $s \equiv x + y - 1 = 0$. Determine the general equation of…
A ray of light travels along the line $r \equiv 2x - y + 4 = 0$ and hits a plane mirror located on the line $s \equiv x + y - 1 = 0$. Determine the general equation of the line $r'$ containing the reflected ray, knowing that this line is the reflection of $r$ across $s$.
Examples of movements in sequence
Identifying a movement. If you are given two points and their images:
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If $A'-A=B'-B$ (same vector), it is a translation.
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If the midpoints of $AA'$ and $BB'$ coincide, it is a point reflection with that center.
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Always check that $|AB|=|A'B'|$ (isometry).
In a video game animation, the geometric center of a character located at $A(1, 3)$ undergoes a translation by vector $\vec{t} = (4, -2)$ followed immediately by a point…
In a video game animation, the geometric center of a character located at $A(1, 3)$ undergoes a translation by vector $\vec{t} = (4, -2)$ followed immediately by a point symmetry (point reflection) with respect to point $C(2, 0)$. Find the final position of the character and verify analytically whether the result would be the same by applying the point symmetry first and then the translation.
Given the parallel lines $r \equiv 2x + y - 3 = 0$ and $s \equiv 2x + y + 7 = 0$, find the translation vector $\vec{v}$ of minimum magnitude that transforms the line $r$…
Given the parallel lines $r \equiv 2x + y - 3 = 0$ and $s \equiv 2x + y + 7 = 0$, find the translation vector $\vec{v}$ of minimum magnitude that transforms the line $r$ into the line $s$. Prove analytically that this vector is perpendicular to both lines and that its magnitude coincides with the shortest distance between them.
An isometry in the plane transforms the point $A(1, 2)$ into $A'(5, 4)$ and the point $B(3, -1)$ into $B'(7, 1)$. a) Show that the distance between $A$ and $B$ is the…
An isometry in the plane transforms the point $A(1, 2)$ into $A'(5, 4)$ and the point $B(3, -1)$ into $B'(7, 1)$.
- a) Show that the distance between $A$ and $B$ is the same as that between $A'$ and $B'$.
- b) Determine whether the described transformation corresponds to a pure translation or a point reflection, justifying your answer analytically and finding the corresponding translation vector or center of symmetry.
Prove analytically that the composition of two reflections across parallel lines is equivalent to a translation. To do this, consider the lines $r \equiv x - 2y = 0$ and…
Prove analytically that the composition of two reflections across parallel lines is equivalent to a translation. To do this, consider the lines $r \equiv x - 2y = 0$ and $s \equiv x - 2y + 10 = 0$. Take a generic point $P(x_0, y_0)$, find its reflection $P'$ across $r$, and then the reflection $P''$ of $P'$ across $s$. Determine the translation vector $\vec{v} = \vec{PP''}$ and check that it is perpendicular to the lines and that its magnitude is twice the distance between them.
Summary
Go back to the beginning: the light reflected in the water at the Malecón is the reflection of the light over the waterline (a horizontal axis $y=k$). You just apply $(x, y)\to(x, 2k-y)$.