Transformations and Translations
You're playing pool and the cue ball is nowhere near a straight shot at the 8-ball. You'll have to bank it off the bottom cushion. Where exactly should you aim? A delivery drone that flies across town, a gear that turns, and a laser that bounces off a mirror are all transformations of the plane. In coordinates they come down to a few simple formulas, and by the end of this page you'll use them to aim that bank shot.
Transformations and translations
Rigid motions never change the distance between two points. Here is the formula you'll use to check that.
The simplest rigid motion is the translation: slide every point the same distance in the same direction.
Some problems write a vector with parentheses, like $(-4, 2)$. It means the same thing as $\langle -4, 2 \rangle$.
Triangle $PQR$ was translated by $\langle 3, 2 \rangle$. Every vertex moved 3 right and 2 up, so the three dashed arrows are parallel and equally long.
A delivery drone is at position $A(3, 5)$ and moves along the vector $\vec{v} = (-4, 2)$. What are the coordinates of its final position $A'$?
A delivery drone is at position $A(3, 5)$ and moves along the vector $\vec{v} = (-4, 2)$. What are the coordinates of its final position $A'$?
On a giant chessboard, a piece moves from the square with coordinates $P(-1, -3)$ to the square $Q(4, 2)$. Determine the components of the translation vector $\vec{PQ}$.
On a giant chessboard, a piece moves from the square with coordinates $P(-1, -3)$ to the square $Q(4, 2)$. Determine the components of the translation vector $\vec{PQ}$.
In architectural software (CAD), the plan of a window is defined by the polygon with vertices $A(1, 1)$, $B(4, 1)$, $C(4, 3)$ and $D(1, 3)$. The designer decides to…
In architectural software (CAD), the plan of a window is defined by the polygon with vertices $A(1, 1)$, $B(4, 1)$, $C(4, 3)$ and $D(1, 3)$. The designer decides to translate the window by applying the vector $\vec{v} = (-3, 2)$. Calculate the coordinates of the new vertices $A'$, $B'$, $C'$ and $D'$.
Reflections
A reflection flips a figure across a line of reflection, like a mirror or a fold in paper. Each point and its image are the same distance from the line, on opposite sides. The line of reflection is the perpendicular bisector of the segment $PP'$.
In a logo design, the point $M(-4, 7)$ is reflected across the y-axis. What are the coordinates of the reflected point $M'$?
In a logo design, the point $M(-4, 7)$ is reflected across the y-axis. What are the coordinates of the reflected point $M'$?
In a textile factory, a machine folds a fabric along the horizontal line $y = 2$. If a decorative pattern is printed at point $P(3, -1)$, what will be the coordinates of…
In a textile factory, a machine folds a fabric along the horizontal line $y = 2$. If a decorative pattern is printed at point $P(3, -1)$, what will be the coordinates of the pattern $P'$ once the fabric is folded?
A straight water supply pipe passes through points $A(0, 4)$ and $B(4, 0)$. For design redundancy reasons, a second pipe is built which is exactly symmetrical to the…
A straight water supply pipe passes through points $A(0, 4)$ and $B(4, 0)$. For design redundancy reasons, a second pipe is built which is exactly symmetrical to the first with respect to the x-axis. Find the coordinates of the symmetrical points $A'$ and $B'$, and determine the explicit equation of the line that defines the path of this second pipe.
Rotations about the origin
A rotation turns a figure by a given angle about a fixed point called the center of rotation. By convention, positive angles turn counterclockwise.
The center is the only point a rotation leaves in place. That's why the one point of a rotating circular sign that doesn't move is its own center.
A gear part located at point $K(5, 0)$ rotates $90^\circ$ counterclockwise about the origin $O(0, 0)$. In what position $K'$ will the part be after the rotation?
A gear part located at point $K(5, 0)$ rotates $90^\circ$ counterclockwise about the origin $O(0, 0)$. In what position $K'$ will the part be after the rotation?
A rotation of $180^\circ$ is applied to a circular traffic sign with its center at point $C(3, 4)$ and radius $2$. Which exact point on the sign remains in the same…
A rotation of $180^\circ$ is applied to a circular traffic sign with its center at point $C(3, 4)$ and radius $2$. Which exact point on the sign remains in the same position (is invariant) after the transformation?
A square has its geometric center at the origin $O(0, 0)$ and one of its vertices is $V_1(3, 3)$. To find the other vertices, successive rotations of $90^\circ$…
A square has its geometric center at the origin $O(0, 0)$ and one of its vertices is $V_1(3, 3)$. To find the other vertices, successive rotations of $90^\circ$ clockwise about the origin are applied, obtaining $V_2$, $V_3$ and $V_4$. Calculate the coordinates of the four vertices and prove analytically that the distance between $V_1$ and $V_2$ is exactly equal to the distance between $V_2$ and $V_3$.
Point reflection: the $180^\circ$ rotation
Rotating $180^\circ$ about a center $C$ is also called a point reflection across $C$. Each point is sent "straight through" $C$ to the opposite side. That means $C$ is the midpoint of $PP'$.
Some problems call this central symmetry and the center the "center of symmetry". It is the same idea.
A gardener wants to plant a tree at point $B'$, which must be symmetrical to the tree located at $B(2, -6)$ with respect to the center of the garden $C(0, 0)$. Calculate…
A gardener wants to plant a tree at point $B'$, which must be symmetrical to the tree located at $B(2, -6)$ with respect to the center of the garden $C(0, 0)$. Calculate the coordinates of $B'$.
Two streetlights are located at points $F_1(-2, 8)$ and $F_2(6, -2)$. An electrical transformer is to be placed at their center of symmetry. What are the coordinates of…
Two streetlights are located at points $F_1(-2, 8)$ and $F_2(6, -2)$. An electrical transformer is to be placed at their center of symmetry. What are the coordinates of this center?
A robotic arm articulates at point $C(2, 1)$. Its gripper is initially located at $P(5, 3)$. If the arm performs a rotation of $180^\circ$ about the joint $C$, what will…
A robotic arm articulates at point $C(2, 1)$. Its gripper is initially located at $P(5, 3)$. If the arm performs a rotation of $180^\circ$ about the joint $C$, what will be the new coordinates $P'$ of the gripper?
Rotating about any center
The rules $(-y, x)$ and $(y, -x)$ only work when the center is the origin. For any other center $C$, make $C$ the origin first.
A security camera mounted on a pole at $C(1, 2)$ initially focuses on a door located at $P(4, 2)$. To cover a corridor, the camera rotates $90^\circ$ counterclockwise…
A security camera mounted on a pole at $C(1, 2)$ initially focuses on a door located at $P(4, 2)$. To cover a corridor, the camera rotates $90^\circ$ counterclockwise about its axis at $C$. What are the coordinates of the new point $P'$ the camera will point to? (Assume the focal length remains constant).
In the CNC machining of a part, the cutting tool is initially located at point $H(0, 5)$. The program executes two commands: first, the chuck rotates $180^\circ$ about…
In the CNC machining of a part, the cutting tool is initially located at point $H(0, 5)$. The program executes two commands: first, the chuck rotates $180^\circ$ about the center $C(2, 3)$, bringing the tool to position $H'$. Immediately after, the carriage translates according to the vector $\vec{u} = (-1, -4)$ until it reaches the final position $H''$. Determine the exact coordinates of $H'$ and $H''$.
A triangular piece of a historical mosaic with a vertex at $P(2, 6)$ has detached and, upon falling, has rotated exactly $90^\circ$ counterclockwise about an unknown…
A triangular piece of a historical mosaic with a vertex at $P(2, 6)$ has detached and, upon falling, has rotated exactly $90^\circ$ counterclockwise about an unknown point $C(x, y)$, leaving the vertex in the new position $P'(-4, 4)$. Knowing that in a rotation of $90^\circ$ counterclockwise about the origin a vector $(a, b)$ is transformed into $(-b, a)$, set up the equations corresponding to the transformation of the vector $\vec{CP}$ into $\vec{CP'}$ and solve the system to find the coordinates of the center of rotation $C$.
Moving into 3D
Everything so far extends to space, where a point has three coordinates $(x, y, z)$.
In an automated warehouse, a package moves in a straight line from shelf $A(10, 5, 2)$ to the loading zone $B(2, 8, 0)$. Determine the translation vector $\vec{AB}$. If…
In an automated warehouse, a package moves in a straight line from shelf $A(10, 5, 2)$ to the loading zone $B(2, 8, 0)$. Determine the translation vector $\vec{AB}$. If another package located at $C(4, 4, 4)$ makes exactly the same movement, what will be the coordinates of its final position $C'$?
A 3D printing program first translates a vertex of a solid using the vector $\vec{v} = (0, 0, 4)$ and then applies a central symmetry with respect to the origin…
A 3D printing program first translates a vertex of a solid using the vector $\vec{v} = (0, 0, 4)$ and then applies a central symmetry with respect to the origin $O(0, 0, 0)$. If the original vertex was at position $V(2, -3, 1)$, what is its final position $V'$?
An underwater drone is at position $D(4, -2, -5)$ (the xy-plane represents sea level). First, it ascends vertically a certain distance by applying the translation vector…
An underwater drone is at position $D(4, -2, -5)$ (the xy-plane represents sea level). First, it ascends vertically a certain distance by applying the translation vector $\vec{v} = (0, 0, 5)$. Next, it emits a sonar pulse that reflects specularly off the seabed, which is completely flat and defined by the equation $z = -10$. What are the coordinates of the symmetrical point of the drone's new position with respect to the seabed?
Composing transformations
You can compose transformations: apply one, then apply the next to the result.
Two translations just add their vectors. Moving by $\vec{u}$ and then by $\vec{w}$ is the same as moving once by $\vec{u} + \vec{w}$.
Two reflections across parallel lines also give something simple.
A robot vacuum cleaner moves from its base along the vector $\vec{u} = (2, 5)$ and then follows the vector $\vec{w} = (1, -3)$. If the base is at the origin $O(0,0)$,…
A robot vacuum cleaner moves from its base along the vector $\vec{u} = (2, 5)$ and then follows the vector $\vec{w} = (1, -3)$. If the base is at the origin $O(0,0)$, what are the coordinates of its final position and which single vector $\vec{v}$ represents the total displacement?
A sailboat moves at sea driven by the wind, which gives it a displacement according to the vector $\vec{v} = (12, -5)$ miles. Simultaneously, an ocean current drags it…
A sailboat moves at sea driven by the wind, which gives it a displacement according to the vector $\vec{v} = (12, -5)$ miles. Simultaneously, an ocean current drags it according to the vector $\vec{c} = (-3, -2)$ miles. If the sailboat left the port at $P(-4, 10)$, what is its final position $P'$ and at what straight-line distance is it from the starting point?
A ray of light passes through a thick crystal and undergoes a double internal reflection. First, it undergoes a reflection across the vertical line $x = 1$, and then…
A ray of light passes through a thick crystal and undergoes a double internal reflection. First, it undergoes a reflection across the vertical line $x = 1$, and then another reflection across the line $x = 4$. If an impurity in the crystal is at point $M(-2, 3)$, find its position after the first reflection ($M'$) and after the second ($M''$). What direct translation vector is equivalent to the composition of these two reflections?
Identifying a transformation
Given a figure and its image, you can tell which transformation was used by looking at how the coordinates change.
| What you see | Transformation |
|---|---|
| $P' - P$ is the same vector for every point | translation by that vector |
| $(x, y) \to (x, -y)$ or $(-x, y)$ | reflection across the $x$-axis or $y$-axis |
| $(x, y) \to (-x, -y)$ | $180^\circ$ rotation about the origin (point reflection) |
| $(x, y) \to (-y, x)$ | $90^\circ$ counterclockwise rotation about the origin |
| $(x, y) \to (y, -x)$ | $90^\circ$ clockwise rotation about the origin |
Two quick tests help. A translation moves every point by the same vector. A rotation about the origin keeps every point the same distance from the origin.
To justify your answer, show that one rule works for every vertex. For a point reflection, you can also show that the midpoints of $AA'$, $BB'$ and $CC'$ are all the same point.
In a study of geometric transformations, the vertices of a triangle are $A(1, 2)$, $B(3, 1)$ and $C(2, 4)$. After applying a single transformation in the plane, the new…
In a study of geometric transformations, the vertices of a triangle are $A(1, 2)$, $B(3, 1)$ and $C(2, 4)$. After applying a single transformation in the plane, the new vertices are $A'(-1, -2)$, $B'(-3, -1)$ and $C'(-2, -4)$. Identify what type of transformation it is (translation, rotation or reflection) and specify its characteristic elements (vector, center and angle, or axis). Justify your answer analytically.
Applications: bank shots and mirrors
Reflections solve problems where something bounces. When a ball hits a cushion or light hits a mirror, it leaves at the same angle it arrived. Here is the trick:
Back to the pool table: reflect the target across the cushion, aim straight at that "ghost" ball, and the cushion does the rest.
Reflections also help when you know the path but not the mirror. The mirror is the perpendicular bisector of $PP'$. If a laser leaves $(3, 8)$ and arrives at $(3, 2)$, the $x$-coordinate stays the same, so the mirror is horizontal, and it sits halfway: $y = \dfrac{8 + 2}{2} = 5$.
A laser beam is emitted from point $E(1, 5)$ and, after reflecting off a flat mirror parallel to the x-axis, hits the receiving point $R(1, -1)$. Knowing that the…
A laser beam is emitted from point $E(1, 5)$ and, after reflecting off a flat mirror parallel to the x-axis, hits the receiving point $R(1, -1)$. Knowing that the x-coordinate does not change and the y-coordinate of the symmetrical points is equidistant from the mirror, what is the equation of the line that defines the position of the mirror?
On a billiard table, a ball is at position $A(2, 3)$. A player wants to hit it so that it strikes the right cushion (located along the line $x = 8$) and, after…
On a billiard table, a ball is at position $A(2, 3)$. A player wants to hit it so that it strikes the right cushion (located along the line $x = 8$) and, after rebounding, hits another ball located at $B(4, 7)$. To calculate the trajectory, they use the symmetrical point of $B$ with respect to the cushion, let's call it $B'$. Find the coordinates of $B'$ and find the equation of the line joining $A$ and $B'$. At what exact point $P$ on the cushion will the ball rebound?