Translations: The Geometry of Sliding
What Is a Geometric Transformation?
Before diving into translations, we need to understand transformations. A transformation is an operation that moves or changes a geometric figure (called the pre-image) to create a new figure (called the image). Think of it as taking a picture of a shape, and then moving or altering that picture. There are four main types of rigid motions (transformations that preserve size and shape): translations, rotations, reflections, and dilations3. Translations are our focus here.
A translation has three key features:
- Distance: How far the shape slides.
- Direction: Which way it slides (e.g., left, right, up, down, or diagonally).
- Parallel Movement: Every point moves along a path that is parallel to the path of every other point.
Because every point moves identically, the pre-image and the image are congruent4—they have exactly the same size and shape.
Describing a Slide with Vectors
How do we give precise instructions for a slide? We use a mathematical tool called a vector. A vector tells us both direction and magnitude (length). On the coordinate plane, we describe a vector with two numbers, usually written in angle brackets or as a column. For example, the vector $\langle 3, -2 \rangle$ means "move 3 units to the right and 2 units down."
Let's see an example. Suppose we have a triangle with vertices at $A(1, 1)$, $B(3, 1)$, and $C(2, 3)$. We want to translate it by the vector $\langle 4, 2 \rangle$.
- Point $A$: $(1 + 4, 1 + 2) = (5, 3)$
- Point $B$: $(3 + 4, 1 + 2) = (7, 3)$
- Point $C$: $(2 + 4, 3 + 2) = (6, 5)$
The new triangle $A'B'C'$ has vertices at $(5, 3)$, $(7, 3)$, and $(6, 5)$. The shape and size of the triangle have not changed at all; it has simply been relocated.
| Starting Point (Pre-image) | Translation Vector | Resulting Point (Image) | Description of Slide |
|---|---|---|---|
| $(2, 5)$ | $\langle -3, 0 \rangle$ | $(-1, 5)$ | Move 3 units left. |
| $(-1, -2)$ | $\langle 0, 4 \rangle$ | $(-1, 2)$ | Move 4 units up. |
| $(0, 0)$ | $\langle 5, -5 \rangle$ | $(5, -5)$ | Move diagonally to the right and down. |
Properties of a Translation
Translations have special properties that make them easy to identify and work with:
- Congruence: As mentioned, the pre-image and image are congruent. All corresponding side lengths and angle measures are equal.
- Parallelism: Corresponding line segments in the pre-image and image are parallel. For example, if side $AB$ in the original shape is horizontal, side $A'B'$ in the translated shape will also be horizontal.
- Orientation Preservation: The order of vertices (clockwise or counterclockwise) does not change. A shape is not flipped over. This distinguishes translations from reflections.
- No Fixed Points: In a pure translation (where the vector is not $\langle 0, 0 \rangle$), no point remains in its original position.
Visualizing Translations in Real-World Applications
Translations are not just abstract math; they are everywhere in the real world. Let's explore a few scenarios where the concept of 'sliding along a plane' is crucial.
Animation and Video Game Design: When an animator wants a character to walk across the screen, they use a series of translations. Each frame of the character's position is a translation of its position in the previous frame. In a 2D video game, when you press the right arrow key to move your character, the game's code is essentially applying a translation vector, like $\langle 10, 0 \rangle$, to the character's coordinates many times per second.
Architecture and Engineering: When designing a building with repeating patterns, such as windows on a facade, architects use translations. The design for one window is created, and then it is translated multiple times to the correct positions along the wall. This ensures consistency and saves design time.
Manufacturing: In an assembly line, a robotic arm might pick up a part and place it onto a product. The path the arm takes to move the part from the supply bin to the exact correct spot on the product can often be described as a translation.
Important Questions
Q1: What is the difference between a translation and a reflection?
A translation slides a shape without flipping it. Every point moves the same distance in the same direction. A reflection flips a shape over a line (like a mirror image). In a reflection, the orientation of the shape is reversed. For example, if you write the letter "R" on paper and slide it, it's still an "R". If you reflect it in a mirror, it looks like a backward "R".
Q2: What happens if you add two translation vectors together?
Adding vectors results in a single, combined translation. If you first translate by $\langle a, b \rangle$ and then by $\langle c, d \rangle$, the overall effect is the same as one translation by $\langle a+c, b+d \rangle$. The order of translations does not matter; the final image will be the same. This property is called commutativity.
Q3: How do you translate a shape that is not on a coordinate plane?
You can still perform a translation using geometry tools. You would need to know the distance and direction. Using a ruler and a protractor, you could measure the required distance. More commonly, you can use a vector defined by two points. For example, if a shape is to be translated such that point $A$ goes to point $A'$, then the vector for the translation is defined by the arrow from $A$ to $A'$. You would then apply that same vector to every other point in the shape.
Footnote
- Rigid Motion (or Isometry): A transformation that preserves all distances and angles. The shape's size and shape do not change. Translations, rotations, and reflections are all rigid motions.
- Coordinate Plane: A two-dimensional surface formed by two perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical). Any point's location is given by an ordered pair $(x, y)$.
- Dilation: A transformation that changes the size of a figure by a scale factor but preserves its shape and proportions. It is not a rigid motion because size changes.
- Congruent: Having exactly the same size and shape. All corresponding sides and angles are equal.