Basic Rigid Transformations: Reflections, Rotations & Translations

Instructor: David Karsner
Moving around a two-dimensional shape is called transformation. This lesson explains the three basic rigid transformations: reflections, rotations, and translations.

Transformations in Video Games

Have you ever wondered how your favorite video game gets its characters to move around? It's through the use of transformations. The computer in your gaming system interprets the input from the controller and then picks which transformation to use.

Basic Rigid Transformations

A transformation is when you take a shape and you move it in some way. A basic rigid transformation is a movement of the shape that does not affect the size of the shape. The shape doesn't shrink or get larger. There are three basic rigid transformations: reflections, rotations, and translations.

There is a fourth common transformation called dilation. Since dilation entails the shrinking or enlarging the shape; dilation is not a rigid transformation. It also possible to combine several transformations into one movement.

Reflections

A reflection, as its name suggests, is a movement that results in the the shape flipping across some line. The line acts like a mirror. The shape is usually drawn on a x,y coordinate grid. Most often, the line that is used as the mirror is either the x-axis or the y-axis, but any line will work.

Imagine that you have a rectangle drawn on a coordinate grid with the vertices of (3,5), (3,10), (6,5), and (6,10) and you want to reflect it across the x-axis. The x-axis will act like the mirror. The point (3,5) is 5 units from that mirror so the reflection of that point will be 5 units on the other side (3,-5). The y value of each of the points will change signs. The reflection of the original rectangle will have the vertices of (3,-5), (3,-10), (6,-5), and (6,-10).

Reflection
ReflectedRectangle

Rotations

A rotation rotates the shape around a center point. Every rotation has a direction (clockwise or counter-clockwise), center point, and the degree of rotation. The center of the rotation is the point that the shape will rotate around. The center can be the center of the shape, the origin of the x,y coordinate grid, or any other point.

The degree of rotation is the amount of turning that the shape will do. If you turn a shape completely around until it is exactly where it started, that is a 360 degree turn. The degree of rotation is usually anywhere from 1 to 360 degrees, but it can be more. On occasion you will see a negative degree of rotation. This means that you will rotate the shape counter-clockwise.

Let's go back to our original rectangle with vertices at (3,5), (3,10), (6,5), and (6,10). Imagine that this time you want to rotate your rectangle 180 degrees clockwise around the origin (0,0). The rectangle was originally in quadrant I. Ninety degrees of rotation puts it in quadrant IV. Ninety more degrees of rotation (totaling 180 degrees of clockwise rotation) will put the rectangle in quadrant II. The vertices of the rotated rectangle will be (-3,-5),(-3,-10),(-6,-5) and (-6,-10). Notice that the only difference between these vertices and the original vertices is that these are negative. That means they have all maintained the same distance from (0,0).

Rotation
RotatedRectangle

Translations

A translation is a sliding of the shape. Imagine that you have a bookcase sitting against your wall. You decide that you are going move the bookcase 18 inches to the right. This scenario would be a translation. With translations, the direction and the length of the move must be given. If the shape is placed on a x,y coordinate grid, then the movement of the x coordinate and the y coordinate must both be given.

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