Isometry of the Plane
What is an Isometry?
Isometry is a transformation that preserves distances.
Definition: A transformation of the plane is an isometry if, for any two points and , the distance between their images and is equal to the distance between and .
The shape of the image must remain congruent to the original object.
Every isometry in a 2D plane can be categorized into one of four types.
A. Translation
A translation moves every point in the plane a same distance in a same direction. It is defined by a vector.
Use the grid box below to create a point and perform a translation using a vector of your choice.
Next, repeat it by creating a shape.
Observe its property.
Grid Box
B. Rotation
A rotation turns the plane around a fixed point (the center) by a specific angle.
Use the grid box below to create a point and perform a rotation by an angle of your choice.
Next, repeat it with a shape.
Observe its property.
C. Reflection
A reflection "flips" the plane across a fixed line (the axis of reflection).
Use the grid box below to create a point and perform a reflection on a line of your choice.
Next, repeat it with a shape.
Observe its property.
Hint: If you label the vertices of a triangle A, B, C clockwise, their images A', B', C' will appear counter-clockwise.
Tip!
How to create an isometry transformation on the grid.
- Create a point: Click on 'point' tool and mark a point on the grid.
- Translation: Click on 'translate by vector' tool, select the point made earlier, then mark another two points to indicate the vector.
- Rotation: Click on 'rotate about point' tool, select the point made earlier, then mark a point for rotation, input an angle value, and click 'enter'.
- Reflection: Click on 'line' tool', mark two points to draw a line. Click on 'reflect on line' tool, select the point made earlier, then click on the drawn line.
- Create a shape: Click on 'polygon' tool, mark the vertices to form a shape.
Exploration
Self-Directed Task (Use the grid below)
- Create a segment . Apply a rotation and then a reflection. Measure the new segment . Does it match the original length?
- Create any shape. Can you create a single reflection that results in the same image as two translations?
- Create a triangle. Reflect it over a horizontal line. Then, translate it +3 units along the axis. Try to achieve this same result using only one of the other three isometries.
Grid Box
Exploration
You will realise it is not possible.
This is why Glide Reflection is a distinct fourth type of isometric transformation.
D. Glide Reflection
A glide reflection is a composition of a reflection and a translation, where the translation vector is parallel to the line of reflection.
Analogy: Think of footprints in the sand. To get from a left footprint to a right footprint, you must reflect it across a center line and then slide it forward.
Try this out:
- Create a shape (object) on a grid.
- Translate / Slide your shape according to a given vector. Lightly sketch the image prime (the first transformation image).
- Reflect / Flip the image prime over the reflection line. Ensure the reflection line is parallel to the vector in Step 2.
- Now, repeat the transformation for the same shape at its original position but perform the reflection followed by translation.
- Observe its property.
Self Evaluation
- From the exploration above, can you summarise which type of isometric transformation preserves its orientation and which does not?
- If a transformation stretches a square into a rectangle, is it an isometry? Why or why not?
- Translation and rotation preserve orientation; reflection and glide reflection reverse the orientation.
- No, it is not an isometry. By definition, an isometry must preserve distance between all points. If a square is stretched into a rectangle, the side lengths (distances) change, and the object is no longer congruent to the original shape. This is known as a dilation or enlargement.