Composition of 3 reflections - General position

What is the result of the composition of these 3 reflections Ra'

Rb'

Rc' ? Step 1:

Reflect the given shape in the order given above.
Format the intermediary images and the final image (color, transparency) to make the resulting isometry visually clean.
Observe the preimage (original brown shape) and the final image. Can you see what the resulting isometry is?

******************* Step 2: Use the sliders to change the postion of the lines without changing the resulting isometry. Keep in mind that if you rotate two intersecting lines about their intersection point withouth changing the angle between them, they will still generate the same rotation.

Use the red slider to rotate b' and c' until b' is perpendicular to a' .
Use the blue slider to rotate a' and b' until a' || c'.

Think about the situation thoroughly. 1. Why is the new combination of lines giving us the same isometry as the original line arrangement? 2. Can you see how this procedure is helping us see that any "slide flip" or "flip slide" can be turned into a glide reflection (with the vector parallel to the line)? Write your ideas down and bring them for a discussion in class.

Extension

Can you use this applet to explain why we don't have a "turn reflection" in our list? Hint: The original line arrangement can be though of as a turn reflection: Ra'

Rb' generate a rotation (What's its center and angle? Make sure you see it in your intermediary images). Adding Rc' makes it a "turn reflection". Now show that by rotating the lines as we did in step 2 will replace the first rotation Ra'

Rb' by a different isometry.

Composition of 3 reflections - General position

Extension

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