# Spherical Geometry: Isometry Exploration

Objective:

Extend concepts of reflection, rotation and translation to the sphere

1. On the sphere below sketch several images of the smiley face under translation. Can we translate the image to the back of the sphere? Why or why not? What happens if our translation distance is greater than the circumference of the circle?
2. Rotate the smiley face through an angle of 45 degrees about K in a counter clockwise direction.
3. Reflect the smiley face across line segment r.
4. Create a glide-reflection using line segment r as the translation direction.
• How are translations on the sphere different from those in the plane? How are they similar?
• How are rotations on the sphere different from those in the plane? How are they similar?
• How are reflections on the sphere different from those in the plane? How are they similar?
• How are glide-reflections on the sphere different from those in the plane? How are they similar?
5. Translations are compositions of reflections. Draw a sketch showing how we can replace one translation by multiple reflections and get the same end result.
6. Rotations are compositions of reflections. Draw a sketch showing how we can replace one translation by multiple reflections and get the same end result.

Handin: A sheet with answers to all questions.