Spherical Easel Exploration

Objective: Discover principles of geometry on the sphere.

Go to Spherical Easel. This program will allow you to explore the geometry of the sphere.

Some features of Spherical Easel to note:

• In Spherical Easel angles are measured in radians. $\pi$ radians $=180^{o}$. $\pi$ is approximately 3.14.
• To measure angles using "Measurements-Angle Measure" be sure to select the line segments in a counterclockwise fashion around the vertex.The drawn arc at the vertex will indicate an acute angle.
• Also to measure the angles in a triangle using "Measurements-Triangle" select your line segments of the triangle in a counterclockwise fashion. The first three numbers on the pop-up are the lengths of the sides. The next three are the angle measurements.
• One can move a point of a triangle and the measurements will change with it.

Answer the following questions as completely as possible.

1. What do “lines” look like on the sphere?
2. What do polygons look like on the sphere?
3. Create some triangles on the sphere and measure their angle sums. Do the angles always add up to 180° (=π radians)? Do they ever add up to 180°? What is the smallest angle sum you can make? What is the largest angle sum you can make?
4. Can you create rectangles and squares on the sphere? Why or why not?
5. Can you create parallel lines? Why or why not? Can you create parallelograms on the sphere? Explain why or why not.
6. Create a rhombus, with four equal sides and with opposite angles equal.
7. Create a polygon with two sides - a biangle.
8. Create the height or altitude of a triangle by constructing the perpendicular line between a side and the opposite vertex. In order for Spherical Easel to recognize this segment you will have to highlight it using the Segment button. Additionally sometimes the altitude does not connect with the base and you will have to add a line through the base and then find its intersection with the base.
9. Draw a triangle and compute ½bh in the three possible ways. Do we get the same value no matter what side we choose as a base? Do the values for ½bh correspond with the area that the software computes for us? Do you think that the area of a triangle on the sphere could be given by the same formula we use in the Euclidean plane? To illustrate this draw a picture of your triangle with its altitudes and sides and indicate their lengths and the values of ½bh.
10. On the plane we had four types of isometries we looked at: translations, rotations, reflections and glide-reflection (where the last one is of course a composite of the translation and a reflection). Do these same isometries work on the sphere? Can we translate by any distance? Can we always reflect over a mirror line? Can we rotate through any angle? How are translations and reflections related on the sphere? Are all translations reflections and are all reflections translations? Explain.Draw pictures of the results of both on an object on the sphere.

Handin: A sheet with answers to all questions.