Progress Report 3
Introduction and Instructions: In this project you will explore how well
the area axioms the class has developed apply to the surface of a sphere. Some
work, some do not. You may wish to use the figure provided here.
Write brief answers to each of the following questions on a separate sheet of
paper. Please include your group number and the names and roles of each person
in the group on the first page of your write-up. Fold the paper(s) lengthwise
and have each group member sign the write up on the outside page these signatures
will be taken to mean that each group member had a good understanding of each
answer and could present it with some assistance from the group.
Questions:
- Of the figures we have been investigating (e.g., triangle, rectangle, trapezoid
and so on), are there any that can be constructed in a plane but not on a
sphere? Explain why or why not.
- Of the axioms for area in a plane, which appear to also work for computing
area in a sphere?
- Does the procedure for finding the area of a polygon in the plane work
for polygons on the sphere? (What adjustments, if any, do you have to make
in the area axioms?)
- Does the formula for the area of a triangle work? (In particular, do different
choices of base and altitude for the same triangle yield the same area?)
- Arrive at a formula for finding the area of a lune and then describe a
triangle as the intersection of lunes.
- Find a formula for finding the area of a triangle on a sphere.
- Of the norms for cooperative behavior, select the norm where your group
has shown the most improvement since the group was formed. Give an example
or two to demonstrate the improvement. For what norm has there been the least
change?
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