# Course:Harris, Fall 08: Diary Week 9

Mon:

• No class, Fall break

Wed:

• Preliminary sketches for the Tessellation Art Project are due on Friday
• Comparison of Euclidean planar geometry and spherical geometry:
• point:
• Same in both.
• line:
• "spherical line"
• Definition: a circle on the sphere whose plane goes through the center of the sphere.
• Alternative definition: a circle on the sphere of largest possible radius (hence, "great circle").
• Different from planar line:
• It has a finite length.
• Anything else?
• Similar to planar line:
• The shortest path between any two points lies along a spherical line.
• Anything else?
• line segment:
• Arc of a great circle: much the same as for planar geometry (except worrying about what "between" means).
• length of a line segment:
• Same as for planar geometry.
• angle (and its angle measure) between two segments sharing an endpoint:
• Use the tangent plane at that common endpoint to project the spherical segments onto that plane.
• Then it's exactly the same as planar geometry.
• distance:
• Length of the line segment between two points.
• Does the same definition work in both?
• between:
• In the plane, P is between Q and R on the line L means P, Q, and R are points on L and
• When traversing L from one infinite "end" to the other we come across first Q and then P and then R, or first R and then P and then Q.
• That doesn't apply on the sphere, since a line doesn't have infinite ends.
• dist(Q,P) + dist(P,R) = dist(Q,R).
• We can at least state that on the sphere--but is it what we want "between" to mean?
• Topic for discussion next time!

Fri:

• Dr. Anneke Bart took over the class, as I was away at Notre Dame for a relativity conference.
• Discussion of the axiomatic approach to geometry, with emphasis on the differences between
• the plane and
• the sphere.