# Course:Harris, Fall 08: Diary Week 9

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- 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?

- Definition: a circle on the sphere whose plane goes through the center of the sphere.

- "spherical line"
*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?

- Length of the line segment between two points.
*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!

- 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.

- In the plane, P is between Q and R on the line L means P, Q, and R are points on L and

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.

- Discussion of the axiomatic approach to geometry, with emphasis on the differences between