Notes for 2/4/04

1) What is a “straight”?

• Shortest distance between point/places
• How can you make a straight path?

(1) Stretch a string around a sphere: why does this give us a great circle?

(2) Visually, ex… light rays.

(3) If you are walking and blindfolded: have your steps parallel and equal

(4) In a car: the wheels must be going at equal speed so it won’t curve

(5) For spheres: roll and trace the points of contact. You should come back to where you started from

(6) Symmetry across a curve

• Which of these gives us a great circle? 3 and 6. ex.. use a mirror, both right and left sides are the same

2) Definition of Geodesic: means straight path

• On a sphere this produces the great circle, where both the right and left sides are the same, symmetric

3) What are lines on a plane?

• Lines are constructed by two points
• 3 points construct a plane

4) Which points do you need?

5) How many points do you need for a sphere?

6) Given any 2 points is there a great circle?

• Yes.

7) Is there exactly one great circle through two points? Why?

• No
• There are infinity numbers of great circles because there are infinity numbers of opposite/antipodal points.

8) If you have a point on a sphere, what is its opposite/ antipodal point?

• The furthest distance from the point
• Draw a line from the point to the center of the sphere; the opposite point is the intersection of the line to the sphere on the other side.

9) What are the only circles that if you take the opposite point, you get the same circle?

• A great circle. Opposite points are always on the great circle.

10) Are there parallel great circles?

• No, great circles intersect each other

11) Elliptic Geometry:

• Point = pair of antipodal points on a sphere

12) Why are only great circles straight?

• Counterexample: try to tie a ribbon around a smaller circle around the sphere, can you do it? Is the path straight?

§ No, the path is curved

§ There are infinity number of paths that are curved on a sphere, the only straight path are great circles because it goes through the antipodal points.

13) How do you measure the distance on a surface of a sphere from points A to B?

• Measure the arc length A to B.

14) What is its unit of measurement?

• Degrees or radians

15) Distances: have a point and antipodal point

16) Duel Polar, “equator”= Great Circle