Math 445 Class Wed 1/29/2003

What is straight? (Class #2 – on the sphere)

Straightness in general

• Straightness and symmetry
• Straightness and development into the plane
• Straightness and shortest distance
• Straightness and unique path for each direction

Straightness on the sphere

• Distance – intrinsic and extrinsic

• Straightness by ribbon and rolling
• Tangent cones and cylinders and circles
• Geodesics are great circles

Circles on a sphere

• Circles and planes, cylinders and cones
• Circles and spherical distance'
• Length and radii of circles

Great circles

• Great circles and planes of symmetry
• Great circles through two points
• Poles of great circles (and polars/equators of points)
• Great circles and antipodal points
• Angles between great circles
• Great circles as geodesics – why other circles are not

Visualization and Mapping

• Cross sections, side views and parallel projections
• Stereographic projection and central projection
• Spherical Geometry via stereographic projection on the plane