Download ASSIGNMENT 4

ASSIGNMENT 4
1. What are the geodesics on a cone? Look at cones of different cone angles (make
models and do the ‘ribbon test’).
2. Can geodesics intersect themselves on a cone? How many times? Here again,
look at cones of different cone angles (make models and do the ‘ribbon test’). Use
cone with the following cone angle measures: 0 – 59 degrees, 60 – 89 degrees, 90
– 179 degrees, 180 – 360 and more than 360 degrees.
3. Describe the terms ‘1-sheeted covering of a cone’, 2-sheeted covering of a
cone’,…, n-sheeted covering of a cone’.
4. Define ‘branch point of a covering’.
5. Define the ‘lifts of a point on a cone’
6. Can there be more than one geodesic joining two points on a cone? How many are
there?
7. Explain the term ‘Locally Isometric’.
Results on Local Isometry
i.
ii.
iii.
iv.
A cylinder and the plane are locally isometric at each point.
A cone and the plane are locally isometric except at the cone point.
2 cones are locally isometric at the cone point iff their cone angles are the same.
A sphere is not locally isometric with the plane.
8. When is a surface said to be ‘smooth’? ‘complete’?
Results in Differential Geometry
i.
If a surface is sooth, then a geodesic on the surface is always the shortest path
between “nearby” points
ii. If a surface is smooth and complete, then any two points can be joined by a
geodesic that is the shortest path between them.
Euclid’s definition of right angle
When a straight line intersects another straight line such that the adjacent angles are equal
to one another, then the equal angles are called right angle.
9. Explain: ‘On a cone, right angles are not equal’