Euclidean and Non Euclidean Geometry

Problem 4.1 Straightness on Cylinders and Cones

a. What lines are straight with respect to the surface of a cylinder or a cone? Why?

Why not?

Ans:

The straight is when it is symmetrical in half turn.

If a bug were to walk a cylinder

without changing direction, it would

be either vertical alongside the axis

line, or at an angle. The straight lines

will be helical or great circles. Any

other lines would not be a straight line

as it will have to change direction at a

certain point.

In a cone, the bug can walk in a straight

line either through the tip or around the

tip. Depending on the angle of the

cone, the straight lines around the tip

sometimes cross themselves. In one of

the models with less than 90 degrees, it

crosses itself twice.

Examine:

Can geodesics intersect themselves on cylinders and cones?

Ans:

In cylinders, the geodesics do not

intersect. However, the great circles

repeat on themselves.

In cones, depending on the angle of the

cone, the straight lines around the tip

sometimes cross themselves. In one of

the models with less than 90 degrees, it

crosses itself twice.

Can there be more than one geodesic joining two points on cylinders and cones?

Ans:

Yes, there could be more than one geodesic connecting two points. One creating a

vertical generator and another creating a helical in the cylinder or around the tip in

a cone. The following models prove this.

What happens on cones with varying cone angles, including cone angles greater than

360°? These are discussed starting in the next section.

Ans:

Geodesics behave differently when the cone angle varies. Although generators do

not change much when the angle varies, the geodesics that goes around the tip

changes drastically. As the angle decreases the geodesics get narrower and narrower.

It finally crosses itself. If the angle of the cone is to be reduced further, the geodesics

that go around the tips cross themselves multiple times.

360 degrees is just a plane. When the angle increases above 360, the straight line

might not be the shortest distance between two points. When two points are

considered, due to added surface area, it might be faster to take a curved line than

going straight.