Straight Lines on Cones and Cylinders – Initial Experiments

A straight line on a cone or cylinder means straight from an ant's view, not the view of an observer in space.  You can simulate this a number of ways.  We will take an approach with ribbons and (initially) with rolling.

Cylinder and Cone Definitions (right circular)

Let c be a circle in a plane with center O.  The right circular cylinder on c is the set of all points on lines perpendicular to the plane that pass through c.  A right circular cone on c with vertex P is the set of all points on lines though c and also through a point P on the line through O perpendicular to the plane.  The point P is called the vertex of the cone.

The lines in the definitions are called generating lines of the cylinder or the cone.  The intersection of the cylinder or cone with a plane parallel to the plane of c is called an equatorial circle (for the cone, one special case is does not give a circle – the parallel plane through P).

Notice that this definition of a cylinder or cone describes a surface of infinite extent.  But when we make a model we are of course confined to making a bounded piece of the cylinder.

How to make a Model of a Right Circular Cylinder

Here are two ways of making a model that we shall use:

How to make a Model of a Right Circular Cone

Here are two ways of making a model that we shall use:

 Straight Lines on a Cylinder or Cone by Experiment