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:

• Make a model of (a piece of a cylinder) from a rectangle by taping two opposite sides together.
• Roll up a rectangle so that the paper goes around the cylinder several times.  This turns out to be an important way of thinking of a cylinder.

How to make a Model of a Right Circular Cone

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

• Make a model of (a piece of a cylinder) from a circle by cutting out a sector and then taping two opposite sides together.
• Cut a slit in a circle and roll it up so that the paper goes around the cone several times.  This turns out to be an important way of thinking of a cylinder.

 Straight Lines on a Cylinder or Cone by Experiment

• Make a model of a cylinder.
• Draw a straight line on a table or the floor.  Roll the cylinder along the straight line and observe which points touch the line.  These points will be the points of an "ant's path" on the cylinder.
• To see this better, stretch a ribbon on the line and roll up the ribbon as you roll the cylinder.
• Do experiments to answer questions 4.3 and 4.5 on Assignment Sheet 4B.