What Is a Straight Line?
Straight is that which the middle is in front of both extremities.
—Plato, Parmenides
A straight line is a line which lies evenly with the points on itself.
—Euclid, Elements, Definition 4
having the same direction throughout its length; having no curvature or angularity;
—Webster's New Twentieth Century Dictionary
I.When do you call a line straight?
Consider your experiences. How would you explain straightness to a young child? (How would a young child explain straightness to you?) If you use a ruler, how do you know if the ruler is straight? What properties do straight lines have that distinguish them from non-straight lines?
Think about the question in four related ways:
1.How can you check in a practical way if something is straight? (Do not assume you have a
ruler, for then we ask, “How can you check that the ruler is straight?”)
2.How do you construct something straight—lay out fence posts in a straight line, or draw a
straight line?
3.What symmetries does a straight line have? A symmetry of a geometric figure is a reflection,
rotation, translation, or composition of them which preserves the figure.
4.Can you write a definition for “straight line”?
II.How do you construct a straight line?
Find practical methods for constructing a straight line. How do you know the line is straight?
III.On a sphere, when do you call a line straight?
Imagine an ant crawling around on a sphere. The ant can neither fly off the sphere nor dig into the sphere. The ant's world is just the surface; it never leaves it. What is a straight line to the ant? What will the ant see or experience as straight?
Show that the great circles on a sphere are straight with respect to the sphere, and that no other circles on the sphere are straight with respect to the sphere.
Try to consider only the surface of the sphere. We often find it difficult to separate the actions on the sphere from the 3-dimensional properties we observe. Try to avoid extrinsic 3-dimensional vision of curves on a sphere.
How can the ant check if it is traveling in a straight line?
Use balls, rubber bands, tape, string, etc. to help model the problem.
Can a straight line ever intersect itself? If so, how many times? Can there be more than one straight line joining two points? How many? Is there always at least one?
IV.On a cone or cylinder, when do you call a line straight?
What lines are straight with respect to the surface of a cone or a cylinder? Why? Why not? Does the cone angle affect whether a line is straight or not? Assume the cone or cylinder continues indefinitely, that is with no top or bottom.
Imagine an ant crawling around on a cone or cylinder, as before with the sphere.
If we make a cone or cylinder by rolling up a sheet of paper, will straight remain the same for the ant if we unroll it? If we draw a straight line on a sheet of paper and roll it up, will it continue to be straight for the ant crawling on it?
On a cylinder or cone can a straight line ever intersect itself? If so, how many times?
Can there be more than one straight line joining two points on a cylinder or cone? How many? Is there always at least one?
V.On the surface of the earth, when do you call something straight?
A parent says to their child, “Come straight home young man (lady)!”
The shortest path between two points is a straight line. What is the shortest path between two cities on opposite sides of a mountain range?