I am reading "Applied Differential Geometry" by William Burke. While I always had no trouble in standard math classes (diff. eq. and calc.) modern math (diff. geom.) is another story. This level of abstractness eludes me. So here is my question:

The author writes:

"The parametrization along the line passing through two points in an affine space is not unique." -p. 14.

This is perfectly clear to me. The scale is arbitrary as well as where I start and stop on the line. (It's similar to arc length along a curve. E.g. I can use feet or inches, cm, or m along the curve. Also, where I start on the curve is completely arbitrary.)

He continues:

"If we single out the parametrization that runs from zero to one between two points, then the structure of the affine space A is given by a map

with conditions

"

Now there is a figure which shows four points on a straight line. Point 'a' is labeled , and point 'b' is labeled . This makes sense according to the above definition of the map. However, the other two points, and do not work.

I think I understand them correctly - that they are two other (arbitrary) scalings for the parametrization of the line. But seeing that in detail completely - that is applying them - eludes me.