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

$\displaystyle \Lambda_{.} : A A R \rightarrow A; (a,b,k) \mapsto \Lambda_{k}(a,b)$

with conditions

$\displaystyle \Lambda_{0} = a, \Lambda_{1} = b$"

Now there is a figure which shows four points on a straight line. Point 'a' is labeled $\displaystyle \Lambda_{0}$, and point 'b' is labeled $\displaystyle \Lambda_{1}$. This makes sense according to the above definition of the map. However, the other two points, $\displaystyle \Lambda_{\frac{1}{2}}$ and $\displaystyle \Lambda_{2}}$ 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.