Affine structure, Lambda_k

**livingdog** · July 6th 2010, 05:07 AM

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

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

with conditions

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

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

**maddas** · July 6th 2010, 06:33 AM

The image is here: Applied differential geometry - Google Books

I don't understand what your question is. Are you thinking of a specific application?

An affine space is just a linear space without distinguished origin. If you chuse an origin then you can write explicitly $\Lambda : (a,b,t) \mapsto (1-t)a + tb$ .

**livingdog** · July 6th 2010, 09:15 AM

Originally Posted by maddas

The image is here: Applied differential geometry - Google Books

I don't understand what your question is. Are you thinking of a specific application?

An affine space is just a linear space without distinguished origin. If you chuse an origin then you can write explicitly $\Lambda : (a,b,t) \mapsto (1-t)a + tb$ .

EDIT: let me restate this (and it's my fault I cannot be more clear through my confusion)

There is a figure (thx to maddas for posting it) which shows four points on a straight line.
Point 'a' is labeled $\Lambda_{0}$ , and point 'b' is labeled $\Lambda_{1}$ .
This makes sense according to the above definition of the map.

However, I do not know how to apply (use?) the other two scalings: $\Lambda_{\frac{1}{2}}$ , and $\Lambda_{2}}$ .

I think I understand them conceptually - that they are just two other arbitrary scalings for the parametrization of the line. But seeing that in detail - i.e. applying them to the line - completely eludes me. So how does one apply $\Lambda_{\frac{1}{2}}$ , and $\Lambda_{2}}$ ?

If I crank out the first one, then $\Lambda_{\frac{1}{2}}(a,b) = \frac{1}{2}(b + a)$

and if I crank out the second one, then $\Lambda_{2}(a,b) = 2b - a$ .

Where does that get me? Are the $\Lambda_{k}$ just points on the line - and not scalings??

I apologize b/c I don't think I asked a ask...

...maybe this is my question: WHAT THE HEC IS $\Lambda_{k}$ ???

...oh wait a minute, I think I see it now. Let me try to explain this.

$\Lambda_{k}$ chooses the start and end point of the scale - the 'k' factor - that one is using for the line. Thus the entire line can be located using these different scales, $\Lambda_{1}$ for scaling in units of '1', $\Lambda_{0.75}$ for scaling in units of 0.75, etc., where k $\in$ R.

See the attached image:

I apologize if I am still too thick to understand.

**maddas** · July 6th 2010, 03:25 PM

Errrr, let me write $\Lambda_k(a,b)$ as $\Lambda(a,b,k)$ . Then fixing a and b, $\Lambda(a,b,\cdot):\mathbb{R} \to A$ is a parameterized (constant speed, I assume) line in the affine space. For a given value of k, $\Lambda(a,b,k)$ is a point on the line. A (constant speed) reparamterization of the line is a map $\gamma : \mathbb{R} \to A, t \mapsto \Lambda_k(a,b,c_1+c_2t)$ for constants c_1 and c_2. Intuitively, if you start at a at t=0 and you get to b at time t=1, then $\Lambda(a,b,k)$ is where you are at time t=k...

**livingdog** · July 7th 2010, 01:45 AM

Originally Posted by maddas

Errrr, let me write $\Lambda_k(a,b)$ as $\Lambda(a,b,k)$ . Then fixing a and b, $\Lambda(a,b,\cdot):\mathbb{R} \to A$ is a parameterized (constant speed, I assume) line in the affine space. For a given value of k, $\Lambda(a,b,k)$ is a point on the line. A (constant speed) reparamterization of the line is a map $\gamma : \mathbb{R} \to A, t \mapsto \Lambda_k(a,b,c_1+c_2t)$ for constants c_1 and c_2. Intuitively, if you start at a at t=0 and you get to b at time t=1, then $\Lambda(a,b,k)$ is where you are at time t=k...

I think I see now:

take a bare line
attach, and fix, two points: 'a' and 'b'
to locate any point on the line, use $\Lambda_k$

So e.g. $\Lambda_\frac{1}{2}(a,b)$ locates the $\frac{1}{2}$ point between 'a' and 'b'; $\Lambda_{2}(a,b)$ locates the $2$ point using 'a' and 'b' as the scale of the line.

If that's it then I got it.

Sorry for being so dense. But abstract math was never my forte'. Beyond linear algebra I get bogged down not being able to "see" the concepts without many questions/answers - as my posts in this thread demonstrates.

Thank you very much for your explanations and patience with me!

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