Understanding Affine Geometry and Space: Origins, Definitions, and Applications

[pmb_phy]

Some of the posters here are in my "ignore list", ...

I'm sorry but I don't understand why this fact is important enough to be stated in this post. It does not help anyone with understanding what you wrote.

Pete

 

[Cantab Morgan]

The origin of a vector space has a special meaning. It's the (unique) additive identity operator. In other words, I can add the vector zero to any vector and not change it. Addition of vectors has all the nice and expected properties like associative, commutative...

However, the addition of elements of an affine space is not well defined. It makes sense to write v + w, for a pair of vectors. But it makes no sense to write P + Q for a pair of points in an affine space. By analogy, one can drive a mile east, then turn right and drive another mile. But what does it mean to "add" the location of the city of Boston to that of the city of Los Angeles?

To me, saying that an affine space has no origin is saying that there's no additive identity element. An observation that follows directly from the idea that one can't "add" affine points.

(The vector that represents the "difference" between the locations of those cities can be defined. Drive thataway to get from Boston to Los Angeles. But even if you represent points with ordered tuples of numbers and vectors as ordered tuples of numbers, they are not the same kind of mathematical objects. The "drive thataway" vector is not an element of the affine space of points.)

 

[sundried]

Affine spaces are characterized by the fact that you have a notion of line, and a notion of parallelism of lines. These are the two main ingredients, and they allow you to take affine linear combinations of points.

 

[Cantab Morgan]

I'm enjoying this thread immensely and learning a lot. Nobody is yet on my ignore list.

Affine spaces are characterized by the fact that you have a notion of line, and a notion of parallelism of lines. These are the two main ingredients, and they allow you to take affine linear combinations of points.

I believe you are quite on target; parallel lines are the heart of affine geometry. Can you help me understand what you mean by "affine linear combinations of points"?

 

[sundried]

With pictures it is easier to grasp what convex combinations are: Have a look at the YouTube video: WildTrig35: Affine geometry and barycentric coords

 

[AmphibiousBat]

I've been meaning to think of this for some time. I have a deep feeling that Affine methods
are the natural ones for Newtonian physics and for Minkowsky physics. Affine methods are a nice marriage of the manifold picture and the vector space picture, that works in flat spaces.
When you use affine space, you don't forget about the vector space "its always there floating in the back ground" What you do, is imagine you have geometric flat space
no preferred origin, and when you take two points in that space, you get a difference vector (the same as you's do in the vector space method) you can't however "add two points"
but you can add a vector to a point to get another point. To emphasize: an affine space is a "pair" of a geometric flat space (whaterver flatspace may mean) and a vector space.

Naturally, Minkowsky space has no origin, we just imagine an origin to be there for simplicity.
by the way if you have a parametrized curve in an affine space you may define its derivative,
which "feels" just like the abstract tangent vector to a curve on a manifold.--(in fact that is what it is if you look at the affine space as a manifold) this derivative may be defined directly without the use of coordinates, because you can take differences between points.

I know this much, but I have'n t explored it any further.

 

[Studiot]

I would recommend

Introduction to Differentiable Manifolds
by Auslander and MacKenzie

Now available very cheaply in from Dover

for anyone studying or reading round most of the topics raised here, plus more.