# What is the affine space and what is it for?

These two topics already exist:

(preface: got in contact with affine space through computer graphics subject in university)

What are affine spaces for?

What are differences between affine space and vector space?

However, people stop explaining when it comes to the point. So I'd like to ask at that point: this post says:

vector spaces are the natural generalization of translations of spaces affine spaces are important, because they recover the concept of points which the "arrows" (vectors) of a vector space move.

Which means: Vector spaces exist to handle space movement and the affine space exists to handle the coordinates of a vector - so is basically one meta level below the vector space?

this post states:

An affine space is an abstraction of how geometrical points (in the plane, say) behave. All points look alike; there is no point which is special in any way. You can't add points. However, you can subtract points (giving a vector as the result).

Which means: I can basically imagine a coordinate system, where objects like arrows (vectors) do not exist. I only have points to work with. In addition I have only a small number of operations, I can do on them (e.g. subtracting)?

this very nice post did not explain, "why" P2 is in fact an affine space. It just says "it is":

In fact, P2 is a classical example of an affine space.

My conclusion: It's basically like a vector space, but without the need of an absolutely specified origin. It's like as if you have a vector space, but moved away from the origin and there is no vector space anymore (like P2 from the post). So if the "user" wants to make use of the affine space, he has to define it's own origin. All points, that are used later (set by the point coordinates) are based on that "self-made"-origin's position

In total I imagine the affine space like a coordinate system with no lines, no vectors, no squares, etc, but only dots. The dot's positions are always relative to the "initially set" origin point. In programming-style words: it's basically a kind of "super class" for the vector space?

@usage (in Computer graphics): From my understanding: It's made for having a sub-coordinatesystem for objects in the space, so that things don't need to be manipulated on the base of the "global" coordinate systems? E.g. "local transformations" in 3D programs (like blender) would be realized through the concept of the affine space?

## my professor says:

If you don't have a vector space, you can't have an affine space

This statement actually destroys every understanding I gained through the posts here on stackexchange...

• What's the question, exactly? – Hans Lundmark Nov 25 '15 at 14:35
• It may be more fruitful to compare groups of transformations. Speaking of groups acting on a Cartesian space, with the analogous questions in parentheses: orthogonal transformations ("What is an inner product space?"), linear transformations ("What is a vector space?"), affine transformations ("What is an affine space?"). – Andrew D. Hwang Nov 25 '15 at 15:04
• @Hans Lundmark: one part of questions would be, if my understand is right, regarding the affine space (for that I put the questionsmarks in) another part would be the question as stated: what is it (/for)? why is it really necessary? why is a "usual" space not enough? – TheTrowser Nov 25 '15 at 16:07
• @TheTrowser did you get your answer from somewhere ? Can you share it here ? – Martin Spasov Jul 8 '18 at 16:24