# Affine Space An affine space can be thought of as what is left of a vector space after you have forgotten about which point is the origin. This simply means that we have added [Translations](Translations.md) to [Linear Transformations](Linear%20Transformations.md) (i.e. linear maps). ### Affine Structure and Affine Space To start, read the [wikipedia information description](https://en.wikipedia.org/wiki/Affine_space#Informal_description) and look at your [whiteboard notes](https://photos.google.com/photo/AF1QipNc5Tk0a6o0UwuuMOYFyiwzhhtD1SkzQ9ZvO82z), *affine combinations* notebook, and convex and affine combinations in notability. With that said, I can summarize the key ideas as follows: * Our goal is to add **translations** to **linear maps**. Linear maps make specific use of a fixed origin. This fixed origin implies/ensures certain structure. If we remove this, we will have less structure to work with. If we have removed this, we are left with affine structure, known as **affine space**. * To start, consider two fixed points in space, $a$ and $b$. * We will define CS1 as our first coordinate system, with the traditional fixed origin at $0$. From the perspective of CS1, $a$ and $b$ are the red vectors shown below:  * Now, from the perspective of CS2, which has an origin of $P$, $a$ and $b$ are the respective blue vectors. * So, to be clear, $a$ and $b$ are fixed points in space, irrespective of coordinate system. Based on the coordinate system we pick (specifically, where we select the origin to be), we will have different *vectors* corresponding to moving *from the origin* to our points $a$ and $b$. * It is worth mentioning at this point: irregardless of the coordinate system we pick (irregardless of origin), the *distance* between points $a$ and $b$ will be the same! This already should provide us some intuition that there is structure that is mutually present in each coordinate system view. * With that said we can clearly see that origin does matter at first. Compare the red $a$ vector and the blue $a$ vector. By using different origins we can see that the vectors to get to the point $a$ are very different. * We can say that there is a transformation rule to take a vector in CS1 that is describing a point, and then describe it in CS2 (with a new origin). For instance, point $a$ is described in CS1 with the red vector $a$. To determine the vector that describes it in CS2 (with an origin $p$) we simply do: $a - p$. In general, given a vector $v$ described in CS1, if we wish to see how it is described in CS2 that has a different origin, we simply do: $v_{cs2} = v_{cs1} - p$ * This is a general transformation rule that allows us to move between coordinate systems with different origins * Now, if we wanted to look at a rule for adding to vectors it would look like: translate your two vectors into the new CS representation: $a_{cs2} = a - p$ $b_{cs2} = b - p$ * Then just be sure to account for the initial movement via $p$, resulting in a transformation rule: $(a + b)_{cs2} = p + a_{cs2} + b_{cs2} = p + (a - p) + (b - p)$ * Again, this will hold for any two vectors. This is a rule that provides information and structure. It is consistent whenever we wish to simply change our origin. * Now, after changing our origin, our vectors to get from origin $O$ and origin $p$ to points $a$ and $b$ respectively are indeed different. * So, generally, if we change our origin and then take a linear combination of the resulting vectors $a$ and $b$ (that take us from new origin to points $a$ and $b$), it will be a different linear combination than in the original coordinate system (with original origin). * However, remember, changing the origin still leaves a great deal of structure in place. We are only following a simple transformation rule, one that leaves many things unchanged. So, with this in mind, is there any type of constraint we could come up with that would *guarantee* that whether we add the original vectors for $a$ and $b$, or the new representations as they are described in the new CS, we get the same result? If we could come up with this, we would have *effectively found a constraint under which origin doesn't matter*! * The big key idea here is that by only looking at **affine combinations** we are effectively ensuring that origin doesn't matter! Because in an affine combination, we will end up with the same result irregardless of what origin we picked! * Specifically, regardless of what origin we pick, we will *describe the same point* with our affine combination, even in spite of using different origins.   --- Date: 20210618 Links to: [Affine Transformations](Affine%20Transformations.md) References: * [Wikipedia](https://en.wikipedia.org/wiki/Affine_space) * [Affine Subspaces and transformation](https://www.youtube.com/watch?v=fWRm9dISpNk&t=211s) * [What is an affine space?](https://youtu.be/c-iA43vPL9M?t=507)