Make a vector space to "house" a parabola I've had this idea I find interesting.
A line on a plane or in space that goes through the origin of the system is a vector space because if you add up or multiply with a scalar any of it's elements, the result remains on the line. Similarly, a plane in space is a vector space.
Now, a parabola evidently isn't. If you take two points on it and add them together, they will rarely result in a point that's still on the parabola. 
So I was thinking if there was a way to construct a vector space (redefine addition and scalar multiplication) that can be translated to a parabola in the standard coordinate system.
Well. My original idea was to simply have the line $u = v$ and then say that there's a mapping between that vector space and a parabola. Something like $y = au^2 + bu + c$ but then some of the things I wanted to work don't work.
For example, if I wanted to find the distance between a parabola and a point, I'd like to be able to make an orthogonal projection of the point to the parabola and then calculate the distance between those two points.
I obviously can't just project a point onto the $u = v$ line since the point needs to be in the same space as the parabola is, but there's no way (as far as I know) to translate any point from the parabolas special space to the standard coordinate system unless the point is actually on the parabola.
Maybe there is a way to construct a coordinate so it would be kind of warped resulting in the x axis becoming a parabola?
What I would essentially like to do is have a point in standard coordinates, but placed in the warped system. Then "unwarp" the coordinate system so I get the same point in the actual standard coordinate system. Then project the point onto the x axis for example and warp the coordinate system again so I get the point on the parabola. Then calculate the distance between the original point and the new point.
This isn't related to any specific problem or anything. Just something I wanted to know.
 A: I'd encourage you to read Hartshorne's Algebraic Geometry for a rigorous form of reasoning about this idea, as there is much more to this than meets the eye, including rigorous ways in which a parabola would generalize a vector space.
Take the parabola to be the set of zeros of $y-x^2$. Now, if $y-x^2=0$ and $y-d=0$, then $d-x^2=0, \implies x=\sqrt{d}.$ Thus you can identify each points on the parabola (in this naiive model) with a single real coordinate, as the other is dependent on it. Take a vertical line to be the set of zeros (in two coordinates) of $x-c, c \in \mathbb{R}$. Now, if you determine the intersection of that line with the parabola, you get a point in standard coordinates for the intersection point, one of those coordinates gives you the coordinate in the parabola's coordinate system, and your choice of $c$ is the coordinate for the projection of the line onto the $x$ axis.
$y-x^2=0, x=c \implies y-c^2=0 \implies y=c^2,$ which gives us $[c^2]$ in the parabola's coordinate (for which I chose to use $y$), $[x,y]=[c,c^2]$ in standard coordinates, and $[c]$ the coordinate to the projection onto the $x$ axis (and $[c^2]$ in $x-c$ coordinates).
Then I suppose you would do this replacing $x-c$ with each line in the pencil of $[c,0]$ to project onto the orthogonal line to it, searching for one which is the normal line to a point on the parabola, and hence makes the shortest line to it.
But from trying to answer your question it doesn't seem like you really need a solution so much as a place to explore these things. Perhaps you would also be interested in The Geometry of Schemes, which takes some ring theory to truly understand, but can also be read in reverse, as an occult dictionary from ring theory to geometry. If it ever hinders your construction of the machinery, you can explore most things from much from elementary standpoints like synthetic geometry and systems of linear equations. For example, you can access the tangent bundle to the parabola through basic calculus. You can't, though, find the extra hidden point on the real line, which is a shame.
