What are the relationships (connections) among:
- Euclidean (or Plane) geometry
- Analytic geometry
- Affine geometry
How do these things relate?
I know that this is a very general question, so I'm looking for a simple overview.
Thanks.
[Math] Euclidean Geometry versus Analytic Geometry versus Affine Geometry
affine-geometryanalytic geometryeuclidean-geometry
Related Solutions
Is the camera plane the projective space of the real world?
It depends.
Usually a physical camera has a limited sensor, so the thing where the image appears is too bounded to be an affine or projective plane. You can say “well, simply extend that plane infinitely”. Then it depends on how exactly you define that extension, whether you end up with an affine or a projective space. If you say “well, take any line on it, and for any distance $d\in\mathbb R$ take the point at position $d$ along that line to be a point of my plane as well” then the result is affine. If you say “well, take any pair of lines on it, and include their point of intersection as well” then it's projective.
And for something planar to be “the projective space” of “the real world”, that real world has to be planar, too. So if you have a camera image of things happening on the flat surface spanned by your table top, then yes, you could obtain an image of that surface in your projective camera plane. But if you take a camera image of a 3d scene, you get a projection, which is a reduction of information and usually not what projective geometry describes. Usually, a projective space is used as something more complete than the underlying affine space: you add points (at infinity), you don't remove them (by combining multiple points in the line of projection into a single point on the image plane).
Is the line which is the image of the horizon the distinguished line?
If you have your geometry sketched on a real paper lying on your real (horizontal) table, then the horizon of your real world is the distinguished line. You draw two parallel lines on your table, you look from an angle and you can almost see them converging on the horizon. Of course, that horizon is infinitely far away, so if you were to go there, the fact that the earth is not flat may break the simile.
If you take a camera image of that situation, then your projective camera sensor plane has its own distinguished line at infinity, namely where two parallel lines in the camera sensor plane intersect. If the image of one distinguished line is the other distinguished line, then your camera plane is parallel to the real world table plane, and the resulting transformation would be called affine. Otherwise it's a generic projective transformation.
Whenever we do an Affine transform do we need to look out for a distinguished line?
It depends on how and why you do affine transformations.
If you do an affine transformation via a multiplication of a matrix of the form $$\begin{pmatrix}*&*&*\\*&*&*\\0&0&1\end{pmatrix}$$ then the form of the matrix already guarantees that points with a $z$ coordinate of zero will get mapped to points which again have a $z$ coordinate of zero, i.e. using the $z=0$ line as the distinguished line in both your spaces is already implied by the form of the transformation.
If you do affine transformations as $$\begin{pmatrix}x\\y\end{pmatrix}\mapsto\begin{pmatrix}ax+by+c\\dx+ey+f\end{pmatrix}$$ then you are using the same scheme computation as above, plus you are using non-homogeneous coordinates, so you might not even see what this $z$ coordinate is about since the 2d affine coordinates don't even allow you to express points at infinity.
If, on the other hand, you have some weird choice of coordinate systems where the distinguished line has a more complicated form, and that form perhaps isn't the same for source and target space of your transformation, and you need to ensure that your transformations are indeed affine, then yes, you need to be explicit about the lines at infinity, and verify that they get mapped correctly. One common example would be trilinear coordinates, where $x+y+z=0$ represents the line at infinity. Mapping from carthesian to trilinear coordinates and back is a situation where you want to make sure that the line at infinity gets transformed correctly.
Why does just a distinction of the geometry a line in the perspective plane make the geometry an Affine geometry?
As the other answers by Joseph Malkevitch and by AdiPiratla have already indicated, there are different ways to treat the relation between affine and projective geometry. One is to say that you get projective geometry from affine geometry if you add a point at infinity for every bundle of parallel lines, and a line at infinity made up from all these points. In this sense, a projective space is an affine space with added points.
Reversing that process, you get an affine geometry from a projective geometry by removing one line, and all the points on it. By convention, one uses the line $z=0$ for this, but it doesn't really matter: the projective space does not depend on the choice of coordinates, and removing any line will turn it into an affine space.
But often one wants to exploit the machinery from projective geometry to perform affine operations. Perhaps you want to combine affine and projective transformations, or some such. In such a setup, you can say that as long as you keep track of which line is the line at infinity, you know how to get from there to affine geometry, so you are already doing affine geometry in a different representation. Affine geometry is like projective geometry with one line (the “distinguished line”) labeled “remove this to obtain an affine plane”. In this sense, an affine space is a projective space with additional information.
I write $P + v$ for the action of the vector $v$ on the point $P$.
An affine space is, in fact, probably the very first visualization you had of vector spaces: e.g. you thought of the plane full of points, and that vectors were arrows that went from one point to another.
The concept of affine space I know requires the action of $V$ on $X$ to be transitive and faithful: this means that, in an affine space, we can define subtraction: $P - Q$ is the unique vector $v$ such that $Q + v = P$. The pair $(Q, v)$ can be pictured as an arrow from $Q$ to $P$.
We can even define nearly arbitrary linear combinations of points: the restriction is that the coefficients have to sum to zero (thus giving us a vector) or sum to one (thus giving us a point).
e.g. if I write $P + \frac{1}{2}Q - \frac{3}{2} R$, I 'really' mean the vector
$$P + \frac{1}{2}Q - \frac{3}{2} R = (P - R) + \frac{1}{2} (Q - R)$$
or any other similar rearrangement into "legal" operations. (they would all give the same answer)
Similarly, if I write $\frac{1}{2}P + \frac{1}{3} Q + \frac{1}{6} R$, I mean the point
$$ P + \frac{1}{3} \left(Q - P \right) + \frac{1}{6} \left(R - P \right) $$
I believe I've seen some definitions of affine space that don't make reference to vectors at all: they are instead axiomatized in terms of the arithmetic of points.
The notion of vector space is, in fact, equivalent to the notion of a pair consisting of an affine space and a point on the space. A vector space already has the structure of an affine space; it just comes equipped with a distinguished point $0$. Conversely, given any affine space and a choice of a point $O$, we can complete its vector space structure by treating every other point $P$ as the vector $P - O$.
Beside from thinking like vectors, convex combinations (where the coefficients sum is one and are all positive) can be thought of as averaging between points. For example,
- $\frac{1}{3} P + \frac{2}{3} Q$ is the point on the line segment $PQ$ that lies two thirds of the way from $P$ to $Q$
They're called convex combinations, because of the relation to convex sets: any convex combination of points in a convex set is again in the set. In fact, the convex hull of a set of points is precisely the set of all convex combinations of those points.
e.g. the set of all points inside the triangle $\Delta PQR$ are of the form $aP + bQ + cR$ where $a+b+c = 1$ and $a,b,c$ are all positive. (if we allow $a,b,c$ to be zero as well, then this set of points would include the triangle itself)
More general affine combinations (the coefficients sum to one) are similar:
- $2P - Q$ is the point on the line $PQ$ that lies on the other side of $P$ from $Q$ that is twice as far from $Q$ as it is from $P$.
All convex combinations, actually, can be made out of binary ones: for example,
$$ \frac{1}{2} P + \frac{1}{3} Q + \frac{1}{6} R = \frac{5}{6} \left(\frac{3}{5} P + \frac{2}{5} Q \right) + \frac{1}{6} R $$
(how did I find this? Let $S$ be this point. I picked points on the plane for $P$, $Q$, and $R$ and found where the line $RS$ met the segment $PQ$: that point is the point that appears in the parentheses on the right hand side)
For linear combinations, you can just pretend the points are vectors. It doesn't matter which point you choose in the affine plane as the origin, you'd get the same result. e.g. my earlier example satisfies
$$P + \frac{1}{2}Q - \frac{3}{2} R = (P - O) + \frac{1}{2} (Q-O) - \frac{3}{2} (R - O)$$
no matter which point you choose for $O$.
Best Answer
Felix Klein defined geometry to be the study of invariants of group actions.
Euclidean geometry looks at things unchanged by Euclidean transformations, i.e. rigid transformations, i.e. rotations, translations (and sometimes reflections).
Affine geometry looks at things unchanged by affine transformations. This usually means non-singular linear transformations, i.e. $\det M \neq 0$. An important subset, equi-affine geometry, studies linear transformations which preserve area, i.e. $\det M = 1$.
Projective geometry does the same with projective transformations.
Analytic geometry doesn't really fit in to this. For me, "analytic geometry" just means "using coordinates". All of Euclidean, affine and projective geometry can be done using coordinates.
Maybe I misunderstand "analytic geometry".