Euclidean geometry is a very good mathematical framework for describing various properties of shapes and motions. Except it's got an exceptional case at its very foundation -- parallel lines, and when we move up to 3D, planes. Parallel lines don't intersect, while any other pair of (different) lines intersect at exactly one point. Of course, being lazy as we are, we hate handling exceptional cases.
Well, it turns out that we can get rid of parallelism and still obtain quite usable geometry.
We won't be giving axiomatic definitions here. Instead we will state some of the properties of projective planes and projective spaces. Some of the properties are axioms and some are theorems. It is not important to us which is which.
Any object that satisfies these properties is called a projective plane.
- For any two distinct points, there is exactly one line that passes through both of them.
- For any two distinct lines, there is exactly one point that is common to both of them.
- There exist at least three points, not all lying on the same line.
- There exist at least three lines, not all passing through the same point.
- Every line contains at least three points.
- Every point lies on at least three lines.
If you're bright enough (of course you are), you have already noticed some kind of symmetry here. Odd-numbered properties can be obtained from even-numbered properties if we replace the word "point" by the word "line" and vice versa, and also replace phrases like "point lies on line" with "line passes through point" and vice versa. This is a very handy property of projective planes. It's official name is duality. Duality means that for every theorem we can automatically obtain another theorem, called its dual, by exchanging points with lines and vice versa.
In order to simplify things even further, instead of saying "point lies on line" or "line passes through point", we will say "point and line coincide ". This phrase is symmetric w.r.t. points and lines, which makes turning a proposition into its dual a completely automatic process.
Why is this useful? It turns out that if we arbitrarily choose a single line (together with all the points that coincide with it) and call it "line at infinity" or "ideal line" and just throw it away, the rest of the projective plane turns into our familiar Euclidean plane. That is, any two lines that were intersecting at ideal line no longer intersect and become "parallel lines", and all axioms of Euclidean geometry hold.
Conversely, if we take an Euclidean plane and complement it with an object called "ideal line", and postulate that any family of parallel lines have their "intersection point" lying at the ideal line, we will get a projective plane. By the way, points on ideal line are called ideal points.
We will go by this route when deriving a coordinate representation of projective geometry. But first, a few words on projective space.
There are more properties akin to properties 5 and 6 of projective plane, but we'll not discuss them here.
- Three points not all coincident with the same line are coincident with a unique plane.
- Three planes not all coincident with the same line are coincident with a unique point.
- For a line and a plane not coincident with it, there's exactly one point that is coincident with both.
- For a line and a point not coincident with it, there's exactly one plane that is coincident with both.
- Two distinct planes are coincident with exactly one common line.
- Two distinct points are coincident with exactly one common line.
Again, we can see that there's a symmetry between odd-numbered and even-numbered properties (we've made it apparent by talking about coincidence right from the start). The difference is that now points are dual with planes, not lines. You can guess what will happen if we move forward to higher dimensions.
In addition, there's a property of projective spaces which says that every plane is a projective plane, in the sense already defined.
Needless to say, a trick similar to that of ideal line will move us back and forth between projective space and Euclidian space, only now we introduce an ideal plane instead of ideal line.
A point on plane is represented by a pair of co-ordinates (x, y). Let's add a third co-ordinate at the end. We postulate that
To arrive from homogenous co-ordinates back to Euclidean, we simply divide by the third co-ordinate: (x, y) = (X/Z, Y/Z). It is immediately clear that there are more "points" than the Euclidean plane has : (X, Y, 0) maps to nothing because we can't divide by zero! Not-so-amazingly, it turns out that such triples precisely correspond to ideal points of projective plane.
- (x, y, 1) represents the same point as the pair (x, y);
- (X, Y, Z) represents the same point as (αX, αY; αZ) for any scalar α
- (0, 0, 0) is not allowed.
What does this buy us? Let's see how we would represent lines. We start with the familliar equation for a line in Euclidean plane:
ax + by + c = 0
Noting that this equation is not affected by scale, we arrive to
aX + bY + cZ = 0, or
uTp = pTu = 0
where u = [a, b, c]T is the line and p = [X, Y, Z]T is a point on the line. Surprise: points and lines have the same representation in homogenous co-ordinates! No wonder, because they are dual concepts. It is easy to derive a formula for intersection point of two lines: p = u1 × u2, and for a line that passes through two points: u = p1 × p2. Again, thanks to duality, the two formulas are identical. More fun with formulas: three points lie on the same line if det[p1 p2 p3] = 0. How would you determine whether three lines all go through the same point?
In three dimensions we will of course have 4-tuples for points and planes.
You might wonder, what parallels (pardon the pun) in the real world these highly abstract concepts may have? Yet many people can see an ideal line with their naked eyes, without even realising it. You can too, if you live on a vast plain or near sea shore. Yes, it's the horizon.
Of course, railroad tracks or edges of a highway don't really intersect, but we perceive them intersecting at the horizon. That's because the world around us undergoes projective transform in our eyes. Photographs of tall buildings often exhibit the same phenomenon. If you take one and continue images of a bunch of lines that ought to be parallel in the real life, you will see that they all intersect at the same point. Another bunch of parallel lines will intersect at another point. All these points lie on a straight line -- the horizon. The horizon is the image of the ideal line in our eyes or on a film.
In the next part we will take closer look at projective transforms.
End of part 1.