Homogeneous Coordinates

Let us invent a new way to represent a number. Suppose the number 2. We can write it as 4/2. Let as write the numerator and denominator as a pair, so we have (4,2). Now, if we adopt the interpretation that any pair of number (a,b) is to represent the number a/b, then, we have a system where a number is represented by a pair of numbers.

What is the point of this?

Consider the real number line. To the right of 0 are 1, …, 1.1,…2, … Infinity. To the left similar to -Infinity. Sometimes, in mathematics we consider the positive infinity and negative infinity as a single point, we call it a Point At Infinity. Why?

For example, consider a circle in the plane, with radius 1/2 and centered on {0,1/2}. Consider the point at the top of the circle {0,1} as the North Pole, N. Consider a point on the real line P with coordinate {p,0}. Let there be a line from N to P. This line will intersect the circle somewhere, call this point Q. This way, every point P has a corresponding point Q on the circle. However, we couldn't say that every point on the circle has corresponding point on the real line, because in our process, the point N has no corresponding point on the real line. However, conceptually, we can think of N actually corresponding to certain point at Infinity.

This is a example, where it seems reasonable to consider that on the real line, there is a point called Infinity, such that positive Infinity and negative Infinity are the same point.

This may seem a bit weird or unnatural, but at least in our context, it makes sense. In mathematics, there are many concepts that at first seems weird or unnatural. For example, negative numbers, irrational numbers, complex numbers, or non-euclidean geometry have all in one time or another being regarded as unnatural and rejected. However, if we disregard our preconceptions and simply define our system carefully, so that there are no contraditions, we should proceed to whatever makes sense in our context.

So, in our context, it makes sense to consider that, if we move along the real line, far far to the right there is a single point called Infinity, which is the same point if we move to the left of the real line.

2006-08

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