单位、透视和缩放

What's the "unit length"? Alternatively, what does it mean for something to have "length $1$"?

Is the unit length an inch? a kilometer? How do we decide?

Your science/physics teachers would probably get mad at me for saying something like this, but in some sense, it doesn't matter. Once we've fixed/set our unit length to be something (like an inch or a centimeter), we can start talking about objects of "length $2$" (two unit lengths), "length $0.5$" (half a unit length), etc.

(Your science/physics teachers would be less mad if I clarified that this scaling idea is useful in applications: it's precisely why telescopes, microscopes, and scale drawings are so useful!)

In other words, "the" unit length (at least in math contexts) is only "well-defined up to" perspective, scaling, or dilation. This is OK because pretty much everything we care about in geometry only depends on relative lengths: intuitively, as you zoom in or out on an object, its "shape" still stays the same, even though the "size" changes. (Later we'll carefully discuss what it means for the "shape" to stay the same, in topics like "Dilation properties".)

Euclid's system did not give a high profile to actual numbers. For instance, Euclid's statement of the Pythagorean theorem ultimately amounts to the assertion that the square drawn on the hypotenuse can be dissected into pieces which can then be reassembled to form the two squares on the other two sides. There is no mention of numbers representing the lengths or areas. In fact, algebra in contrast to geometry was not worked out in Greece, and its development in the Western world had to await contact with Islamic thought in the Middle Ages.

As usual, there was a reason for the Greek attitude. They took the view that the only things that could properly be called numbers were the whole numbers $1$, $2$, $3$, $\dots$ and the fractions or rational numbers ${m\over{n}}$ derived from them. Quantities like the (ratios of) lengths of lines, which they called proportions, were considered to be of quite a different nature. The knew, in fact, that the proportion of the diagonal of a square to its side could not be represented as a fraction ${m\over{n}}$. We would express this by saying "$\sqrt{2}$ is irrational," but they did not use this kind of language.

In these posts, though, it will be assumed that you are familiar with the real number system: the number system which includes positive and negative, rational and irrational numbers, but not imaginary numbers like $i$ (the square root of minus $1$). It is possible to derive the properties of real numbers from those of other simpler mathematical objects, such as the whole numbers that the Greeks used, and such derivations ultimately go back to the discussion of the theory of proportion in Book V of Euclid; but nowadays we think of them as belonging to analysis rather than to geometry.

Using our knowledge of the real number system, we can state some further axioms. These say that every line in our geometry looks like the standard "number line" $\mathbb{R}$; or to put that a bit more precisely, there is a one-to-one correspondence between the line and $\mathbb{R}$, which we can think of as giving a numerical "coordinate" to each point on the line. To specify such a correspondence, our intuition suggests that we must choose an origin and a scale: in other words, we must select one point $A$ on the line to have coordinate $0$ and another point $B$ to have coordinate $1$. Once this has been done, the coordinates of all other points are uniquely determined.

In order to make this intuition precise, we will introduce a new undefined term: a ruler on the line $\mathcal{L}$. A ruler on $\mathcal{L}$ is a one-to-one correspondence $\mathcal{L} \to \mathbb{R}$ which should be thought of as giving a "uniform scale" along $\mathcal{L}$. The real number corresponding to a point $X$ will be called the coordinate of $X$ (relative to the given ruler).

Axiom (Ruler Axiom). Let $A$ and $B$ be distinct points, and let $\mathcal{L}$ be the line through $A$ and $B$. Then there is a ruler on $\mathcal{L}$, in terms of which $A$ corresponds to $0$ and $B$ corresponds to $1$.

A ruler in terms of which $A$ has coordinate $0$ and $B$ has coordinate $1$ will be called a ruler based on the points $A$ and $B$.

What would happen if we had selected a different origin and scale? This question brings out the important distinction between the point $X$ itself, which is a geometric object having nothing to do with any choice of ruler, and its coordinate, which is a real number depending not only on $X$ but also on the choice of ruler or "coordinate system." As long as we stick to one coordinate system this distinction is blurred; but in more advanced geometry it is often necessary to "transform the coordinates." Our present interest is in the one-dimensional case of rulers on a line; but the same principles apply in higher dimensions.

In these posts we will usually present quantities which depend on a choice of "coordinate system" by small letters. Thus the coordinate of a point $X$, relative to some ruler, is a real number $x$. If we are also considering a second ruler, $x'$ will denote the coordinate of $X$ relative to this new ruler. The two rulers are related by a change of origin and a change of scale: we expect, therefore, that the relationship will be of the form$$x' = U(x) = lx + m,\text{ }(l \neq 0).$$For example, one could think of the line as temperature scale, with the first ruler given by Celsius measurement and the second by Fahrenheit measurement; then the coordinate transformation $U$ is given by$$U(x) = {9\over5}x + 32.$$The next axiom makes explicit our assumption about the relationship between two rulers on the same line.

Axiom (Ruler Comparison Axiom). Two different rulers on the same line are related by a transformation of the form $$x' = U(x) = lx + m,\text{ }(l \neq 0).$$Rulers allow us to talk about the ratios of distances between points on a line. We cannot yet talk about distances themselves, because we have not assumed that there is any absolute standard of distance; but we can talk about one pair of points on a given line being twice as far apart (or half as far apart or whatever) as another. More formally, we can make the following definition:

Definition. Let $A$, $B$, $C$, and $D$ be points on a line $\mathcal{L}$. We define the ratio $AB: CD$ to be$$AB: CD = (a-b):(c-d)$$where $a$, $b$, $c$, and $d$ are the coordinates of $A$, $B$, $C$, and $D$ relative to some ruler on $\mathcal{L}$.

(A ratio $x:y$ is just given by a couple of real numbers, not both of which are zero; and two ratios $x:y$ and $x':y'$ are equal if $xy' = yx'$. A ratio is just like a fraction, except that either one of $x$ and $y$ is allowed to be zero.)

There is, however, a possible problem hidden in this innocuous-looking definition. We have defined the ratio $AB: CD$ in terms of some particular choice of ruler on the line $\mathcal{L}$. How do we know that if we had chosen some other ruler, we would have got the same result? Only if our concept of ratio does not depend on the choice of ruler can it be said to be truly geometrical. Similar questions arise in mathematics whenever the definition of some concept involves an arbitrary choice. One has to check that the concept is well-defined.

Of course, this kind of check is what we have the ruler comparison axiom for. If we work with a second ruler, it tells us that we have$$a' - b':c' - d' = U(a) - U(b):U(c) - U(d)$$$$=(la + m) - (lb + m): (lc + m) - (ld + m) = a-b:c-d.$$So the ratio does not depend on the choice of ruler, and this makes our definition a good one.

Example. Distance ratios along a line $\mathcal{L}$ satisfy various algebraic relations which are easily proved in terms of coordinates. For instance, if $AB: CD = x:y$ and $AB:DE = x:z$, then $AB:CE = x:y + z$. To give a formal proof of this, introduce a ruler on $\mathcal{L}$. The equation $AB:CD = x:y$ is equivalent to $y(a-b) = x(c-d)$, and the equation $AB: DE = x:z$ is equivalent to $z(a-b) = x(d-e)$. These equations can easily be combined to give$$(y+z)(a-b) = x(c-e)$$which is equivalent to $AB:CE = x:y + z$. We will not bother to work out all possible relations of this kind now; later on we will refer to this sort of calculation as "using the properties of ratios."

If $A$ and $C$ are thought of as fixed points, the ratio $AB:BC$ determines the point $B$. This is part of the content of the next proposition.

Proposition. Let $A$ and $C$ be fixed points on a line $\mathcal{L}$. Then for every possible ratio $x:y$ except $1:-1$, there is exactly one point $B$ on $\mathcal{L}$ for which $AB:BC = x:y$. There is no point $B$ for which $AB:BC = 1:-1$.

Proof. Choose a ruler relative to which $A$ has coordinate $0$ and $C$ has coordinate $1$. For a point $B$ with coordinate $\lambda$, the ratio $AB: BC$ equals $\lambda: 1 - \lambda$. If we put this equal to $x:y$, we get $y \lambda = x(1 - \lambda)$, which has the unique solution $\lambda = x/(x+y)$ provided that $x+y \neq 0$, i.e. $x:y \neq 1:-1$.$\,$ $\square$

Using the properties of ratios, one sees that if $AB:BC = x:y$, then $AB:AC = x:x+y$ and $AC:BC = x+y:y$. Any of these ratios therefore suffices to determine the point $B$. The ratio $AB:AC$ cannot equal $1:0$, and the ratio $AC: BC$ cannot equal $0:1$.

An enthusiast for ratios might regard it as a blemish that there is one ratio which does not correspond to any point on the line. One way of remedying this might be to regard the ratio $AB: BC = 1:-1$ as defining $B$ to be some new kind of "ideal point" or "point at infinity" on the line $\mathcal{L}$. It turns out that this can be done, and has many interesting properties; for instance, two parallel lines "meet at a point at infinity." This leads to the subject of projective geometry, which we won't talk about in this post.

Distance ratios allow us to define what we mean by a line segment.

Definition. Let $A$ and $C$ be two distinct points. The line segment $AC$ is the set of all those points $B$ on $\overleftrightarrow{AC}$ such that $AB: BC = x:y$, where $x,\,y \ge 0$.

If we work with the ruler where $A$ has coordinate $0$ and $C$ has coordinate $1$, then $B$ has coordinate $x/(x+y)$. If $x$ and $y$ are both non-negative this ranges between $0$ and $1$; so another way of defining $AC$ is as the set of all those points on $\overleftrightarrow{AC}$ that have coordinates between $0$ and $1$ in this particular ruler. One refers to such points $B$ as being between $A$ and $C$. If $A$ and $C$ are the same, then we define $AC$ to be the set consisting just of the single point $A$.

Definition. Let $AC$ be a line segment. The midpoint of $AC$ is defined to be the unique point $B$ on $AC$ such that $AB:BC = 1:1$. (If $A$ and $C$ are the same, then we let $B = A$.)

Notice that the midpoint of $AC$ is the same as the midpoint of $CA$. This is necessary for our definition to make sense, since you cannot tell just from the set of points $AC$ which end is $A$ and which end is $C$.

Rulers allow us to speak of the ratios of distances along a particular line. But they still leave all the lines in our geometry rather isolated from another; we do not yet have any way to compare distance ratios between two different lines. Our final axiom in this section, the similarity axiom, makes such a comparison possible.

Axiom (Similarity Axiom). Let $\mathcal{L_1}$ and $\mathcal{L_2}$ be two parallel lines in a plane, and let $\mathcal{M}$ and $\mathcal{N}$ be two other lines, meeting the parallel lines at points $M_1$, $M_2$, $N_1$, and $N_2$, and meeting each other at $X$. Then the ratios $XM_1: XM_2$ and $XN_1: XN_2$ are the same.

In the language of classical geometry, $XM_1N_1$ and $XM_2N_2$ are similar triangles; they have the same shape, even though they do not have the same size. One triangle is a "scaled" version of the other and so the scale factor between pairs of corresponding sides is the same.

There is a converse to the similarity axiom: if the ratios are the same, the lines are parallel. The proof uses the similarity axiom together with the parallel axiom:

(In progresss, to be continued...)