Bielefeld, May 1997

Bogenmessung.

In elementary school, we learn to compare angles and to measure them by means of a protractor — a kind of yardstick bent into an arc. Later we discover the connection between angles and the proportions of right triangles, which is the basis of almost all indirect planar and spatial measurement. In that context we may, with some luck, find out about addition formulas like sin(α + β) = sin(α) cos(β) + cos(α) sin(β).

Toward the end of our school career, we are introduced to the functional perspective, in which (cos(t), sin(t)) are the coordinates of a point moving counter-clockwise around the unit circle at unit speed — yielding the famous wavy graphs which snake along the entire number line. Whether as time or as arclength, however, the independent variable t of this scenario dissolves into thin air as soon as we seriously try to play by the rules of the Cartesian plane. After all, the latter consists of nothing but pairs (x, y) of numbers such as (0.6, 0.8) for instance. But where do we find t in this case?

To answer this question, let us retrace our steps through the addition, comparison and measurement of angles. Instead of a black-board or a paper page, our canvas will be the Cartesian plane, and our drawing tools will be vectors and matrices. Since the measurement of arcs is our ultimate aim, we must also presuppose some knowledge of real numbers — but no more than is assumed in the most modest calculus book.

1. Initially, an angle is simply a point on the unit circle, in other words, a pair of numbers (x, y) such that x^{2} + y^{2} = 1. Such a pair already has the two possible meanings of point or vector, and now gets a third: that of angle. While the vector V = [x, y] indicates a “shift”, the corresponding angle θ = ⟨x, y⟩ indicates a “turn”, namely the rotation given by the matrix

R(θ) = [ \begin{matrix}x & −y \\ y & x \end{matrix} ] = x [ \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ] + y [ \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ]

= xI + yJ, where J is the last of the matrices displayed above. You should verify on a piece of graph paper that R(θ) geometrically tilts the whole plane, swinging the horizontal axis into alignment with the vector [x, y]. Also note: det R(θ) = x^{2} + y^{2} = 1.

In this view, θ is just an abbreviation for ⟨x, y⟩, and cos θ, sin θ are just fancy names for the coordinates x, y. The reason for this notational extravaganza is that vectors and angles must be read very differently. For instance, the negative of the vector V = [x, y] is of course −V = [−x, −y], but the negative of the angle θ = ⟨x, y⟩ is given by −θ = ⟨x, −y⟩, which yields the reverse turn R(θ)^{−1}.

“Adding” an angle θ_{1} = ⟨x_{1}, y_{1}⟩ to an angle θ_{2} = ⟨x_{2}, y_{2}⟩ means performing the appropriate turns one after the other, that is, multiplying the matrices R(θ_{1}) = x_{1}I + y_{1}J and R(θ_{2}) = x_{2}I + y_{2}J. This works out to R(θ_{1})R(θ_{2}) = (x_{1}x_{2} − y_{1}y_{2})I + (x_{1}y_{2} + x_{2}y_{1})J, the minus sign in the first term coming from the fact that J^{2} = −I. Since the product still has determinant = 1, it is indeed a rotation — and that is the one we take to be the “sum” of these angles. In other words, we define R(θ_{1} + θ_{2}) := R(θ_{1})R(θ_{2}). Likewise we have R(θ_{1} − θ_{2}) = R(θ_{1})R(θ_{2})^{−1} or

⟨x_{1}, y_{1}⟩ − ⟨x_{2}, y_{2}⟩ = ⟨x_{1}x_{2} + y_{1}y_{2} , x_{2}y_{1} − x_{1}y_{2}⟩ .  (∗)

For instance, if θ_{1} = ⟨3/5, 4/5⟩ and θ_{2} = ⟨12/13, 5/13⟩ we have θ_{1} + θ_{2} = ⟨16/65, 63/65⟩, which is quite different from what we would get by vector addition.

All this was pure algebra, applicable to any “number” system S in which one can add, subtract and multiply under the usual laws of arithmetic. If S also allows division by any non-zero element (and if 1 + 1 ≠ 0) it is easily shown that the “unit circle” consists of (−1, 0) and all (x, y) such that

x = \frac{1 − s^{2}}{1 + s^{2}} , y = \frac{2s}{1 + s^{2}} ,

for some s ∈ S with s^{2} ≠ −1. The parameter s can be interpreted as the slope of the line through (x, y) and (−1, 0). In particular, S could be the system of rational numbers — in which case every rotation R ≠ ±I, ±J spawns an infinite sequence of powers R^{n}, which keeps wrapping around the circle without ultimately leaving any visible gaps.

2. Now to the main problem. If x^{2} + y^{2} = 1, and we think of (x, y) as an “angle” with the associated rotation matrix R = xI + yJ, there is no mystery about the numbers x and y: though pompously named cos(R) and sin(R), respectively, they are just data. But — within the Cartesian context — how do we measure the “arc” through which R rotates the plane? To study this question, we may stay inside the first quadrant, where sines and cosines are never negative. This does not really restrict our range, since we can move from one quadrant to the next by using J as a multiplier (i.e., “adding” a right angle).

Rotations R, S in the first quadrant can be compared by saying that R “is greater than or equal to” S (written R ≥ S) if RS^{−1} is still in the first quadrant — otherwise R is “less than” S (written R < S). This relation is conveniently reflected by the size of the sines: sin(R) ≥ sin(S) implies cos(S) ≥ cos(R), and together these give sin(RS^{−1}) = sin(R) cos(S) − cos(R) sin(S) ≥ 0, with analogous implications in the case of “less than”. The cosines remain non-negative both times, since cos(RS^{−1}) = cos(R^{−1}S) = cos(R) cos(S) + sin(R) sin(S).

For calibration, we shall construct a string of rotations B_{1}, B_{2}, … whose arcs are known fractions of the quadrant. Starting with B_{0} = J, every successive B_{m} is obtained from the preceding one by bisection, and therefore has half the arc.

Bisecting a rotation matrix A = uI + vJ, with v > 0, means finding a matrix B = xI + yJ such that y > 0 and B^{2} = A. Since (xI + yJ)^{2} = (x^{2} − y^{2})I + (2xy)J, we must have x^{2} − y^{2} = u and 2xy = v. Since det A = 1 is to be the square of det B = x^{2} + y^{2}, we further get x^{2} + y^{2} = 1, whence 2x^{2} = 1 + u. If both v and y are to be positive, 2xy = v implies that x must also be positive. The resulting equations and inequalities are uniquely solvable by

x = \sqrt{\frac{1 + u}{2}} ,  y = \frac{v}{2x} .  (†)

Since this bisection process is unique, it makes perfect sense to write B = A^{1/2}.

The matrices B_{1} = J^{1/2}, B_{2} = J^{1/4}, …, B_{m} = J^{1/2^{m}} so obtained have arcs of 1/2, 1/4, …, 1/2^{m} quadrants. As k runs from 1 to 2^{m}, the powers B_{m}^{k} produce 2^{m} equal segments in the first quarter circle. Therefore we shall say that the k-th power of B_{m} has an arc of k/2^{m} quadrants, and write B_{m}^{k} = R(k/2^{m}). For such measures of arc μ = k/2^{m} and μ′ = k′/2^{m}, we clearly have

R(μ)R(μ′) = R(μ + μ′) ,  (∗∗)

where the + now stands for the actual addition of numbers, rather than the symbolic addition of angles. Based on the obvious rule B^{k}B^{k′} = B^{k+k′}, this works for any integers k and k′ without restriction — as long as we keep in mind that the circle eventually starts over again, i.e., that R(4) = J^{4} = I. Equation (∗∗) looks even more convincing if written as J^{μ+μ′} = J^{μ}J^{μ′}.

Thus, our unit of measurement is the quadrant, i.e., the right angle, which in conventional terms equals 90 degrees or π/2 radians. To use our calculators, we must of course revert to one of these traditional scales. Conceptually, however, they are harder to handle, as they involve the awkward division of angles by 3 and 5 or the delicate notion of arclength.

3. So far, exact arc measures have been assigned only to the rotations B_{m}^{k}. Other angles must be approximated by these “binary” ones. The idea is to sneak up on a given R by a product of the form

P_{m} = B_{1}^{e_{1}}B_{2}^{e_{2}}B_{3}^{e_{3}} ⋯ B_{m}^{e_{m}}  with  P_{m} ≤ R < P_{m}B_{m} ,

where each of the exponents e_{1}, e_{2}, e_{3}, … is either 0 or 1. Every step of this process goes as follows: having successfully constructed P_{m−1} (where m can be as low as 1, with P_{0} = I), we put

P_{m} = \{ \begin{matrix}P_{m−1}B_{m}; & if P_{m−1}B_{m} ≤ R, \\ P_{m−1}; & if R < P_{m−1}B_{m}. \end{matrix}  (‡)

In the first case (e_{m} = 1) we have P_{m}B_{m} = P_{m−1}B_{m}^{2} = P_{m−1}B_{m−1}, which is greater than R by construction of P_{m−1}. In the second case (e_{m} = 0) we get R < P_{m}B_{m} by default.

It is easy and fun to write a computer program based on this process. For R = .6I +.8J and m = 21, it yields the exponent string e_{1}e_{2}e_{3} … e_{21} = 100101110010000000101. This is the binary expansion of k_{m}/2^{m} = .5903344 — which multiplied by 90 (to go from quadrants to degrees) amounts to 53.1301^{∘} — a very nice approximation to the arcsine of .8.

It is fairly clear that P_{m} = B_{m}^{k_{m}} = R(k_{m}/2^{m}), where k_{m} = 2^{m−1}e_{1} + ⋯ + 2e_{m−1} + e_{m}. Let us note incidentally that the coordinates of any B_{m}^{k} (for 0 < k < 2^{m}) are irrational. Indeed, if they were rational, so would be those of the 2^{m}-th power J^{k}. Since k may be assumed odd, say k = 2κ + 1, this would imply the rationality of (−1)^{κ}J^{1/2} and hence of \sqrt{2}.

4. The proviso P_{m} ≤ R < P_{m}B_{m} will ensure that P_{m} approaches R — because B_{m} converges to I — as m grows indefinitely. Visually this may seem obvious, but ever since the Ancient Greeks found invisible cracks in their number line, purely optical evidence has been inadmissible in mathematical courts. Let us take time out to settle this matter.

For B_{m} = u_{m}I + v_{m}J and all m, we shall prove u_{m}^{2} ≥ 1 − 1/2^{m} (and hence v_{m}^{2} ≤ 1/2^{m}). As this is obvious for m = 0 and m = 1, suppose it established up to some m. The trick is to remember that w ≥ w^{2} for any w between 0 and 1 — which now gives u_{m} ≥ 1 − 1/2^{m}. Adding 1 and dividing by 2, Eq.(†) takes us to the next rung of the ladder, namely u_{m+1}^{2} = (1 + u_{m})/2 ≥ 1 − 1/2^{m+1}, and so on and on. Choosing m large enough, we can clearly make 1 − u_{m} ≤ 2^{−m} and v_{m} ≤ 2^{−m/2} as small as we wish.

If continued indefinitely, the recursive process shown in Eq.(‡) yields an infinite binary expansion — i.e., a real number — for the exact arc measure of R. Conversely, any infinite string {e_{m}} of zeroes and ones will define a sequence {P_{m}} of rotations as above, which converges to something because {sin(P_{m})} forms a non-decreasing sequence of real numbers < 1. Thus any real number μ between 0 and 1 leads — via its binary expansion — to a rotation R(μ) which may be thought of as an infinite product

R(μ) = B_{1}^{e_{1}}B_{2}^{e_{2}} ⋯ B_{m}^{e_{m}} ⋯ = J^{μ}

The addition law (∗∗) remains intact, because it holds for every stage of the approximation, and matrix multiplication is compatible with limits. It also shows that R(μ) varies continuously with μ, because R(μ′) gets arbitrarily close to I for small enough μ′.

All in all we have constructed a continuous homomorphism from the additive group of real numbers onto the group of 2 × 2 (real) rotation matrices. It is essentially unique: if R̃ is another candidate, it easily follows that R̃(μ) = R̃(1)^{μ} = R(s)^{μ} = R(sμ) for suitable s. Hence R is unique up to rescaling of its independent variable.