Reference / Paper · 1998
Geometry of the Prytz Planimeter (Foote)
The mathematics of the hatchet (Prytz) planimeter.
Reference / Paper · 1998
The mathematics of the hatchet (Prytz) planimeter.
arXiv:math/9808070v1 [math.DG] 16 Aug 1998 GEOMETRY OF THE PRYTZ PLANIMETER Robert L. Foote Wabash College Abstract. The Prytz planimeter is a simple example of a system governed by a nonholonomic constraint. It is unique among planimeters in that it measures something more subtle than area, combining the area, centroid and other moments of the region being measured, with weights depending on the length of the planimeter. As a tool for measuring area, it is most accurate for regions that are small relative to its length. The configuration space of the planimeter is a non-principal circle bundle acted on by SU(1, 1) (≈ SL(2, R)). The motion of the planimeter is realized as parallel translation for a connection on this bundle and for a connection on a principal SU(1, 1)-bundle. The holonomy group is SU(1, 1). As a consequence, the planimeter is an example of a system with a phase shift on the circle that is not a simple rotation. There is a qualitative difference in the holonomy when tracing large regions as opposed to small ones. Generic elements of SU(1, 1) act on S 1 with two fixed points or with no fixed points. When tracing small regions, the holonomy acts without fixed points. Menzin’s conjecture states (roughly) that if a planimeter of length ℓ traces the boundary of a region with area A > πℓ2 , then it exhibits an asymptotic behavior and the holonomy acts with two fixed points, one attracting and one repelling. This is obvious if the region is a disk, and intuitively plausible if the region is convex and A ≫ πℓ2 . A proof of this conjecture is given for a special case, and the conjecture is shown to imply the isoperimetric inequality. A planimeter is a mechanical instrument used to determine the area of a region in the plane. The user moves a “tracing point” around the boundary of the region. When the tracing point returns to its starting point, some feature of the instrument does not quite return to its initial position. This can be interpreted as holonomy, and is related to the area of the region. Most planimeters consist of a rod, one end of which is the tracing point T . A wheel is attached to the rod that partially rolls and partially slides on the paper as the tracer point is moved. The most familiar such instrument is the polar planimeter (Figure 1(a)), 1991 Mathematics Subject Classification. 70F25, 53B15, 53A99, 53C65. Key words and phrases. planimeter, non-holonomic, SL(2, R), parallel translation, principal bundle, holonomy, phase shift, isoperimetric inequality. The author would like to acknowledge helpful discussions on this topic with Felix Albrecht, Larry Bates, Richard Bishop, Lance Drager, and Jeff Lee. To appear in Reports on Mathematical Physics. Typeset by AMS-TEX 1 invented by Jacob Amsler in 1854, in which the end of the rod opposite the tracer point is hinged to a secondary rod, restricting its motion to a circle. In the linear planimeter (Figure 1(b)) the other end of the tracing rod is restricted to move along a linear track. The “roll” of the wheel is recorded on a scale, which reads out the area when the tracer point returns to its starting point. These work because the wheel mechanically integrates a 1-form that differs from 12 (−y dx + x dy) by an exact form (see [Fo1] for details and other references). Figure 1(a). Polar Planimeter. Figure 1(b). Linear Planimeter. The Prytz planimeter (Figure 2), in contrast, contains no internally moving parts. It consists simply of a rod with its ends bent at right angles. One end, the tracer point T , is sharpened to a point. The other end, C, is sharpened to a chisel edge parallel to the rod. The chisel edge is usually slightly rounded, making it look similar to a hatchet, and consequently the device is also known as a “hatchet planimeter.” It was invented in about 1875 by Holger Prytz, a Danish cavalry officer and mathematician, as an economical and simple alternative to Amsler’s planimeter. Prytz referred to it as a “stang planimeter,” “stang” being Danish for “rod.” An amusing account of the history of the Prytz planimeter is given by Pedersen [Pe]. For a very complete history of planimeters through 1894, see Henrici [He]. Figure 2. Prytz planimeter. Figure 3. Standard tractrix. 2 To use the planimeter, grasp it at the end with the tracer point, keeping its ends perpendicular to the plane. Move the tracer point along some curve taking care not to apply any torque to it. As T moves along the curve, C follows a pursuit curve that is always tangent to the rod. The paths of T and C are similar to those followed by the front and rear wheels of a bicycle. Note that if T is moved along a straight line, the path of C is a standard tractrix (Figure 3). Thus when T moves along an arbitrary curve γ, the path of C is called a tractrix of γ. Tractrices of various curves, particularly of circles, are considered in [L,Mo,Pou]. To measure the area of a region Ω, start T at some base point B on ∂Ω, and note the initial position of C. As T moves around ∂Ω, C describes a zig-zag path (Figure 4), which Richard Bishop has likened to the motion of parking a car. When T gets back to B, the chisel edge C does not return to its starting point—it has undergone a displacement. Multiply the displacement σ of C by the length ℓ of the rod and you have the area. Actually it’s only an approximation, but the error turns out to have a very nice geometric description. We have implicitly assumed here that ∂Ω consists of a simple, closed curve, so Ω is bounded and simply connected. To precisely describe the behavior of the planimeter, we will also need to assume that ∂Ω is piecewise C 1 . These assumptions about Ω will be made for the rest of the paper without further comment. Figure 4. Motion of the Prytz planimeter. Despite its simplicity and inherent inaccuracy as a measuring tool—in fact, because of its inherent inaccuracy—the Prytz planimeter is more interesting mathematically than the polar planimeter. After giving the elementary theory of the Prytz planimeter, we will discuss Menzin’s conjecture on how the planimeter behaves when tracing large regions and its connection to the isoperimetric inequality, the analytic approach of the inventor and one of his contemporaries, the behavior of the planimeter as a non-holonomic system—most specifically 3 as parallel translation for a connection on a principal SU (1, 1)-bundle and for the induced connection on an associated circle bundle—and finally, a proof of a special case of Menzin’s conjecture. 1. Elementary Theory Let p and q be distinct points in R2 , and consider the segment joining them. Let N be the unit vector perpendicular to the segment so that q −p and N form a positively-oriented frame. Figure 5. Figure 6. If the segment moves slightly, an infinitesimal oriented area is swept out (Figure 5). If m = 12 (p + q) is the midpoint of the segment, then this area is given by dA = ℓN ·dm = 1 2 ℓN ·(dp + dq), where ℓ is the length of the segment (which can be variable), and dp, dq, and dm are the infinitesimal displacements of p, q, and m, respectively. Note that N points in the direction of motion for which the area is counted positively. Let dσ = N ·dp be the component of dp in the direction of N . Noting that the infinitesimal rotation of the segment is dθ = 1ℓ N ·(dq − dp), where θ is the angle of inclination of the segment relative to some fixed direction, the expression for the infinitesimal area can be written as dA = ℓ dσ + 12 ℓ2 dθ. These expressions for dA are intuitively plausible. If dp = dq, then dA = ℓN ·dm = 1 2 ℓN ·(dp + dq) is just the area of a parallelogram. If dp and dq are parallel to the segment, they are orthogonal to N and no area is swept out. If the segment rotates about its midpoint, then dm = 0 and the oriented areas swept out by the two halves of the segment cancel. These motions account for all four dimensions of the configuration space of the moving segment, and so any infinitesimal motion of the segment is a linear combination of these. A similar intuitive argument can be made for dA = ℓ dσ + 12 ℓ2 dθ. More rigorously, the expression dA = ℓN ·dm is the integrand in the two-dimensional version of Guldin’s formula for the area swept out by a moving segment [Cou,Fo1]. The formula for infinitesimal rotation (which will be needed later) is easily obtained by differentiating N ·(p − q) = 0 and noting that dN/dθ is the unit vector u = 1ℓ (p − q). We get 0 = u dθ·(p − q) + N ·(dp − dq) = ℓ dθ + N ·(dp − dq). If the endpoints of the moving segment trace out curves, the expressions for dA can be integrated, yielding the total oriented area swept out. In particular, if the endpoints trace out closed curves (Figure 6), it is easy to show [Fo1] that the oriented area swept out is 4 Aq − Ap , where Ap and Aq are the oriented areas enclosed by curves traced out by p and q, respectively. The moving segment becomes a Prytz planimeter if we fix its length ℓ and use the endpoint q as the tracer point T . The behavior of the planimeter is then given simply by the non-holonomic restriction dσ = 0, that is, the infinitesimal movement of the chisel point C has no component in the direction of N . As a consequence, dA = 21 ℓ2 dθ and the total oriented area swept out by the planimeter at any moment is A = 12 ℓ2 ∆θ, where ∆θ is the net rotation of the planimeter. This gives a simple proof that the area under the standard tractrix (Figure 3) is 41 πℓ2 . Now consider what happens when the tracer point is moved around the boundary of a region Ω in the positive direction, starting and ending at some base point B ∈ ∂Ω (Figure 4). The chisel point C does not return to its original position. For regions that aren’t too large relative to ℓ (and for which the planimeter gives the most accurate results, as we will see), the chisel point comes to rest at a point close to its original position. Now imagine the user fixing the tracer point T at B and rotating the planimeter about B (this motion violates the non-holonomic constraint) so that C moves along a circle back to its initial position (we will call this circle—with center B and radius ℓ—the “initial circle”). As both endpoints would now have traced out closed curves, the oriented area swept out is Aq − Ap = AΩ − Aγ , where AΩ is the area of Ω and Aγ is the oriented area enclosed by the curve γ formed by the zig-zag path of C during the normal use of the planimeter plus the arc of the initial circle from the final position of C back to its initial position. The oriented area Aq − Ap also equals the integral of dA = ℓ dσ + 21 ℓ2 dθ. As the initial and final angles of the planimeter are the same, dθ integrates to 0. Along the part of γ followed by the chisel we have dσ = 0, but along the part of γ that is the arc of the initial circle, dσ integrates to σ, the length of that arc. Thus we have AΩ − Aγ = ℓσ, or AΩ = ℓσ + Aγ , and we see that the error made by the approximation AΩ ≈ ℓσ is Aγ . In a typical use of the planimeter, the curve γ encloses a number of “triangular” regions. These regions are generally small compared to Ω. Moreover the boundaries of those regions inside the initial circle are traced with the opposite orientation of those outside, and so their oriented areas have opposite signs in their sum Aγ . Keeping this in mind, a good starting position is one that will result in the chisel edge spending part of its time inside the initial circle and part of its time outside. For example, one could start with the planimeter perpendicular to a line that roughly bisects the region. Another way to minimize the error, suggested by several authors [Ba,Cr,Hi,K,Pr14,Sa,St], is to draw a line segment from B to the centroid Bc of the region. Instead of starting the tracer point at B, start at Bc . Trace along the segment BBc to B, then 5 around the curve, and finally back along BBc , stopping at Bc , as suggested in Figure 2. This, of course, leads to the problem of locating the centroid, which is at least as hard as computing the area! In practice, one simply guesses. Prytz and Hill are the only authors who give any argument that the centroid should be used as the base point (outlined in Section 3). The other authors simply appeal to this as a geometrically plausible way to balance the triangular regions inside and outside the initial circle. Similar arguments can be made to explain how other planimeters work, and have appeared in a variety of forms dating from the early 1800’s (see [Fo1] for references). The argument given above to explain the behavior of the Prytz planimeter in particular is due to Henrici [He], who was the first to give a common theoretical setting for most of the planimeters invented up to 1894. This argument also appears in a paper by Kriloff [K], and may have been discovered independently by him. It is interesting that this simple geometric argument can be made without referring to the exact relationship between the movement of the tracer point and the angle θ between the planimeter rod and some fixed direction. This relationship is the focus of much of the rest of the paper. Recall the expression for the infinitesimal rotation of a moving line segment, ℓ dθ = N ·(dq − dp), which for the planimeter becomes ℓ dθ = N ·dq, since dσ = N ·dp = 0. If the tracer point T = q follows the curve γ : R → R2 , the differential ′ equation governing θ is then ℓ dθ dt = N ·γ (t). If θ measures the angle between the planimeter dy dx and the x-axis, i.e., p−q = ℓ(cos θ i+sin θ j), the equation becomes ℓ dθ dt = sin θ dt −cos θ dt . dθ = sin θ, If the tracer point moves along the x-axis with x = t, as in Figure 3, we get ℓ dx the solution of which is easily seen to be tan θ = Aex/ℓ . 2 (1) This can be rewritten as tan θ = − csch(a + x/ℓ), where a = log A. If the planimeter starts perpendicular to the x-axis when x = 0, then a = 0, and so tan θ = − csch(x/ℓ). This is the slope of the line tangent to the tractrix when that curve is parameterized by the point where the tangent meets the axis (see, e.g., [Cox]): τ (x) = x − ℓ tanh(x/ℓ), ℓ sech(x/ℓ) . (2) Variations of the Prytz planimeter have been made, the most notable ones being by Goodman and Scott. As we have seen, the length of the arc along the initial circle between the initial and final locations of the chisel edge is the important quantity to measure, and not simply the distance between these points, although the latter is usually adequate. The variations due to Goodman and Scott allow the direct measurement of this arc. Goodman [G] incorporated a curved scale with radius ℓ into the planimeter rod, so the scale lies along the initial circle when applied to the points (Figure 7). Scott’s variation [Sc] implements 6 the idea in the discussion above that brings the chisel end of the planimeter back to its initial position. He put a wheel next to the chisel edge with its axis parallel to the rod. This wheel does not contact the paper during the tracing of ∂Ω, but rides slightly above it. When the tracing point returns to the base point, the planimeter is tilted slightly, bringing the wheel into contact with the paper. The user then rotates the planimeter around the tracer point. The wheel rolls along the initial circle between the points, measuring the arclength. Both inventors, particularly Scott, seemed to believe that they were addressing the cause of the instrument’s error, but the proof above and the analysis by Prytz and Hill in Section 3 show that the error is more complicated (and more interesting) than this. Figure 7. Goodman’s planimeter. The planimeters of Prytz, Goodman, and Scott were all marketed (the original Prytz planimeter was produced by the firm of Knudsen in Copenhagen [Pe]). The additional details of the latter two, however, defeated Prytz’s original purpose of economy relative to Amsler’s planimeter. In reaction to Goodman and Scott, Prytz [Pr3] advised engineers “rather than use the ‘improved stang planimeters,’ let a country blacksmith make them a copy of the original instrument.” A collector has sent the author two photographs, one of a Prytz planimeter manufactured in the Netherlands and one of a Goodman planimeter, which are posted on the author’s web page (URL at the end of this paper). The author would be very interested to learn of other instruments of this type that still exist. 2. Menzin’s Conjecture and the Isoperimetric Inequality When the planimeter traces the boundary of a polygonal region Ω with edges that are large compared to ℓ, the length of the planimeter, it is intuitively clear that the path of C asymptotically approaches a particular tractrix that is closed, and that when the planimeter follows this closed tractrix, it makes a complete rotation. This also happens when Ω is a disk of radius R > ℓ; it is easily shown that the closed tractrix is a circle of √ radius R2 − ℓ2 . Furthermore, it is intuitively plausible for large convex regions. After much experimentation, Menzin [Me] conjectured that this happens whenever AΩ > πℓ2 . The last section contains a proof of Menzin’s conjecture when ∂Ω is a parallelogram. 7 In spite of its title (having to do with Amsler’s planimeter), the paper by Morley [Mo] goes into considerable detail on the tractrices of circles. In addition to the case where R > ℓ, he notes that when R = ℓ the asymptotic tractrix reduces to the center of the circle, but it is attractive only from one side. When R < ℓ each tractrix is made up of regularly spaced cusps. It turns out that Menzin’s conjecture implies the isoperimetric inequality. To see this, note that when C follows the closed tractrix, dA = ℓ dσ + 12 ℓ2 dθ integrates to πℓ2 , since dσ = 0 and the planimeter makes a complete rotation. Since C traces a closed path, the total area swept out is AΩ − AC , where AC is the area enclosed by the path of C (which does not include an arc of the initial circle). Furthermore, note that dA can be written as dA = ℓ dσT − 21 ℓ2 dθ, where dσT = N ·dq is the component of dq (the infinitesimal displacement of the tracer point) in the direction of N . This expression for dA integrates R to ℓσT − πℓ2 , where σT = dσT . (Note that if a wheel is mounted on the rod at T similar to the wheel on a polar or linear planimeter, then σT is the total signed distance the wheel rolls.) Thus we have πℓ2 = AΩ − AC = ℓσT − πℓ2 . By Menzin’s conjecture this holds under the assumption that AΩ > πℓ2 . In the limiting case where AΩ = πℓ2 , we have AC = 0. A little algebra yields σT2 = 4πAΩ . Noting that dσT measures only a component of the infinitesimal distance the tracer point moves (dσT ≤ ds), we have σT ≤ L∂Ω , where L∂Ω is the length of ∂Ω. The isoperimetric inequality follows: L2∂Ω ≥ σT2 = 4πAΩ . Furthermore, L2∂Ω = 4πAΩ implies dσT = ds, that is, the tracer point only moves in the direction perpendicular to the rod. In this case the chisel edge does not move at all, and the tracer point describes a circle of radius ℓ. Thus the isoperimetric inequality is an equality only if ∂Ω is a circle. For other connections between planimeters and isoperimetric inequalities, see [Fo1&2]. 3. Analysis by Prytz and Hill Prytz and Hill studied the “stang” planimeter analytically, rather than geometrically. Prytz’s original paper [Pr1] was published anonymously in Danish under the pseudonym ‘Z’ (see [Pe] for the story behind this). His subsequent short notes on this are very tersely written [Pr2&4]. Hill’s account is much more readable [Hi]. 8 Figure 8. Both authors use (1) to write an infinite series expansion for the rate at which θ changes when the tracer point moves along an arbitrary curve (Figure 8). Hill, who keeps more terms, gets 2 3 r4 r6 r5 r r 2 dθ ℓ = + 2+ +... + + + . . . cos(ϕ − θ). dϕ 2 8ℓ 144ℓ4 3ℓ 30ℓ3 Here (r, ϕ) are polar coordinates of the tracer point T about the base point B. Their analysis depends on the function r/ℓ being small on the path followed by the tracer point, which is to say, that Ω is small and that the base point B is not far from Ω, both relative to ℓ, the length of the planimeter. As the tracer point moves around ∂Ω, the dominant terms not depending on θ integrate to Z Z Z r4 IB 1 r2 r 2 dA = 2 , dϕ = A and dϕ = 2 2 2ℓ Ω 2ℓ ∂Ω 8ℓ ∂Ω 2 where A is the area of Ω and IB is the second moment (the moment of inertia) of Ω about B. Assuming the initial direction, θ = 0, of the planimeter is the positive x-axis, and that |θ| stays small (which follows from the assumption that r/ℓ is small), Hill writes the remaining most dominant term as Z Z 1 1 2 1 3 3 r cos(ϕ − θ) dϕ = r cos ϕ + θ sin ϕ − θ cos ϕ + . . . dϕ, 3ℓ ∂Ω 3ℓ ∂Ω 2 and observes that 1 3ℓ 1 r cos ϕ dϕ = ℓ ∂Ω Z 3 Z Ω x dA = Ax , ℓ where x is the x-coordinate of the centroid. Prytz and Hill thus have the approximation 2 IB x RB Ax 2 + 2 =A 1+ + 2 , ℓσ = ℓ ∆θ ≈ A + ℓ 2ℓ ℓ 2ℓ 9 2 where RB is the mean-square distance of points in Ω from B. Although they don’t say it explicitly, it is easily seen that the error in this approximation is O (d/ℓ)3 , where d is the diameter of the set Ω ∪ {B} (or simply the diameter of Ω if B ∈ Ω). Thus the error made by the approximation A ≈ ℓσ is O(d/ℓ). Note that when measuring small regions, ∆θ is small but non-zero, that is, the chisel edge does not come to rest in the place where it started. As we will see in the next section and at the end of the last section, this can happen if the planimeter traces the boundary of a large region or a curve that bounds an oriented area of 0. Prytz and Hill recommend measuring the region twice with the same base point but with opposite initial directions, and averaging the results. Prytz’s reasoning on this is not clear, but Hill notes that the x term will have opposite signs in the two measurements, and so will drop out in the average, giving the approximation 2 RB IB ℓσ ≈ A + 2 = A 1 + 2 , 2ℓ 2ℓ where σ is the average of the displacements in the two measurements. From this, the error in the approximation A ≈ ℓσ is O (d/ℓ)2 . Note that x also vanishes if the centroid is on the line through B perpendicular to the initial position of the planimeter, agreeing with the intuitive observation in Section 1 that a good starting position is one that is perpendicular to a line bisecting the region. 2 The term IB = ARB is minimized when B is the centroid, which accounts for the recommendation of Prytz and Hill that the tracing start and end at the centroid. From this analysis it appears that the holonomy ∆θ of the Prytz planimeter is some combination of all of the moments of Ω about the base point B. It is hoped that the modern approach taken in the next sections will lead to a better understanding of the relationship between these. 4. Motion of the Prytz Planimeter as Parallel Translation in a Fiber Bundle The configuration space for the Prytz planimeter is E = R2 × S 1 , where the first factor is the location of the tracer point and the second factor is the angle giving the direction of π the planimeter. We will consider E −→ R2 as a trivial circle bundle, where π is projection onto the first factor. If the tracer point follows a curve γ in R2 , the resulting motion of the planimeter defines a section of this bundle along γ. We will see that this motion is π described as parallel translation of an Ehresmann connection on E −→ R2 . Thus, when the planimeter traverses the boundary of some region, its net rotation ∆θ is an example of the holonomy of this connection. Our main source for connections and parallel translation in fiber bundles is [KMS], but also see [Poo]. Many of the results in this and the next section illustrate the general theory in [KMS]. 10 Intuitively, the motion of the tracer point along a curve induces a one-parameter family of diffeomorphisms of the fiber S 1 (in which the fibers over different points are identified by projection E → S 1 onto the second factor, i.e., by the triviality of the bundle). It is clear that these diffeomorphisms are not simply rotations. For example, if the tracer point is moved along a straight line, the diffeomorphisms have two common antipodal fixed points, one attracting and one repelling, the repelling fixed point being the fiber element in the direction of motion. As another example, consider the holonomy determined by tracing the boundary of a region that is small relative to the length of the planimeter. As seen in Sections 1 and 3, the resulting diffeomorphism of the initial circle (which is the fiber over the base point where the tracing begins and ends) is only approximately a rotation, π since the net rotation of the planimeter depends on its initial direction. Thus E −→ R2 should not be viewed as a principal bundle in this context, since the group G acting on the π fiber is not the group of rotations (more precisely, the connection on E −→ R2 , defined below, is not principal). In the next section we will determine the group G, and see that π E −→ R2 is, in fact, an associated bundle of a G-principal bundle, and that the connection π on E −→ R2 is induced by a connection on this principal bundle. Take (x, y, θ) as coordinates on E, where (x, y) are base coordinates and θ is the fiber coordinate. The tracer point and chisel edge then have coordinates q = (x, y) and p = (x + ℓ cos θ, y + ℓ sin θ), respectively, and the forward-pointing normal (from Section 1) is ∂ ∂ − cos θ ∂y . N = sin θ ∂x The expression dσ = N ·dp, defined in Section 1, is a 1-form on E. It is not exact, and henceforth will be denoted simply as σ. The coordinate expression for σ is σ = sin θ dx − cos θ dy − ℓ dθ, and so the motion of the planimeter is governed by the non-holonomic condition σ = 0, or ℓ dθ = sin θ dx − cos θ dy. (3) For e = (q, θ) ∈ E, let He = ker σe ⊂ Te E. The vector fields X= 1 ∂ ∂ + sin θ ∂x ℓ ∂θ and Y = ∂ 1 ∂ − cos θ ∂y ℓ ∂θ (4) form a basis for He at each e ∈ E, and π∗ |He : He → Tπ(e) R2 is clearly an isomorphism, where π∗ is the differential of π. It follows that H = {He | e ∈ E}, viewed as a sub-bundle of π T E → E, forms the distribution of horizontal subspaces of a connection on E −→ R2 (see ∂ [KMS, §9.3]). The connection form is the 1-form Φ = − 1ℓ σ ⊗ ∂θ , which takes values in the bundle V of vertical vectors, where Ve = ker(π∗ )e ⊂ Te E. More precisely, Φe : Te E → Ve , ∂ , is projection onto Ve with kernel He . If the tracer defined by Φe (W ) = − 1ℓ σ(W ) ∂θ e 11 point of the planimeter follows a piecewise smooth curve γ in R2 , the resulting motion of the planimeter defines a curve γ̃ in E covering γ such that γ̃ ′ (t) ∈ Hγ̃(t) for all t, by virtue of (3). Thus the motion of the planimeter is parallel translation for this connection. The connection is also determined by its Christoffel form Γ, which is a 1-form on R2 with values in X(S 1 ), the Lie algebra of smooth vector fields on S 1 . (See [KMS,§9.7]. There needs to be just one Christoffel form since the bundle is trivial.) Suppose the tracer ∂ ∂ + b ∂y . The induced point of the planimeter is at q ∈ R2 moving with velocity v = a ∂x ∂ 1 vector field w = Γ(v) = c(θ) ∂θ on S is given by applying (3): ℓ c(θ) = ℓ dθ(wθ ) = (sin θ dx − cos θ dy)(v) = a sin θ − b cos θ. Consequently, Γ(v)θ = wθ = 1 ∂ (a sin θ − b cos θ) . ℓ ∂θ (5) ∂ ∈ T(q,θ) E. Then w = Γ(v) is defined by More directly, given θ ∈ S 1 , let vθ = v + 0 ∂θ ∂ 1 Γ(v)θ = −Φ(vθ ) = ℓ σ(vθ ) ∂θ (an identification is being made here between the fiber Eq = ∂ . {q}×S 1 and the model fiber S 1 ). In either case we can write Γ = 1ℓ (sin θ dx−cos θ dy)⊗ ∂θ In general, a connection on a fiber bundle need not be complete, that is, parallel translation need not be defined for all elements of the bundle along all piecewise smooth curves. However if the fiber is compact, as in the present case, then the connection is necessarily complete, and so is properly called an Ehresmann connection. This follows from a comment in [KMS, §9.9], or by the following standard argument. Suppose γ : [0, 1] → R2 is a piecewise smooth curve. Let ǫ > 0 and extend γ to have domain I = (−2ǫ, 1 + 2ǫ), and consider the pull-back connection on the pull-back bundle γ ∗ E → I. Let f : I → R be a function with support in (−ǫ, 1 + ǫ) that is identically 1 on [0, 1]. Let T be the horizontal d to γ ∗ E. Then T has compact support (by the compactness lift of the vector field f dt of the fiber), and so is a complete vector field. As the flow of T over [0, 1] represents parallel translation along γ from γ(0) to γ(1), the connection is complete. Note that this implies the intuitive observation made at the beginning of this section that the motion of the tracer point along a curve induces a one-parameter family of diffeomorphisms of S 1 , or equivalently, a family of diffeomorphisms between the fibers of E over points along the curve. ∂ The vector fields X and Y , given in (4), are, respectively, the horizontal lifts of ∂x and 1 ∂ ∂ ∂y for this connection. The curvature is given by the Lie bracket R(X, Y ) = [X, Y ] = ℓ2 ∂θ ∂ (see [KMS, §9.4]). The vector fields X, Y , ∂θ form a frame on E. The dual frame is easily ∂ 1 . We see that seen to be dx, dy, − ℓ σ. In terms of this frame we have R = ℓ12 (dx ∧ dy) ⊗ ∂θ if the tracer point moves counterclockwise around an infinitesimal rectangle with edges ∂ 1 ∂ ∂ ∂x = π∗ (X) and ∂y = π∗ (Y ), then the infinitesimal motion of the fiber is ℓ2 ∂θ , which is also counterclockwise. It follows that the holonomy group (which will be determined in 12 the next section) must contain the rigid rotations, although from the results of Sections 1 and 3, it contains more than this. Although the connection defined here on E → R2 is not principal, it’s interesting to observe that the curvature can be computed using a “covariant exterior derivative” as on a principal S 1 -bundle [KMS, §11.5]. If α is a 1-form on E, then dh α is defined to be the horizontal component of dα, that is, dh α(X, Y ) = dα(P X, P Y ) for X, Y ∈ Te E, where ∂ P : Te E → Te E is projection onto He with kernel Ve . Let ω = − 1ℓ σ so that Φ = ω ⊗ ∂θ . 1 1 One computes that dω = − ℓ ω ∧(cos θ dx+sin θ dy)− ℓ2 dx∧dy. The horizontal component of this is Ω = dh ω = − ℓ12 dx ∧ dy. The curvature form Ω is related to the curvature tensor ∂ R in the usual way, namely, R = −Ω ⊗ ∂θ . π 5. E −→ R2 as an Associated Bundle π We will now see that the bundle E = R2 × S 1 −→ R2 is an associated bundle of a principal bundle, and that the connection on E is induced from a connection on this principal bundle. As a comparison, and to further Bishop’s analogy of the motion of the planimeter with that of a car, see [Fe]. Our main references for principal bundles are [KMS, KN]. As noted in the previous section, since the connection on E → R2 is complete, motion of the tracer point along a piecewise C 1 curve in R2 induces a one-parameter family of diffeomorphisms. The diffeomorphisms induced by all piecewise C 1 curves lie in some smallest subgroup of Diff(S 1 ), the group of all diffeomorphisms of S 1 . Our first task is to determine this subgroup. Theorem 1. The diffeomorphisms of S 1 induced by moving the tracer point of the planimeter along arbitrary piecewise C 1 curves in R2 form a group, namely, the group M0 (S 1 ) of Möbius transformations that preserve S 1 and its orientation. The proof consists of a number of steps. In the proof and the rest of the paper we will write elements of S 1 as eiθ and identify R2 with C when convenient. First we show that the collection of diffeomorphisms is a group. The only question is whether the collection is closed under composition. If γ : [0, 1] → R2 is a piecewise C 1 curve, let ψγ denote the diffeomorphism of S 1 induced by moving the tracer point along γ from γ(0) to γ(1). Since the connection on E → R2 is invariant under translations of R2 , then ψγ+v = ψγ for every v ∈ R2 . If γ1 , γ2 : [0, 1] → R2 are two such curves, then ψγ2 ◦ ψγ1 is the diffeomorphism induced by moving the tracer point along the curve γ defined by γ(t) = γ1 (2t) for t ∈ [0, 12 ] and γ(t) = γ2 (2t − 1) + γ1 (1) − γ2 (0) for t ∈ [ 21 , 1]. Thus the collection of diffeomorphisms is closed under composition, and hence is a group. Before continuing with the proof of Theorem 1, we note some facts related to M0 (S 1 ). (1) M0 (S 1 ) is the same as the group of Möbius transformations that preserve the unit disk D, which is, of course, the group of orientation-preserving isometries of the 13 Poincaré disk model of the hyperbolic plane. These Möbius transformations can be written as z 7→ az+b . This is a standard fact from elementary complex analysis, bz+a of course, but is done particularly nicely in [Se]. a b 2 2 (2) The “mixed signature” special unitary group SU (1, 1) = : |a| − |b| = 1 b a 1 is a double cover of M0 (S ), the homomorphism being the usual map taking . We will use this to identify M0 (S 1 ) A = ab ab to the transformation A·z = az+b bz+a with the projective group PSU (1, 1) = SU (1, 1)/{±I}, and consequently to iden iγ β : γ ∈ R, β ∈ C . This tify the Lie algebra of M0 (S 1 ) with su(1, 1) = β −iγ is an easy exercise, and is done implicitly in [Se]. It is completely analogous to the more common identification of SL(2, R) as a double cover of the Möbius transformations preserving the upper half-plane (see, e.g., [Bo]). SU (1, 1) and SL(2, R) are conjugate subgroups of SL(2, C), related by any matrix representing a Möbius transformation mapping D onto the upper half-plane. (3) By restricting to S 1 , M0 (S 1 ) injects into Diff(S 1 ). The differential of this is a Lie algebra injection ϕ : su(1, 1) → X(S 1 ). More precisely, let X ∈ su(1, 1). To compute the vector field ϕ(X) at z0 = eiθ0 ∈ S 1 , let eiθ(t) = etX ·z0 . Then ∂ ϕ(X)z0 = θ (0) = ∂θ z0 ′ d −iz 0 etX ·z0 dt 0 ∂ . ∂θ z0 (6) Next we determine how the Christoffel form Γ : T R2 → X(S 1 ) and the map ϕ : su(1, 1) → X(S 1 ) are related. Lemma 2. The Christoffel form Γ takes values of ϕ. In particular, if v = in the image 0 a+ib ∂ 1 ∂ a ∂x + b ∂y , then Γ(v) = ϕ(X), where X = − 2ℓ a−ib 0 . ∂ ∂ Proof. Suppose the tracer point is at q ∈ R2 moving with velocity v = a ∂x + b ∂y . From 1 (5), the induced vector field on S is Γ(v)z0 = ∂ 1 , (a sin θ0 − b cos θ0 ) ℓ ∂θ z0 (7) where z0 = eiθ0 . We need to identifyΓ(v) as ϕ(X) for some X ∈ su(1, 1). a(t) b(t) 0 β ∂ . Let X = β 0 , and let b(t) a(t) = etX . From (6) we get ϕ(X)z0 = 2 Im(βz 0 ) ∂θ z 0 ∂ If β = a + bi, then ϕ(X)z0 = 2(b cos θ0 − a sin θ0 ) ∂θ . Comparing this with (7), it follows z0 1 that Γ(v) = ϕ − 2ℓ X . Let γ : [0, 1] → R2 be a C 1 curve. We seek a curve in M0 (S 1 ) (actually its lift in SU (1, 1)) that generates the same one-parameter family of diffeomorphisms as γ. Suppose 14 A : [0, 1] → SU (1, 1) is a C 1 curve, and consider the resulting flow on S 1 . At t = t0 the vector field for the flow is given (in a computation similar to that in (6)) by d A(t)A(t0 )−1 ·z0 −iz 0 dt t0 ! ∂ = ϕ A′ (t0 )A(t0 )−1 z . 0 ∂θ z0 For A and γ to generate the same vector field on S 1 at time t, we must have Γ(γ ′ (t)) = ϕ(A′ (t)A(t)−1 ). Define the su(1, 1)-valued 1-form ω on R2 by ω(v) = −ϕ−1 (Γ(v)), that is, ω(v) is the matrix in su(1, 1) that gives rise to the vector field −Γ(v) on S 1 . From the 0 dz 1 lemma, we see that the formula for ω is ω = 2ℓ . Then the desired curve in dz 0 SU (1, 1) is the solution of the initial value problem A′ (t)A(t)−1 = −ω(γ ′ (t)), A(0) = I. (8) It follows that the group of diffeomorphisms of S 1 generated by the motion of thetracer 0 β 1 point is a subgroup of M0 (S ). By the lemma, the values of ω are of the form β 0 . These generate the entire Lie algebra su(1, 1), and so it follows that the group is M0 (S 1 ). This completes the proof of Theorem 1. From the proof of the theorem we see that the action of the one-parameter subgroup e on S 1 is the same as that of moving the tracer point in a straight line in R2 with constant velocity v. If the tracer point starts at q ∈ R2 with the planimeter in the direction eiθ0 , then the resulting curve in E = R2 × S 1 is (q + tv, e−tω(v) ·eiθ0 ). Identifying C and R2 , the path of the chisel, which is a standard tractrix, is τ (t) = (q + tv) + ℓe−tω(v) ·eiθ0 . ∂ (identified with 1 = 1 + 0i in q + tv), we have e−tω(v) = Letting q = 0 and v = ∂x −tω(v) cosh(t/2ℓ) − sinh(t/2ℓ) − sinh(t/2ℓ) cosh(t/2ℓ) and cosh(t/2ℓ)eiθ0 − sinh(t/2ℓ) − sinh(t/2ℓ)eiθ0 + cosh(t/2ℓ) cosh(t/ℓ) cos θ0 − sinh(t/ℓ) sin θ0 = t+ℓ + iℓ . cosh(t/ℓ) − sinh(t/ℓ) cos θ0 cosh(t/ℓ) − sinh(t/ℓ) cos θ0 τ (t) = t + When θ0 = π/2, this simplifies to τ (t) = t − ℓ tanh(t/ℓ) + iℓ sech(t/ℓ), which is (2) in complex form. Now consider S 1 as the points at infinity of the Poincaré disk model D of the hyperbolic plane. Acting on D, e−tω(v) is the one-parameter group of hyperbolic translations that v (note that this is in the opposite moves the origin along a geodesic with initial velocity − 2ℓ direction of the motion of the tracer point). This has the following nice interpretation. Suppose that a hyperbolic stargazer goes walking with constant velocity v in D. It is 15 natural for the stargazer to think of herself as always being at the center of the celestial circle, with the hyperbolic plane passing beneath her feet with constant velocity −v. If she fixes her gaze on a particular star, then the retrograde motion of that star is the same as the motion of a (Euclidean!) Prytz planimeter of length ℓ = 1/2. This discussion shows how a geodesic in R2 can be developed into a geodesic in D. Generalizing this, one can develop any polygonal path in R2 to a polygonal path in D (and by polygonal approximations, any piecewise C 1 path). One needs to be careful, however. A sequence eX1 , . . . , eXn of translations that moves the origin around a closed path in D determines a rotation (relative to the origin) of the points at infinity, whereas the tracer point of the planimeter following a closed loop typically does not result in a pure rotation of the initial directions of the planimeter. Evidently, the motion of the tracer point that induces the same sequence eX1 , . . . , eXn is not generally a closed path in R2 . Similarly, the motion of the tracer point in R2 around a closed path does not generally develop to a closed path in D. As a consequence of Theorem 1, we view E → R2 as an M0 (S 1 )-bundle or as an SU (1, 1)-bundle [KMS, §10.1]. The form ω in the proof is, of course, a “local frame representation” of a connection on some principal bundle [KN, pg. 66] (a “physicist’s connection” in [KMS, §11.4]), and (8) is the corresponding parallel translation equation. Writing down the principal bundle and realizing E → R2 as an associated bundle is now straight forward. For the remainder of this section, let G be M0 (S 1 ) or SU (1, 1). Consider the principal G-bundle P = R2 × G → R2 . As G acts on S 1 , we have the standard construction of the associated S 1 fiber bundle Ẽ → R2 (see [KN, pg. 54; KMS, §10.7]). The space Ẽ is given by (P × S 1 )/∼, in which (q, A, eiθ ) ∼ (q, I, A·eiθ ). The map E → Ẽ that takes (q, eiθ ) to the equivalence class of (q, I, eiθ ) is clearly a fiber bundle equivalence. Each element (q, A) in the fiber {q} × G of P → R2 represents a “frame” for the fiber {q} × S 1 of E → R2 , that is, a diffeomorphism from the model fiber S 1 to {q} × S 1 given by eiθ 7→ (q, A·eiθ ). As both bundles are trivial, it is easiest to make computations relative to the “standard frame,” that is, the “identity” diffeomorphism given by projection {q} × S 1 → S 1 , represented by (q, I) ∈ P . The standard frame is more than a computational convenience, however, since it represents the Euclidean geometry of the plane. The identification it makes of the fibers of E → R2 is by Euclidean translation. Writing the connections on E → R2 and P → R2 using the “local descriptions” of Γ and ω amounts to describing how the motion of the planimeter differs from Euclidean parallel translation of vectors. The connection form ω̃ : T P → su(1, 1) along the ident