Try
A counterclockwise circle turns its tangent once during one traversal. Traversing it twice gives total signed curvature 4π while tracing the same geometric set. A self-intersecting regular loop can have a different turning number again. Global curvature remembers traversal, not just the outline of a drawing.
For a regular closed C² plane curve, use arc length on [0,L] with matching endpoint tangents. A continuous tangent-angle lift satisfies θ(L)−θ(0)=2πm for an integer m, the turning number. Integrating signed curvature gives the same difference.
Predict
Predict whether ∫k ds always equals 2π. Only under additional hypotheses, such as a positively oriented simple closed curve and the turning theorem, can that value be concluded.
Predict whether signed area and unsigned enclosed area are always identical. For a simple oriented curve their signs can be controlled; for self-intersecting curves the line integral records signed winding-weighted area. State which object an inequality concerns.
Experiment
Compare exact examples.
| Curve and traversal | Turning number | Total signed curvature |
|---|
| Circle once counterclockwise | 1 | 2π |
| Same circle clockwise | −1 | −2π |
| Circle twice counterclockwise | 2 | 4π |
For an ellipse (a cos t,b sin t), a>b>0, curvature is ab/(a²sin²t+b²cos²t)^(3/2). Differentiate its denominator to locate four stationary curvature values at the axis endpoints. This verifies one example of the four-vertex phenomenon; it does not prove the general four-vertex theorem.
Understand
The turning-number identity follows from k=θ′ and endpoint equality of the unit tangent. The lift is taken on an interval; it need not be periodic as a real-valued angle. Total absolute curvature satisfies ∫|k|ds≥|∫k ds|. For a simple positively oriented curve, the topological turning theorem supplies m=1 and therefore the lower bound 2π. The polygon-to-smooth proof below supplies that result, with polygon triangulation stated as its planar-topology input.
A useful global area estimate can be proved with the periodic Wirtinger inequality: a mean-zero scalar function h of period L satisfies ∫h²≤(L/2π)²∫h′². Translate a unit-speed closed curve so both coordinate means vanish. Apply Wirtinger to each coordinate to get ∫||γ||²≤L³/(4π²).
Green’s signed-area formula and Cauchy–Schwarz now give
2|A|=|∫det(γ,γ′)ds|≤(∫||γ||²)^(1/2)(∫||γ′||²)^(1/2)≤L²/(2π).
Thus 4π|A|≤L². For a simple curve this is the isoperimetric bound on enclosed area. The analytic Wirtinger inequality and Green theorem are the stated inputs; the deduction is complete.
The missing simple-curve turning argument
A loop’s arbitrary integer turning number was proved in the previous lesson. We now explain why simplicity forces the value+1 or−1, without using Gauss–Bonnet (whose local proof itself uses this turning result).
First take a positively oriented simple polygon. The finite polygon triangulation theorem says that a polygon with N vertices, including a nonconvex one, can be partitioned by noncrossing diagonals into N−2 triangles. This is the explicit elementary plane-topology input here, not a claim that every polygon is convex. The triangles’ angle sums add to (N−2)π, equal to the sum of all polygon interior angles because the diagonals only split those angles. At a vertex of interior angle α the signed exterior turn is π−α; it is negative at a reflex vertex. Thus the sum of signed exterior turns is Nπ−(N−2)π=2π. Reversing traversal negates it. Omitting negative turns at reflex vertices is precisely the erroneous argument that counts only convex corners.
Now let γ be a C² regular simple closed curve, parametrized by arc length on a circle of length L. The following approximation argument transfers the polygon result and lists the needed control. By continuity of its unit tangent T, choose short parameter arcs on which all tangents make angle less than π/8 with a fixed direction for that arc. The projection of γ onto that direction is strictly increasing along the arc, so the arc is a graph. Compactness gives a finite family of such graph neighborhoods. Simplicity and compactness also separate nonadjacent arcs: once a fixed positive parameter separation is imposed, the minimum Euclidean distance between such pairs is positive, since γ(s)=γ(t) occurs only for equal circle parameters.
Choose an inscribed polygon with sufficiently small mesh, refining that finite family. Its edges are integrals of T, so each nonzero chord direction stays in the same angle cone as the tangents along its short arc. The polygon is simple: edges from separated parameter arcs stay in disjoint small neighborhoods by the positive separation bound; edges in a common graph neighborhood have ordered, disjoint projection intervals except for adjacent endpoints and cannot cross. This combines a local monotonicity statement with a global separation statement; fine sampling alone would not prove simplicity without them.
Replace the piecewise-constant direction field of the polygon by rotating through the shorter signed angle between consecutive edge directions at each vertex. All these rotations have size less than π/2 for a sufficiently fine polygon. The resulting continuous circle-valued direction map has degree equal to the polygon’s total exterior turn divided by 2π. On each short transition interval it stays within an open semicircle of the smooth tangent map T, as do the edge directions on the other intervals. Their normalized straight interpolation is consequently a continuous homotopy with no zero denominator: two vectors inside this angle bound cannot be negatives of each other. Homotopy preserves degree. One can justify that preservation with interval angle lifts: the endpoint angle difference divided by 2π is an integer depending continuously on the homotopy parameter, so it is constant.
The polygon has degree+1 for the orientation keeping its interior on the left, and the smooth curve has the same orientation under this close graph approximation. Therefore γ has turning number+1. The reverse orientation has−1. It follows that ∫k ds=±2π and ∫|k|ds≥2π for a simple regular closed C² curve. The interior/orientation and polygon triangulation facts are elementary planar-topology dependencies; neither is being derived from a curvature plot. The proof does not apply to a double traversal, which is not injective as a map from the parameter circle.
A concave-polygon transfer before smoothing
Consider the counterclockwise polygon (0,0),(3,0),(3,3),(2,1),(0,3). Draw it, verify simplicity, identify the reflex corner and calculate its exterior turns with atan 2(det(u,v),u·v), where u,v are successive nonzero edge directions. Then explain why smoothing the corners in sufficiently small disjoint neighborhoods preserves the total turning number, even though pointwise curvature changes.
Feedback and review criteria
The edge vectors are (3,0),(0,3),(−1,−2),(−2,2),(0,−3). Directly the turns from edge 1 to 2,2 to 3,3 to 4,4 to 5,5 to 1 are approximately 90°,153.435°,−108.435°,135°,90°, totaling 360°. The negative turn is at (2,1). Small corner roundings replace each jump by a continuous tangent rotation of exactly the same signed angle. They keep nonadjacent pieces disjoint when chosen within the separated neighborhoods, so the resulting simple curve retains degree 1. A reviewer should check orientation, the negative turn, sum and nonintersection assumptions; matching only the final 360° is insufficient.The polygon argument uses the finite triangulation theorem; its role is exposed rather than hidden as “obvious from a picture.” For a source comparison, Shifrin’s Differential Geometry notes, Chapter 1, inspected 30 October 2026, discuss the turning-angle/global-curve framework. The argument above makes the noncircular polygon-to-smooth bridge explicit.
Apply
Repair “A finite curvature plot proves the total-curvature theorem.” Sampling can miss sharp turning or sign changes. The proof uses an integral and a global winding argument, with regularity and closure assumptions that numerical data alone do not certify.
For optional depth, the general four-vertex theorem and the classification of regular plane loops by turning number require additional global arguments. The ellipse calculation is practice in differentiating curvature, while the turning and area deductions illustrate the distinct roles of topology and analysis. We do not label all these global theorems proved by one example.
Continue
Keep local curvature, turning number, absolute curvature and signed area in separate columns of a report. Continue to surface charts and regular values to move from one-dimensional traces to two-dimensional geometric objects.
