Curvature: Why a Flat Map of the Earth Must Lie
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Curves, arc length, tangents
A parameterised curve is a smooth map $\gamma:[a,b]\to\mathbb{R}^n$. Its velocity $\gamma’(t)$ is the tangent vector, its speed is $\lVert\gamma’(t)\rVert$, and arc length is the integral of speed:
Parameterisation is a choice, arc length is not — traverse the same track twice as fast and $\gamma’$ doubles while $s$ is unchanged. Reparameterising by arc length gives unit speed, $\lVert\gamma’(s)\rVert = 1$, and then the tangent $T = \gamma’$ only rotates, never lengthens. Curvature is exactly the rate of that rotation:
In the usual non-unit-speed parameterisation $\gamma(t) = (x(t), y(t))$ this becomes $\kappa = \lvert x’y’’ - y’x’‘\rvert/(x’^2+y’^2)^{3/2}$, and for a graph $y = f(x)$,
Check it on a circle of radius $r$: with $\gamma(t) = (r\cos t, r\sin t)$ the numerator is $r^2$ and the denominator is $(r^2)^{3/2} = r^3$, giving $\kappa = 1/r$. Small circles curve hard. The reciprocal $1/\kappa$ is the radius of curvature, and the circle of that radius tangent to the curve is the osculating circle.
Surfaces and the first fundamental form
Parameterise a surface as $\mathbf{r}(u,v)$. The tangent plane at a point is spanned by \(\mathbf{r}_u\) and \(\mathbf{r}_v\), and all intrinsic measurement — lengths of curves drawn on the surface, angles between them, areas — is governed by three functions:
This is the first fundamental form: the inner product of the ambient space, restricted to the tangent plane and written in the $(u,v)$ coordinates. It is precisely what the surface-dwelling ant can measure with a ruler and a protractor, without ever leaving the surface.
The second fundamental form, by contrast, involves the unit normal and records how the surface bends away from its tangent plane — information about the embedding, invisible from inside. Its eigenvalues are the principal curvatures $\kappa_1, \kappa_2$: the maximum and minimum curvature over all normal slices through the point.
Gaussian versus mean curvature
| Surface | $\kappa_1$ | $\kappa_2$ | $K$ | $H$ |
|---|---|---|---|---|
| Plane | $0$ | $0$ | $0$ | $0$ |
| Cylinder, radius $R$ | $1/R$ | $0$ | $0$ | $1/(2R)$ |
| Sphere, radius $R$ | $1/R$ | $1/R$ | $1/R^2$ | $1/R$ |
| Saddle | $>0$ | $<0$ | $<0$ | depends |
The cylinder is the row that carries the argument. It obviously looks curved, and $H \neq 0$ confirms that — but $K = 0$, the same as a flat plane. And indeed you can roll a flat sheet of paper into a cylinder without stretching or tearing it. Bending is free; stretching is not.
Theorema Egregium, stated plainly
Gauss’s Theorema Egregium (1827). The Gaussian curvature $K$ of a surface is determined entirely by the first fundamental form — by $E$, $F$, $G$ and their first and second derivatives. Consequently $K$ is preserved by any local isometry: if two surfaces are locally isometric, corresponding points have equal Gaussian curvature.
“Remarkable” was Gauss’s own word, and the surprise is genuine: $K$ was defined as a product of two extrinsic quantities, yet it turns out to be measurable from inside. The ant can determine $K$ without knowing the surface is embedded in anything at all — for instance by comparing the circumference of a small geodesic circle of radius $\rho$ against $2\pi\rho$, which comes up short on a sphere and long on a saddle.
Recap
- Arc length \(s = \int\lVert\gamma'\rVert\) is parameterisation-independent; curvature of a unit-speed curve is \(\kappa = \lVert T'\rVert\), and for \(y=f(x)\), \(\kappa = \lvert f''\rvert/(1+f'^2)^{3/2}\). A circle of radius \(r\) has \(\kappa = 1/r\).
- The first fundamental form \(ds^2 = E\,du^2 + 2F\,du\,dv + G\,dv^2\) captures everything measurable from inside the surface.
- \(K = \kappa_1\kappa_2\) is intrinsic; \(H = (\kappa_1+\kappa_2)/2\) is extrinsic. Cylinder: \(K = 0\), \(H = 1/(2R)\).
- Theorema Egregium: \(K\) is computable from the first fundamental form alone, hence invariant under local isometry. Sphere \(K = 1/R^2 \neq 0 = K_{\text{plane}}\), so a distance-preserving flat map cannot exist.
Curvature was defined here for surfaces sitting in $\mathbb{R}^3$. Manifolds and tangent spaces drops the ambient space, and Riemannian geometry rebuilds all of this from the metric alone.
References
- do Carmo, M. P. Differential Geometry of Curves and Surfaces, 2nd ed. Dover, 2016.
- Pressley, A. Elementary Differential Geometry, 2nd ed. Springer, 2010.
- Gauss, C. F. Disquisitiones generales circa superficies curvas. Göttingen, 1828 (English translation: General Investigations of Curved Surfaces, Princeton, 1902).