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).