Gauss's Theorema Egregium

Pick a surface and a distance r. An ant walks r along every straight line from one point, and the page measures the ring of end points against the 2πr it would be on paper, then works out the surface's curvature from the difference.

Worked example — Gauss's Theorema Egregium

What it would be on a flat sheet, 2πr	6.2832
The same ring from its formula	6.2832
C / 2πr	1.00000
Shortfall, 2πr − C	0.0000
Radius of a flat circle with the same rim	1.0000
Curvature worked out from the ring alone	0.000 / R²
The two principal curvatures at the centre	1/R and 0
Gauss's curvature K at the centre	0 / R²

These are the values the calculator shows when the page opens. Change any input above to recompute them.

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Lesson

The theory — Gauss's Theorema Egregium

Gaussian curvature K at a point of a surface is the product of its two principal curvatures there: the most and the least the surface bends in any direction through the point, each with a sign for the side it bends towards. Gauss's Theorema Egregium (1827) says that K depends only on lengths measured within the surface, so bending a surface without stretching it leaves K unchanged. One measurement that finds it from inside: the ring of points at distance r from the point has length C = 2πr − (π/3)K r³ + …, so K is the limit of 3(2πr − C)/(πr³) as r shrinks.

What each symbol means

K: Gauss's curvature at the centre, the product κ₁κ₂ of the two principal curvatures; 1/R² on a sphere of radius R
C: the length of the ring of points at distance r from the centre, measured along the surface
r: the distance walked from the centre along the surface

A cylinder is as flat as this page, and an ant can prove it 🖖

Press Cylinder, r = R, then Sphere, r = R. Both surfaces bend by 1/R across the top, and from outside both are plainly round. Walking 1 from the centre in every direction, the ant on the cylinder marks a ring 6.2832 long, exactly what it would mark on a sheet of paper; on the sphere the ring is 5.2871. Along its length the cylinder does not bend at all. Its two principal curvatures are 1/R and 0, and Gauss's curvature K is their product, 0. Gauss proved in 1827 that K can be computed from lengths measured within the surface, so bending a surface without stretching it cannot change K: the result known ever since as the theorema egregium, the remarkable theorem. Rolling paper into a tube stretches nothing. The tube keeps K = 0, and no ant measuring a patch of it can tell that it was ever rolled. It is also why a slice of pizza folded across its crust stops drooping: curved one way, it has to stay straight along its length.

No flat map of the Earth keeps its distances, and the ring says by how much 🖖

Press The Earth, 1,000 km. Every point of the orange ring is 1,000 km from the centre, measured over the ground, and the ring is 6,257.42 km round. A flat circle of radius 1,000 km is 6,283.19 km round. So a map that keeps every distance from the centre true has to draw the rim 25.77 km too long, and a map that keeps the rim true and round has to draw the radii 995.90 km, 4.10 km short. One distance or the other is wrong, on every flat map, and the theorem says why no cleverer projection escapes: the curvature that shortens the ring cannot be removed without stretching. Press The Earth, 2,000 km. Twice the radius, and the rim falls 205.4 km short, 7.97 times as much. The shortfall grows as the cube of the radius, which is why a town plan can ignore the Earth's curve and a map of a continent cannot. Map Projection Distortion Lab shows what each projection gives up instead.

On a saddle the ring comes out too long 🖖

Press Saddle, r = R/2. The ant walks 0.5 in every direction and the ring is 3.2573, longer than the 3.1416 of a flat circle. A sphere shortens the ring and a saddle lengthens it, because a saddle's straight lines spread apart faster than they would on paper, and a sphere's more slowly, until they meet again at the far pole. The sign of 2πr − C is the sign of K, so the ant learns which way its world is curved without seeing either bend. Gauss's K is the limit of 3(2πr − C)/(πr³) as the ring shrinks, which is the expansion of the ring's length that Bertrand and Puiseux published in 1848, solved for K. The table below shows the limit being approached. On the sphere, whose curvature is 1/R² everywhere, the estimate is 0.9876 at r = R/2 and 0.9995 at R/10. On the saddle it is −0.8838 and −0.9945: a ring of radius R/2 already reaches ground where this saddle is curved less than at its centre.

Problem solved in full

How big a circle before a tape measure feels the Earth's curve? 4 steps
A surveyor's tape is good to one part in a million, a millimetre in a kilometre. How large a circle must be paced out on the Earth before its rim comes out measurably short of 2πr? Press The Earth, 1,000 km first: the page gives C/2πr for a disc that size.

Open this problem in the tool
1. At 1,000 km the rim falls short by 1 − 0.99590 of 2πr. On a sphere C = 2πR sin(r/R), and sin x / x = 1 − x²/6 + …, so the fraction is (r/R)²/6 to leading order:
  
  $1 - 0.99590 = 0.00410 \approx \frac{1}{6} (\frac{1 , 000}{6 , 371 . 0088})^{2} = 0.00411$
2. Set that fraction to one part in a million and solve for r/R:
  
  $\frac{1}{6} (\frac{r}{R})^{2} = 1 0^{- 6} \Rightarrow \frac{r}{R} = 6 \times 1 0^{- 6} = 0.0024495$
3. and the radius on the Earth is that times R:
  
  $r = 0.0024495 \times 6, 371.0088 km = 15.606 km$
4. Check it. The flat circle's rim is 2π × 15.606 km, and a millionth of it is
  
  $2 π \times 15.606 km = 98.055 km, 98.055 km \times 1 0^{- 6} = 9.8 cm$
Answer

About 15.6 km in radius, a circle 31 km across. Its rim of 98 km comes out 9.8 cm short of 2πr, which is where the tape starts to notice. The shortfall goes as the cube of the radius, so a circle a tenth that size is short by a thousandth as much, a tenth of a millimetre, and a survey of a town can treat the ground as a flat sheet.

$r \approx 15.6 km, 2 π r - C \approx 9.8 cm$

Learning path

Geometry on a curved surface

References (3)

The measure of curvature as one over the product of the two extreme radii of curvature (Art. 8), and the theorem that it stays unchanged when a surface is developed upon another, with surfaces developable on a plane having curvature zero everywhere (Art. 12): C. F. Gauss, General Investigations of Curved Surfaces of 1827 and 1825, translated by J. C. Morehead and A. M. Hiltebeitel, Princeton University Library, 1902, Arts. 8 and 12, pp. 15 and 20–21 (Project Gutenberg ebook 36856).
The length of the ring traced by a short thread of length s turned about a point on a surface, 2πs − πs³/(3R₁R₂) to fourth order, from which the product of the principal radii, and so Gauss's curvature, is recovered by a measurement made on the surface: J. Bertrand, "Démonstration d'un théorème de M. Gauss", Journal de mathématiques pures et appliquées, 1re série, 13 (1848), 80–82; and V. Puiseux, "Sur le même théorème", same volume, 87–90.
The folded pizza slice as the theorem in the kitchen: a surface with zero Gaussian curvature bent one way must stay flat the other way, so curling the crust stiffens the slice along its length: A. Bhatia, "How a 19th Century Math Genius Taught Us the Best Way to Hold a Pizza Slice", WIRED, 5 September 2014.

By the 42Math team · Reviewed against published sources. Corrections welcome.