 Solution

ID: past-exam-of-the-geometry-course-of-the-university-of-oxford/2008/a/2/i/solution

For a regular parametrization choose the unit normal

n = (r_{u} \times r_{v}) /∣ r_{u} \times r_{v} ∣

I = E d u^{2} + 2 F d u d v + G d v^{2}, E = r_{u} \cdot r_{u}, F = r_{u} \cdot r_{v}, G = r_{v} \cdot r_{v} .

II = L d u^{2} + 2 M d u d v + N d v^{2}, L = n \cdot r_{uu}, M = n \cdot r_{uv}, N = n \cdot r_{vv} .

K = \frac{L N - M ^{2}}{EG - F ^{2}} .

Regularity ensures

EG - F^{2} > 0

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