Normal and Gaussian curvature
General · Mathematics
Study notes
Q: Compute Gaussian curvature of sphere radius R and saddle z = x²-y² at origin. Sphere: L = -Rsin²φ (outward)... use inward: LN-M² = R²sin²φ, EG-F² = R⁴sin²φ. K = R²sin²φ/(R⁴sin²φ) = 1/R²! Saddle: X = (u,v,u²-v²). X_uu = (0,0,2), X_vv = (0,0,-2), n = (0,0,1) at origin. L = 2, N = -2, M = 0; E = G = 1, F = 0. K = (2·(-2)-0)/1 = -4 < 0! (Sphere: positive; saddle: negative!)