Gaussian Curvature - Alternative Formulas

Alternative Formulas

• Gaussian curvature of a surface in R3 can be expressed as the ratio of the determinants of the second and first fundamental forms:

• The Brioschi formula gives Gaussian curvature solely in terms of the first fundamental form:

• For an orthogonal parametrization (i.e., F = 0), Gaussian curvature is:

• For a surface described as graph of a function z = F(x, y), Gaussian curvature is:

• For a surface F(x,y,z) = 0, Gaussian curvature is:

• For a surface with metric conformal to the Euclidean one, so F = 0 and E = G = eσ, the Gauss curvature is given by (Δ being the usual Laplace operator):

• Gaussian curvature is the limiting difference between the circumference of a geodesic circle and a circle in the plane:

• Gaussian curvature is the limiting difference between the area of a geodesic disk and a disk in the plane:

• Gaussian curvature may be expressed with the Christoffel symbols: