Classification by Curvature

Differential Geometry · Axiom Academy

LESSON Classification by Curvature Understanding how Gaussian curvature determines local surface geometry When the Gaussian curvature is positive, both principal curvatures have the same sign. The surface curves in the same direction along both principal directions, creating a bowl-shaped or dome-like geometry. Sphere: K = 1/R² everywhere (perfectly elliptic) Ellipsoid: K > 0 at all points Paraboloid: K > 0 everywhere (bowl shape) When the Gaussian curvature is negative, the principal curvatures have opposite signs. The surface curves upward in one direction and downward in the perpendicular direction, creating a saddle-shaped geometry. Hyperboloid of one sheet: K < 0 everywhere Saddle surface z = x² - y²: Classic hyperbolic point at origin Pringles chip: Natural saddle shape with K < 0 3. Parabolic Points (K = 0, one κ = 0) When the Gaussian curvature is zero and exactly one principal curvature vanishes, the surface is cylindrical . It curves in one direction but remains flat in the perpendicular direction. Circular cylinder: κ₁ = 1/R, κ₂ = 0, so K = 0 Cone (away from apex): K = 0 along generatrices Developable surfaces: Can be unrolled flat without distortion When both principal curvatures vanish, the surface is locally planar . There is no curvature in any direction, and the surface behaves like a flat plane in the neighborhood of the point. Plane: K = 0 everywhere, κ₁ = κ₂ = 0 everywhere Inflection points: Isolated flat points on otherwise curved surfaces

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