Curvature: the idea differential geometry is built on

Differential geometry sounds forbidding, but almost all of its practical content is one question asked carefully: how much does this curve bend here? Answer that well and you can detect features, smooth a mesh, or decide how to shade a surface.

# A regular curve, parameterised with |r'(t)| > 0. Curvature is how
# fast the unit tangent's DIRECTION turns per unit arc length.

T(t) = r'(t) / |r'(t)|                      # unit tangent, where speed is nonzero
κ(t) = |r'(t) x r''(t)| / |r'(t)|^3         # curvature (2-D and 3-D)

# Sanity checks:
#   straight line   -> r'' = 0        -> κ = 0
#   circle radius R -> κ = 1/R        -> smaller circle, bigger curvature

On a smooth surface, one number is not enough because bending depends on tangent direction. At a regular point, the two principal curvatures are the extremal normal curvatures over tangent directions. Their signs classify the local shape (up to the chosen normal/sign convention):

· both nonzero with the same sign → dome-/bowl-like (elliptic)
· opposite signs → saddle-like (hyperbolic)
· one zero and one nonzero → cylinder-like (parabolic)
· both zero → locally flat to second order

This is useful in geometry processing: discrete curvature estimates can help classify smooth regions and surface shape. But mesh-edge detection is not just curvature thresholding — sharp creases are often detected from normal or dihedral-angle changes, and a saddle is a point with opposite-sign principal curvatures, not a “pinch point.”