Differential geometry approaches to curvature and shape analysis

Curvature provides a quantitative measure of how a geometric object bends in space. Differential geometry offers a rigorous framework for defining and computing curvature on curves, surfaces and manifolds. This paper develops a mathematical approach to curvature and shape analysis using differential geometry. Challenges include handling complex surface parametrizations, computing principal curvatures from noisy data, and distinguishing intrinsic and extrinsic curvature. The proposed methodology formulates surfaces as parametric maps, derives the 1st and 2nd fundamental forms, constructs the shape operator and computes Gaussian and mean curvature. Analytical formulas for principal curvatures and curvature classification support algorithmic shape analysis. Results on canonical shapes such as spheres, cylinders and saddles demonstrate how curvature signatures classify geometric structures. The outcomes underscore that differential geometry not only provides elegant theoretical tools but also practical algorithms for shape analysis in computer graphics and geometric modelling.