Differential geometry studies geometric properties of curves and surfaces — quantities invariant under congruence, such as length, angle, and curvature — using differential and integral calculus, in contrast to the finite, corner-based objects of classical Euclidean geometry. A central foundational technique is parameterization: representing a curve as a map from a real-interval parameter to Euclidean space via coordinate functions, from which the tangent vector (the derivative of the parameterization) and arc length (the integral of the speed, i.e., the norm of the tangent vector) are derived; reparameterization, an alternate parameter map yielding the same image curve, preserves the underlying geometric properties.
Differential geometry studies geometric properties of curves and surfaces — quantities invariant under congruence, such as length, angle, and curvature — using differential and integral calculus, in …