Kaarevuus

Kaarevuus, in Finnish often used to mean curvature, is a geometric measure of how much a curve or surface deviates from being flat or straight. It plays a central role in differential geometry, describing how shapes bend in space.

For a regular plane curve drawn as γ(t) = (x(t), y(t)), the curvature can be defined when the curve is parameterized by arc length s as κ(s) = ||d²γ/ds²||. If the curve is not parameterized by arc length, a common formula in three dimensions uses κ = ||γ' × γ''|| / ||γ'||³. In two dimensions, curvature can also be described as the derivative of the tangent direction with respect to arc length, κ = dφ/ds, where φ is the angle of the tangent.

A circle of radius R has constant curvature κ = 1/R, while a straight line has κ = 0. In space curves, curvature is part of the Frenet-Serret apparatus: it measures how quickly the unit tangent vector changes along the curve, together with torsion, which measures twisting out of the plane.

For surfaces, curvature is more varied. Normal curvature depends on the direction on the surface. The Gaussian curvature K is the product of the principal curvatures and reflects intrinsic bending, while the mean curvature H is their average and arises in phenomena such as minimal surfaces and surface diffusion.

Calculating kaarevuus involves derivatives and, frequently, arc-length parametrization. Applications span computer graphics, robotics, architectural design, road and track engineering, and fundamental physics, where curvature describes the bending of space or spacetime.