Kurvust

Kurvust is the Estonian term for curvature, a fundamental concept in geometry that describes how much a geometric object deviates from being flat or straight. It applies to curves in the plane and space, and to surfaces in higher dimensions.

In the plane, for a smooth curve parameterized by arc length s, the curvature κ(s) is the rate of change of the tangent direction with respect to s. A straight line has κ = 0, while a circle of radius R has κ = 1/R, a constant curvature. If a curve is given as y = f(x), the curvature is κ = |f''(x)| / (1 + f'(x)^2)^(3/2). For a space curve r(t), κ = |r'(t) × r''(t)| / |r'(t)|^3, where the cross product expresses bending in space. Signed curvature can be used for oriented plane curves.

On surfaces, curvature becomes more varied. Gaussian curvature K is an intrinsic measure given by the product of the principal curvatures; mean curvature H is the average of the principal curvatures. Geodesic curvature κ_g describes how a curve on a surface deviates from a geodesic. The Frenet–Serret apparatus provides curvature κ and torsion τ for space curves, describing how the curve bends and twists in space.

Applications of kurvust appear across science and engineering, including physics for trajectories and general relativity, computer graphics and animation for rendering smooth curves, robotics and path planning, and civil engineering in arch and rail design. The concept has a long history in differential geometry, with developments by Euler and Gauss, and the Frenet–Serret framework formalizing curvature and torsion for space curves.