curvaturas

Curvaturas (curvatures) is a term used across disciplines to describe the degree to which a geometric object deviates from being flat or straight, and, in biology, to describe the natural bends of anatomical structures.

In geometry, the curvature of a plane curve measures how quickly it deviates from a straight line. For a smooth curve, the curvature kappa is the magnitude of the derivative of the unit tangent with respect to arc length. A circle of radius R has constant curvature 1/R, while a straight line has curvature 0. In space curves, curvature together with torsion forms part of the Frenet-Serret framework.

For surfaces, curvature includes several notions, notably Gaussian curvature, which reflects intrinsic bending, and mean curvature, the average of the principal curvatures. The principal curvatures are the maximum and minimum normal curvatures at a point, describing bending in orthogonal directions.

In anatomy and medicine, curvaturas describe the spine’s natural bends: primary curves (thoracic and sacral) are present at birth, while secondary curves (cervical and lumbar) develop with upright posture. Kyphosis refers to an outward (toward the back) bend, and lordosis to an inward bend. Abnormal curvaturas, such as scoliosis, involve lateral deviations and can affect posture and mechanics.

Curvatura concepts have broad applications in engineering, computer graphics, and physics, influencing stability, design, and modeling. In differential geometry and topology, curvature relates to invariants and tensors that capture how spaces bend intrinsically or in embedding spaces.