page_title: Curvature
section_title: Principal curvature
hierarchical_section_title: Curvature / Surfaces / Curves on surfaces / Principal curvature
context_page_description: In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.
  For curves, the canonical example is that of a circle, which has a curvature equal to the reciprocal of its radius. Smaller circles bend more sharply, and hence have higher curvature. The curvature at a point of a differentiable curve is the curvature of its osculating circle, that is the circle that best approximates the curve near this point. The curvature of a straight line is zero. The curvature of a curve at a point is normally a scalar quantity, that is, it is expressed by a single real number.
  For surfaces, that are embedded in a Euclidean space, the concept of curvature is more complex, as depending on the choice of a direction on the surface or manifold. This leads to the concepts of maximal curvature, minimal curvature, and mean curvature.
  For Riemannian manifolds that are not necessarily embedded in a Euclidean space, one can define the curvature intrinsically, that is without referring to an external space.
context_section_description: All curves with the same tangent vector will have the same normal curvature, which is the same as the curvature of the curve obtained by intersecting the surface with the plane containing T and u. Taking all possible tangent vectors, the maximum and minimum values of the normal curvature at a point are called the principal curvatures, k₁ and k₂, and the directions of the corresponding tangent vectors are called principal normal directions.
caption: Saddle surface with normal planes in directions of principal curvatures