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  2. Curvature - Wikipedia

    en.wikipedia.org/wiki/Curvature

    is equal to one. This parametrization gives the same value for the curvature, as it amounts to division by r 3 in both the numerator and the denominator in the preceding formula. The same circle can also be defined by the implicit equation F(x, y) = 0 with F(x, y) = x 2 + y 2 – r 2. Then, the formula for the curvature in this case gives

  3. Principal curvature - Wikipedia

    en.wikipedia.org/wiki/Principal_curvature

    The product k 1 k 2 of the two principal curvatures is the Gaussian curvature, K, and the average (k 1 + k 2)/2 is the mean curvature, H. If at least one of the principal curvatures is zero at every point, then the Gaussian curvature will be 0 and the surface is a developable surface. For a minimal surface, the mean curvature is zero at every ...

  4. Mean curvature - Wikipedia

    en.wikipedia.org/wiki/Mean_curvature

    For a surface defined in 3D space, the mean curvature is related to a unit normal of the surface: = ^ where the normal chosen affects the sign of the curvature. The sign of the curvature depends on the choice of normal: the curvature is positive if the surface curves "towards" the normal.

  5. Hessian matrix - Wikipedia

    en.wikipedia.org/wiki/Hessian_matrix

    The determinant of the Hessian matrix, when evaluated at a critical point of a function, is equal to the Gaussian curvature of the function considered as a manifold. The eigenvalues of the Hessian at that point are the principal curvatures of the function, and the eigenvectors are the principal directions of curvature.

  6. Curve fitting - Wikipedia

    en.wikipedia.org/wiki/Curve_fitting

    Fitting of a noisy curve by an asymmetrical peak model, with an iterative process (Gauss–Newton algorithm with variable damping factor α).Curve fitting [1] [2] is the process of constructing a curve, or mathematical function, that has the best fit to a series of data points, [3] possibly subject to constraints.

  7. First fundamental form - Wikipedia

    en.wikipedia.org/wiki/First_fundamental_form

    Theorema egregium of Gauss states that the Gaussian curvature of a surface can be expressed solely in terms of the first fundamental form and its derivatives, so that K is in fact an intrinsic invariant of the surface. An explicit expression for the Gaussian curvature in terms of the first fundamental form is provided by the Brioschi formula.

  8. Gaussian curvature - Wikipedia

    en.wikipedia.org/wiki/Gaussian_curvature

    For most points on most “smooth” surfaces, different normal sections will have different curvatures; the maximum and minimum values of these are called the principal curvatures, call these κ 1, κ 2. The Gaussian curvature is the product of the two principal curvatures Κ = κ 1 κ 2.

  9. Radius of curvature - Wikipedia

    en.wikipedia.org/wiki/Radius_of_curvature

    Radius of curvature and center of curvature. In differential geometry, the radius of curvature, R, is the reciprocal of the curvature. For a curve, it equals the radius of the circular arc which best approximates the curve at that point. For surfaces, the radius of curvature is the radius of a circle that best fits a normal section or ...