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Newton's method uses curvature information (i.e. the second derivative) to take a more direct route. In calculus, Newton's method (also called Newton–Raphson) is an iterative method for finding the roots of a differentiable function, which are solutions to the equation =.
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 ...
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.
By using the change of variables log f ↦ u, another commonly found form of Liouville's equation is obtained: =. Other two forms of the equation, commonly found in the literature, [4] are obtained by using the slight variant 2 log f ↦ u of the previous change of variables and Wirtinger calculus: [5] = ¯ =.
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.
Alternatively, the sectional curvature can be characterized by the circumference of small circles. Let be a two-dimensional plane in .Let () for sufficiently small > denote the image under the exponential map at of the unit circle in , and let () denote the length of ().
The mean curvature flow extremalizes surface area, and minimal surfaces are the critical points for the mean curvature flow; minima solve the isoperimetric problem. For manifolds embedded in a Kähler–Einstein manifold , if the surface is a Lagrangian submanifold , the mean curvature flow is of Lagrangian type, so the surface evolves within ...
A point's winding number with respect to a polygon can be used to solve the point in polygon (PIP) problem – that is, it can be used to determine if the point is inside the polygon or not. Generally, the ray casting algorithm is a better alternative to the PIP problem as it does not require trigonometric functions, contrary to the winding ...