China 
Africa 
Explore chapters and articles related to this topic
Chapter 2
Published in Pearson Frederick, Map Projections:, 2018
The second fundamental form provides the means for evaluating the curvature of a surface and determining the principal directions on the surface. The section begins by defining the second fundamental forms for a general surface. The second fundamental form, in conjunction with the first fundamental form, defines the curvature. The second fundamental quantities are then defined. The principal directions on the surface are then obtained by an optimization process. Since the two principal directions are found to be orthogonal, the curvatures in the two principal directions are given by simplified equations.
Multifingered Hand Kinematics
Published in Richard M. Murray, Zexiang Li, S. Shankar Sastry, A Mathematical Introduction to Robotic Manipulation, 2017
Richard M. Murray, Zexiang Li, S. Shankar Sastry
The directional derivative of the Gauss map defines the second fundamental form for a surface. The second fundamental form is a measure of the curvature of a surface. In a local coordinate chart, it is represented by a map IIp : ℝ2 × ℝ2 → ℝ: which has a matrix representation IIp=[cuTnucuTnυcυTnucυTnυ], where p = c(u, υ), nu:=∂n∂u, and nv:=∂n∂v. This matrix describes the rate of change of the normal vector projected onto the tangent plane. It may be interpreted as follows: if p(s) ∈ S is a curve lying on S that is parameterized by arc length and whose tangent vector is p’, then (p’)TIIpp′ gives the usual curvature of the spatial curve p.
Models for ground vehicle control on nonplanar surfaces
Published in Vehicle System Dynamics, 2023
Thomas Fork, H. Eric Tseng, Francesco Borrelli
First we differentiate constraints (5b) and (5c) with respect to time: and expand the time derivatives of , and : The matrix on the left-hand side of (8) is known as the second fundamental form of a parametric surface [21, Ch. 3] which we denote by . This term captures the curvature of the surface. We introduce the symbol for the matrix on the right, due to its interpretation as a Jacobian of the parametric surface as viewed in the body frame. Observe that the above relates motion on the surface (, ) to necessary angular velocity of the body (, ) to remain tangent to the surface.
Modelling of frictionless Signorini problem for a linear elastic membrane shell
Published in Applicable Analysis, 2022
M. E. Mezabia, D. A. Chacha, A. Bensayah
We also give some geometrical preliminaries on the surface (see [22]). The first fundamental form, given as metric tensor in convariant or contravariant components, the second fundamental form given as curvature tensor in covariant or mixed components, the Christoffel symbols , and the mixed components of the covariant derivative of the curvature tensor , are respectively defined by We have and The area element of is , where
Spacelike submanifolds of codimension two in anti-de Sitter space
Published in Applicable Analysis, 2019
Since are spacelike vectors, we have a Riemannian metric (or the first fundamental form) on M defined by where for any We also have a lightcone second fundamental form with respect to the normal vector field defined by for any