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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
At each point on a surface 5, we can define an outward pointing unit normal by taking the cross product between the vectors that define the tangent space. We identify the set of all unit vectors in ℝ3 with S2, the unit sphere in ℝ3. The Gauss mapN:S→S2 gives the unit normal at each point on the surface S. In local coordinates, N(u,υ)=cu×cυ|cu×cυ|. For smooth, orientable surfaces, the Gauss map is a well defined, differentiable mapping. We write n = N(u, υ) for the unit normal at a point on the surface.
Orthogonal Expansions in Curvilinear Coordinates
Published in Gregory S. Chirikjian, Alexander B. Kyatkin, Engineering Applications of Noncommutative Harmonic Analysis, 2021
Gregory S. Chirikjian, Alexander B. Kyatkin
DEFINITION 4.1A Gauss map is a function that assigns to each point on a surface a unit normal vector. Since this may be viewed as a mapping between the surface and the unit sphere, S2, it is also called a normal spherical image.
Fast computation of local minimal distances between CAD models for dynamics simulation
Published in Computer-Aided Design and Applications, 2018
Sébastien Crozet, Jean-Claude Léon, Xavier Merlhiot
Let us first recall the concept of Gauss Map, , of a surface which is a function that maps each point of the surface to its (unit) normal seen as a point on the unit sphere . We extend this concept to faces, edges, and vertices, when considering that maps a point of any of those features to the set of unit vectors on its tangent cone polar. The Gauss Map Image of a subset of a given feature is the union of the Gauss maps of all its points. To address the construction methods of the polyhedral normal cone introduced in section 6.1, is embedded into a Euclidean space and given a local direct coordinate system centered at the origin. Its two poles are located at and . The intersection curve of any plane passing through these poles (and the origin) with is called a meridian (see Fig. 7(a)). Planes orthogonal to the axis (but not necessarily containing the origin) intersect along lines of latitude (see Fig. 7(b)).