Explore chapters and articles related to this topic
Vector Calculus and Differential Geometry
Published in Patrick Knupp, Stanly Steinberg, Fundamentals of Grid Generation, 2020
Patrick Knupp, Stanly Steinberg
At a point on a surface there are two unit normal vectors and an infinity of tangent vectors, i.e., all of the vectors in the plane tangent to the surface. If the surface is described parametrically, then the parameterization defines a unique normal vector and two special tangent vectors. These vectors are defined in this section and used to define the concept of surface area. Two parameters are needed to parametrically describe a surface; the vector form of the parametric equation of a surface is () x=x(ξ,η),0≤ξ≤1,0≤η≤1,
Mesh Parameterization
Published in Kai Hormann, N. Sukumar, Generalized Barycentric Coordinates in Computer Graphics and Computational Mechanics, 2017
Depending on the application, it is often desired to create a parameterization that minimizes deformations between the 2D parametric space and the 3D surface. Using uniform weights wij=1/Ni $ w_{ij} = 1/{\left|{N_i}\right|} $ , where Ni $ {\left|{N_i}\right|} $ denotes the number of neighbors of vertex vi $ \mathbf v _i $ , the obtained parameterization is valid, but can be highly distorted. Figure 6.10 (left) displays the deformations by painting a regular grid in the 2D parametric space and transforming it onto the 3D surface.
Multi-criteria gridshell optimization
Published in Sigrid Adriaenssens, Philippe Block, Diederik Veenendaal, Chris Williams, Shell Structures for Architecture, 2014
Surface parameterization is the two-dimensional (u,v)-coordinate system which is defined on our three-dimensional shell surface, that is, it is a one-to-one mapping between two- and three-dimensional domains. Imagining contour lines of u and v drawn on a shell surface then, a length-preserving parameterization would show u-lines and v-lines forming a square grid (only possible on fat or singly curved surfaces), and an angle-preserving parameterization, or conformal parameterization, parameterization would show u-lines and v-lines intersecting at right angles but the lines themselves could be curved.
Curve parametrization and triple integration
Published in International Journal of Mathematical Education in Science and Technology, 2022
We know that it is possible to produce many parametrizations that could produce the same geometric curve or trace (the trace can be thought of as the path of a moving particle). Composing with a bijective regular function h, which has a regular inverse, is called a reparametrization, and it yields the same trace. Reparametrizing using arc length whenever possible is an example. Different forms of the parametrization may turn out to be more convenient depending on the application. We also know from algebraic geometry, that it is acceptable for a rational parametrization to omit a finite set of points from the curve. For example, a regular polar parametrization of the circle is given by , and a rational parametrization is given by with omitted. Note that for , is a regular curve. Reparametizing by putting into the rational parametrization gives us the regular polar parametrization. This is so as is a bijective function from with a regular inverse on , and therefore, the two parametrizations are equivalent as they describe the same trace. Note that geometric quantities, such as , do not depend on the parametrization. The theory of curve parametrization is a large topic which is important from both the theoretical and the applied points of view – for more details, see Gibson (2001) and Abhyankar (1990).
On probabilistic capacity maximization in a stationary gas network
Published in Optimization, 2020
Given x, for any we define a mapping such that Hence, is continuously differentiable, and, for arbitrary we obtain . Moreover, due to the linear independence of the gradients, the Jacobian matrix has rank 2 in . Thus, there exist indices k,l () such that the according Jacobian sub-matrix is invertible. Without loss of generality let's assume k=1 and l=2. By the Implicit Function Theorem the equation can be resolved in a neighbourhood of equivalently as where are continuous differentiable functions and is a well-defined neighbourhood of . Moreover, the mapping given by defines a parametrization of some surface S in . Clearly, the set is a subset of the surface S and due to (22) we observe that where is the Lebesgue measure in space . In particular, for the according surface measure we obtain that is zero. On the other hand, the union of the family of open sets covers A. Because is separable, a countable selection in A exists, where we obtain Due to the fact that (), we found a union of countable many subsets of S having surface measure zero that covers A. Therefore, from [9, Proposition 4.32] we conclude that .