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Two-Dimensional Finite Element Analysis
Published in Özlem Özgün, Mustafa Kuzuoğlu, ®-based Finite Element Programming in Electromagnetic Modeling, 2018
Triangular elements are widely used in practice, and therefore, it is important to generate a triangular mesh of good quality. To measure the quality of triangular elements, several numerical criteria have been proposed in the literature [9–11]. In general, quality measures favor equilateral triangles, and penalize triangles with one or more small interior angles. One quality measure approach is based on the comparison of the circumscribed and inscribed circles of a given triangle. In computational geometry, the circumscribed circle (or circumcircle) of a triangle is the smallest circle that passes through all vertices of the triangle (see Fig. 5.6(a)). The inscribed circle (or incircle) is the largest circle lying inside the triangle and tangent to the sides of the triangle (see Fig. 5.6(b)). These circles are unique except for degenerate cases. The radii of circumcircle and incircle (denoted by rcc and ric, respectively) can be computed by using the following formulas.
Algebraic proofs for converse theorems for a cyclic quadrilateral
Published in International Journal of Mathematical Education in Science and Technology, 2023
Similarly, and Substituting from (35)–(40) in (32), which can be expressed as upon substituting From (4) for i = 2, j = 1, k = 3, Similarly, for i = 2, j = 4, k = 3 in (4) From (46) and (48), and (43) can be expressed as By definition, x>0 from (44). From Corollary 2.3, it follows that is the only possible a solution of (15) resulting in (26). Hence, D lies on the circumcircle of and ABCD is a cyclic quadrilateral.
Improving template-based CT data evaluation by integrating CMM reference data into a CAD model-based high fidelity triangle mesh
Published in Nondestructive Testing and Evaluation, 2022
Andreas Michael Müller, Tino Hausotte
A second example demonstrating the parametrisation for the STEP entity ‘cylindrical surface’ is shown in Figure 5. The upper image shows the vertices defining the sub-mesh edges in red and the CMM measurement data in blue for the large cylindrical surface of the specimen. The result of the parametrisation into the interpolation space is shown below, with edge points having by design. It is important to note that both and are given in mm although the angular parameter is given in radian (3). In order to ensure interpolation in an isotropic space, the parameter is therefore converted to arc length to have the same unit as . The Delaunay triangulation is created, such that no other vertex is inside the circumcircle of a triangle. If space itself is deformed by choosing different units, those circles become ellipses and the interpolation is biased. The space enclosed by the parameters and is now exactly the area of the cylinder surface.
Bayesian Model Building From Small Samples of Disparate Data for Capturing In-Plane Deviation in Additive Manufacturing
Published in Technometrics, 2018
Arman Sabbaghi, Qiang Huang, Tirthankar Dasgupta
The following illustrative example helps to provide a physical justification for this modular deviation representation. Consider a regular polygon with n edges and size (circumcircle radius) γ1, defined by the nominal radius function where γ2 = (n, γ1) is the parameter vector that uniquely defines this function. A polygon deviation model as hypothesized under this framework is then obtained by setting (corresponding to the polygon’s size), and using in Equation (3). Physically, this corresponds to carving a polygon out of a circle. We can then interpret this model specification by recognizing that, conceptually, the carved polygon deviates globally in a similar fashion as its circumcircle (Figure 4). We characterize δ1 as a global deviation feature shared between the polygon and its circumcircle. The regular polygon’s deviation systematically differs from δ1 due to its straight edges and sharp corners, and so δ2 is introduced to capture its corresponding local deviation feature. We make no further assumptions about relationships between δ1 and δ2, for example, whether one component is larger than the other.