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Concept# Line (geometry)

Summary

In geometry, a line is an infinitely long object with no width, depth, or curvature. Thus, lines are one-dimensional objects, though they may exist embedded in two, three, or higher dimensional spaces. The word line may also refer to a line segment in everyday life that has two points to denote its ends (endpoints). A line can be referred to by two points that lie on it (e.g. \overleftrightarrow{AB}) or by a single letter (e.g. \ell).
Euclid described a line as a "breadthless length" that "lies evenly with respect to the points on itself"; he introduced several postulates as basic unprovable properties from which he constructed all of geometry. Euclidean line and Euclidean geometry are terms introduced to avoid confusion with generalizations introduced since the end of the 19th century, such as non-Euclidean, projective, and affine geometry.
Properties
When geometry was first formalised by Euclid in the Elements, he defined a

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This thesis presents a methodology for the design optimization of hydraulic runner blades. The originality of the methodology comes from the geometric definition of the blade shapes, which uses parametric surfaces instead of a set of profiles. The main advantage of using surfaces is the number of parameters required. The use of surfaces requires a different technique for the blade construction when compared to traditional approaches. NURBS surfaces are used for the geometric representation, the properties provided by such parametric formulation permit to reach the necessary flexibility and accuracy to be attained. Moreover, the surface approach can be seen as a way to liberate the blade design from traditional discrete sectional approaches. Actual blade optimization procedures are a compromise between the quality of the design, its performance analysis and the subsequent computational time-consumed effort. The improvements provided by the above mentioned geometric definition allow the use of more realistic analysis tools for the design evaluation. Thus, Navier-Stokes (k – ε) simulations are integrated in a simple and direct optimization process. The resulting methodology is not penalized by the time-effort required and becomes of interest for its use in industrial applications. Finally, we supply a number of examples to demonstrate the feasibility of the optimization proposal. These examples illustrate the application of the methodology at different levels of geometric complexity. They are interesting not only through the results obtained, but also because they become acceptable in terms of time consumed on daily and industrial applications.

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The Hongrin north dam is a double curvature concrete arch dam located in western Swiss Prealps, which attains 125m high. The right bank abutment of the dam mainly consists of intensively jointed Neocomian limestone and exhibits zones of potential instability.At the time of construction (1965–1969), this slope was reinforced with rock anchors. Subsequent hydrogeological study and groundwater monitoring revealed the presence of water pressure due to a slight seepage flow through the rock joints in the dam foundation. This latter evidence raised an additional concern about the stability of the abutment. In a dedicated study, the stability of the right abutment the Hongrin north dam abutment has been assessed using continuumdiscontinuum numerical analysis. 3DEC (3-Dimensional Distinct Element Code) has been used to model the complicated slope geometry and to explore the role of rock discontinuity in the failure mechanisms. The rock mass is defined as deformable distinct blocks which interact along frictional discrete discontinuities representing the rock joint sets. The water pressure is introduced as fluid pressure boundary condition along the discontinuities, and the rock reinforcement is modeled as structural elements working across the discontinuities. The dam reaction forces, derived from a separate finite element analysis, are evaluated for their possible effects on the stability. The model examines the sensitivity of the abutment stability to the presence of joint water pressure and evaluates the improving effects of rock reinforcement. The results of the analysis allow achieving an enhanced understanding of potential failure mechanisms and helps in proposing further suitable measures to improve the stability of the abutment.