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Some spatial aspects of the analysis of coastal spits
Published in Richard J. Chorley, Spatial Analysis in Geomorphology, 2019
The logarithmic spiral has an increasing radius of curvature away from its initial point, which represents the distal point of the spit. This is usually recurved shorewards as the waves become refracted around the distal end. The equation for a logarithmic spiral is r = eθcotα where r is the length of the radius vector and θ is the angle of the vector from some convenient datum direction. The angle α is the spiral angle. It is the angle between the radius vector and the tangent to the spiral curve at that point. For any one logarithmic spiral it is a constant, and for curves fitted to beach outlines it normally varies between 30 and 90 degrees.
Centrifugal Compressors
Published in Ahmed F. El-Sayed, Aircraft Propulsion and Gas Turbine Engines, 2017
The streamlines for incompressible flow in a vaneless diffuser of constant axial width make a constant angle with the radial direction α. Thus its shape is sometimes called logarithmic spiral. θ3−θ2=tanα2ℓnr3r2
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Published in Wai-Fah Chen, Lian Duan, Bridge Engineering Handbook, 2019
The Fibonacci series is also used to construct a logarithmic spiral, which occurs in nature in snail and ammonite shells, and which is considered particularly beautiful for ornaments. Le Corbusier (1887–1965) used the golden mean to construct his “Modulor” based on an assumed body height of 1.829 m but the Modulor is in itself not a guarantee of harmony.
Discrete element analysis of structural characteristics of stepped reinforced soil retaining wall
Published in Geomatics, Natural Hazards and Risk, 2020
Ke Ma, Longjiang Wang, Liji Long, Yilin Peng, Guangchun He
Specification for design of Highway subgrade in China stipulated 0.3H simplified fracture surface as the fracture surface of the vertical reinforced retaining wall. However, there are no relevant regulations on fracture surfaces of RSRW. It can be seen from Figure 15 that when calculated 10,000,000 steps, the force chains in the frontline of retaining wall suffered a great reduction in both strength and quantity. The strong force chains between particles vanished and the wall body became instable. The massive amount of particles fell off and formed an apparent collapse zone. The black curve marked the potential fracture surface of the retaining wall. It can be seen that the force chain structures are relatively intact at the right side. The shape of the curve is shaped similar to a logarithmic spiral.
Polar eigenvectors: a better way?
Published in International Journal of Mathematical Education in Science and Technology, 2022
There are two classic types of spirals: Archimedean and logarithmic. In essence, Archimedean spirals are of the form and logarithmic spirals are of the form (see Dennis Lawrence, 1972). The spirals resulting from systems of differential equations don’t fall neatly into either of these two types, but they are closest the logarithmic type. They are found when (but ) in Equation (4), which is rewritten here for convenience: and .
Development of a fully planar logarithmic spiral antenna with integrated balun in UWB GPR systems for landmines detection
Published in Electromagnetics, 2022
Narek Grigoor-Feghi, Reza Masoumi, Robab Kazemi
where is the opening angle, is radius of the curve in angle, is the growth rate, and is the angular position. This curve is used to make the planar logarithmic spiral antenna of Figure 2b. Four edges of the antenna are obtained from (1) with = 0°, δ°, 180°, and (180 + δ)°, respectively. This structure is self-complementary with δ = 90° that has the most perfect symmetric beam.