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Scale and Generalization
Published in Terry A. Slocum, Robert B. McMaster, Fritz C. Kessler, Hugh H. Howard, Thematic Cartography and Geovisualization, 2022
Terry A. Slocum, Robert B. McMaster, Fritz C. Kessler, Hugh H. Howard
Figure 6.7 depicts the calculation of one specific measurement, the trendlines, for the Hennepin County data set. The trendlines for a digitized curve are based on a calculation of angularity, or where the lines change direction. Where a curve changes direction—for example, from left to right—a mathematical inflection point is defined (theoretically, the point of no curvature). The connection of these inflection points, which indicates the general “trend” of the line, is called the trendline. The complexity of a feature can be approximated by looking at the trendlines for the entire feature or for the entire data set. A simple measure of complexity derived from the trendline is the trendline/total length of a line or the sinuosity of a feature. Along relatively straight line segments, with little curvilinearity, the trendline will be very close to the curve, and the trendline/total length ratio will be nearly equal to 1.0 (e.g., the relatively straight line near the middle of Figure 6.7). However, a highly complex curve, such as the northern border of Hennepin County, will deviate significantly from the trendline, and the ratio will be distinctly less than 1.0. Thus, the greater the difference between the actual digitized curve and the trendline, the more complex the feature.
Nonlinear Optimization
Published in Michael W. Carter, Camille C. Price, Ghaith Rabadi, Operations Research, 2018
Michael W. Carter, Camille C. Price, Ghaith Rabadi
Many functions that arise in nonlinear programming models are neither convex nor concave. The function pictured in Figure 5.2 is a good example of a function that is convex in one region and concave in another region, but neither convex nor concave over the entire region of interest. Local optima are not necessarily global optima. Furthermore, a point x for which f’(x) = 0 may be neither a maximum nor a minimum. In Figure 5.2, the function f(x) at the point x = e has a zero slope. When viewed from the direction of x = d, it appears that x = e may be a maximum; whereas when viewed from the direction of x = f, the function appears to be decreasing to a minimum. In fact, x = e is an inflection point. For example, the function f(x) = x3 has an inflection point at x = 0.
Process-based approach on tidal inlet evolution – Part 1
Published in C. Marjolein Dohmen-Janssen, Suzanne J.M.H. Hulscher, River, Coastal and Estuarine Morphodynamics: RCEM 2007, 2019
D.M.P.K. Dissanayake, J.A. Roelvink
In the river reaches, the basic unit for describing the geometry was a half-meander (coinciding with a river reach in between two consecutive inflection points). An inflection points is defined as a location where the curvature is equal to zero. The curvature c was determined according to a three-point algorithm on the
Circular time-domain reflectometry system for monitoring bridge scour depth
Published in Marine Georesources & Geotechnology, 2020
Jung-Doung Yu, Jong-Sub Lee, Hyung-Koo Yoon
The TDR instrument (Hyperlabs model HYPERLABS-HL1101, USA) was applied to provide constant voltage and obtain a reflected signal. An input voltage of 250 mV is generated by the device and propagates through a coaxial cable (50 Ω). The electromagnetic waves transmitted through the coaxial cable were delivered to the electrodes using an alligator clip, and the reflected waveform was measured. Fifty signals were stacked to enhance the resolution, and electrical noise was removed. The travel time is determined by the time differences between the first and final inflection points. The inflection point of the signal is selected using the method of tangents (Klemunes 1998), as shown in Figure 1.
An Estimation Method of Surface Defects of LMJ Microshells
Published in Fusion Science and Technology, 2022
V. Dutto, A. Choux, F. C. Chittaro, É. Busvelle, J.-P. Gauthier
Figure 7 shows the second derivatives of the profiles, experimental and simulated, from the curves of Fig. 6b. In Fig. 7, the position of the computed interface is plotted by the black vertical line. The second derivatives of the profiles, on the experimental (blue) and on the simulated (red), are superposed. We note that the vertical line crosses the curves when they are between a minimum at pixel 455 and a maximum at pixel 460. Between these two values, the curves change their sign, which defines an inflection point.
Fixed bed adsorption of water and air contaminants: analysis of breakthrough curves using probability distribution functions
Published in Chemical Engineering Communications, 2023
Khim Hoong Chu, Mohd Ali Hashim
The normal distribution defined by Equation (2) predicts a sigmoid curve, whose shape is determined by its inflection point. An inflection point is where a curve changes concavity; that is, it is a point at which a curve goes from concave up to concave down, or vice versa. For the normal distribution, the location of its inflection point can be determined by finding at what time the second derivative of the function is zero. The second derivative of Equation (2) is given by Equation (3).