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Non-Linear Systems
Published in A. C. Faul, A Concise Introduction to Numerical Analysis, 2018
Geometrically the secant through two points of a curve becomes the tangent to the curve in the limit of the points coinciding. The tangent to the curve f at the point (x(n), f(x(n))) has the equation y=f(x(n))+f′(x(n))(x−x(n)).
Machine Vision
Published in Jerry C. Whitaker, Microelectronics, 2018
David A. Kosiba, Rangachar Kasturi
The slope of the tangent to the curve denoted by the angle θ as a function of s, the position along the curve from an arbitrary starting point, is used to represent curves in many applications. Plots of these functions have some interesting and useful characteristics. Horizontal lines in s–θ plots represent straight lines and straight lines at an angle with the horizontal represent circular arcs whose radii are proportional to the slope of the straight line. An s–θ curve can also be treated as a periodic function with a period given by the perimeter of the curve; hence, Fourier techniques can be used. Other functions, such as the distance to the curve from an arbitrary point inside the curve plotted as a function of the angle with reference to the horizontal, are also used as shape signatures. Figure 19.9(b) shows a simple geometric object and its one-dimensional signature.
Kinematics
Published in George Emanuel, Analytical Fluid Dynamics, 2017
In general, the fluid particles that constitute a vortex filament at one instant of time will not constitute the filament at a later instant of time. The second theorem of Helmholtz provides the condition for the fluid particles to remain fixed with the filament. When this occurs, the filament is a material line. For a vortex filament to be a field line, ω→ $ {\vec{\omega }} $ must be tangent $ {\text{tangent}} $ to the filament, and Equation 4.1 can be used with F=w→ $ F = \vec{w} $ . For the field line to also be a material line, we must have
Pre-service teachers’ understanding of the derivative of a function at a point
Published in International Journal of Mathematical Education in Science and Technology, 2023
María Fernanda Vargas González, José Antonio Fernández-Plaza, Juan Francisco Ruiz Hidalgo
Based on the terms, conventions and notations used by pre-service teachers to delimit the conceptual field, their definitions could be grouped, through content analysis, under five categories: The first category –16 replies– includes replies that defined the derivative of a function at a point solely as the slope of the line tangent to the graph of a function at the point. Examples are listed below.The derivative is the slope of the function.The derivative is the value of the tangent.The derivative is the tangent point of the function.
Nonequivalent definitions and student conceptions of tangent lines in calculus
Published in International Journal of Mathematical Education in Science and Technology, 2022
Winicki-Landman and Leikin (2000) note properties of tangent lines that a student may encounter in the classroom: A tangent has only one point in common with the curve.There is a point at a common distance from all the tangents.All of the curve is on one side of the tangent line.The slope of the tangent line – if it exists – is equal to the value of the derivative of the curve's equation at the point of tangency.A tangent is the graph of the linear approximation of the curve's equation (if it has one) at the point of tangency.A tangent is the limiting position of secant lines passing through the point of tangency.
Effect of spatial resolution and data splitting on landslide susceptibility mapping using different machine learning algorithms
Published in Geomatics, Natural Hazards and Risk, 2021
Minu Treesa Abraham, Neelima Satyam, Prashita Jain, Biswajeet Pradhan, Abdullah Alamri
Slope, aspect, SPI and TWI layers were prepared from the DEM. The difference in altitude values of the two different DEMs will affect the values of these layers as well (Pradhan and Sameen 2017). As depicted in Figure 4, the variation in elevation values is negligible when both layers are compared. However, this minor variation has severe impacts on the DEM derived layers. Hence, the resolution of DEM is a critical factor in determining the quality of results and all these layers were developed using the DEMs of two different resolutions. Slope is a significant parameter in the process of LSM. It is the ratio of vertical distance to horizontal distance between two specified points, expressed using the tangent angle. The slope angle varies from 0 to 90 degrees and studies on LSM supports the consensus on the notion of considering slope as an important parameter in the initiation of landslides. The term aspect indicates the orientation of the slope face, expressed as an angle varying from 0 to 360 degrees, starting from north, in the clockwise direction, they are classified into 8 categories with a difference of 45 degrees each. The literature says that the slope aspect is critical when landslides are triggered after superficial cracks (tension cracks) are formed in clay (Capitani et al. 2013). These types of landslides are detected in the study area, and it is very common that long tension cracks can be identified at the crown of landslides much before the occurrence of landslides. Hence, aspect maps are also prepared using the DEM, for two different resolutions.