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Introduction
Published in Vladimir A. Dobrushkin, Applied Differential Equations with Boundary Value Problems, 2017
By increasing the density of arrows, it would be possible, in theory at least, to approach a limiting curve, the coordinates and slope of which would satisfy the differential equation at every point. This limiting curve—or rather the relation between x and y that defines a function y(x) —is a solution of y′=f(x,y) $ y^{\prime } = f(x, y) $ . Therefore the direction field gives the “flow of solutions.” Integral curves obtained from the general solution are all different: there is precisely one solution curve that passes through each point (x,y) $ (x, y) $ in the domain of f(x,y) $ f(x, y) $ . They might be touched by the singular solutions (if any) forming the envelope of a family of integral curves. At each of its points, the envelope is tangent to one of integral curves because they share the same slope.
Tectonics
Published in Aurèle Parriaux, Geology, 2018
A triaxial test that has caused a sample to rupture and its corresponding Mohr circle describe the maximum strength of the sample under the particular conditions of the test stresses. However, it does not provide the rupture criteria of the rock under other stress states. To obtain this information, it is necessary to determine what rupture criteria best characterize the maximum strength of the rock under other states. The triaxial test makes it possible to construct this function experimentally; to do this, several tests must be done on samples that are as similar as possible, and different values of confining pressures σ3 must be used during the tests. Through these tests, we obtain several rupture circles that allow an envelope called an intrinsic curve to be drawn, because it is characteristic of the material and is not dependent on the geometry of the rupture surface (Fig. 12.11). The interior of this envelope determines the field of non-rupture deformation, the envelope determines the transition to rupture. The shape of the envelope is often a parabolic curve. Its extension to the left of the X-Y axis indicates negative compressions, thus the tensile strength Rt. The equation of the parabola is the following: () τ2−c2Rt⋅σ−c2=0
Combination of Primaries with Flow-Line Secondaries
Published in Julio Chaves, Introduction to Nonimaging Optics, 2017
To calculate the shape of the receiver, we first calculate the positions of E1 and E2 and then, from the equation of the caustic curve, we determine the points whose tangents go through these points. For this, however, we need to know the shape of the caustic curve. A caustic is the envelope of a one-parameter family of light rays. The envelope of a one-parameter family of curves is a curve that is tangent to every curve of the family. Also, each member of the family is tangent to the envelope. In general, a one-parameter family of curves is defined in parametric form by
The influence of atrium types on the consciousness of shared space in amalgamated traditional dwellings – a case study on traditional dwellings in Quanzhou City, Fujian Province, China
Published in Journal of Asian Architecture and Building Engineering, 2019
Xinpeng Wang, Kai Fang, Lin Chen, Nobuaki Furuya
In plane geometry, an envelope of a curve family is a tangent line that shares at least one point with each one of a family of curves. (A curve family is an infinite set of curves where the curves have certain relationships with each other.) In three dimensions, a surface that is a tangent to each one of a family of surfaces is called an envelope surface.