Equations of a plane

Symmetry 50. Intercepts 50. Asymptotes 50. Equations of Slope 51. Tangents 51. Equations of a Straight Line 52. Equations of a Circle 53. Equations of a Parabola 53. Equations of an Ellipse of Eccentricity e 54. Equations of a Hyperbola 55. Equations of Three-Dimensional Coordinate Systems 56. Equations of a Plane 56. Equations of a Line 57. Equations of Angles 57. Equation of a Sphere 57. Equation of an Ellipsoid 57. Equations of Hyperboloids and Paraboloids 58. Equation of an Elliptic Cone 59. Equation of an Elliptic Cylinder 59. 

In the class discrimination methods or hyperplane techniques, of which linear discriminant analysis and the linear learning machine are examples, the equation of a plane or hyperplane is calculated that separates one class from another. These methods work well if prior knowledge allows the analyst to assume that the test objects must... 

Now we consider the case of one-dimensional oscillation. If we neglect the effects due to the volume occupied by the solid particles and scattering, for a gas-solid suspension with solid-to-gas mass ratio of mp, the equation of a plane wave in the mixture may be expressed by [Soo, 1990]... 

The intersection of three planes may be dealt with by using either (1) or (4) as the equation of a plane. With (1) ... 

A simple example of the use of a Lagrange multiplier (constraint) can be seen in the calculation of the equation of a plane ... 

Consider first a system where the observed quantity y depends on two independent variables, and Xj. The resulting relationship describes the equation of a plane in three dimensions and may be written as... 

E. Arbarello, E. Sernesi, The equation of a plane curve, Duke Math. J. 46 (1979). 

Divide through by be rearrange terms and we get the intercept equation of the plane, i.e., the equation of a plane expressed in terms of its intercepts upon the three axes ... 

Using these equations, we can determine the equation of a plane as follows. 

We have also defined "d", the separation between the planes, eind "r", the vector to the nearest neighbor. We find that the equation of a plane is ... 

Equation (44) corresponds to the equation of a plane perpendicular to the vector (1, -Cm, -Dm) and separated by a distance Bm from the origin in a homogeneous 3D space in which 5 / Sz)j, (Bz)y, and Sz)j define the orthogonal x, y, and z directions. The structural factors Cm and Dm are complicated nonlinear combinations of the geometrical factors... 

This is the equation of a plane, describing the transition from ellipsoidal (GPS) height differences to orthometric height differences Sh — (SH). In figure 3-3 this plane is plotted as... 

This is the equation of a plane normal to the vector k and the solution (2.24) therefore represents a plane wavefront travelling with velocity v in the direction k (Problem 2.2) However, the electric and magnetic fields are vector quantities and we therefore assume that the plane-wave solutions of equations (2.19) are of the form ... 

The traveling wave equation can be obtained as the solution of a differential wave equation referred to as the wave equation. Knowing the solution in the forms (2.8.5) and/or (2.8.7) we can find the wave equation itself. Differentiation of the equation of a plane wave (x, 0 twice upon the time and upon the coordinate gives... 

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