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Such a scaling changes the diameter of an object by a factor between the scale factors, the area by a factor between the smallest and the largest product of two scale factors, and the volume by the product of all three. The scaling is uniform if and only if the scaling factors are equal (v x = v y = v z). If all except one of the scale factors ...
In an orthogonal coordinate system the lengths of the basis vectors are known as scale factors. The scale factors for the elliptic coordinates ( μ , ν ) {\displaystyle (\mu ,\nu )} are equal to h μ = h ν = a sinh 2 μ + sin 2 ν = a cosh 2 μ − cos 2 ν . {\displaystyle h_{\mu }=h_{\nu }=a{\sqrt {\sinh ^{2}\mu +\sin ^{2 ...
Prolate spheroidal coordinates μ and ν for a = 1.The lines of equal values of μ and ν are shown on the xz-plane, i.e. for φ = 0.The surfaces of constant μ and ν are obtained by rotation about the z-axis, so that the diagram is valid for any plane containing the z-axis: i.e. for any φ.
Knowing the scale factors, various functions of the coordinates can be calculated by the general method outlined in the orthogonal coordinates article. The infinitesimal volume element is: d V = a 3 ( ζ 2 + ξ 2 ) d ζ d ξ d ϕ {\displaystyle dV=a^{3}\left(\zeta ^{2}+\xi ^{2}\right)d\zeta \,d\xi \,d\phi }
In mathematics, a plane is a two-dimensional space or flat surface that extends indefinitely. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space. When working exclusively in two-dimensional Euclidean space, the definite article is used, so the Euclidean plane refers to the ...
In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cross each other. [9] Such a drawing is called a plane graph or planar embedding of the graph.
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Conversely, it is easily shown that if a, b, c, and d are constants and a, b, and c are not all zero, then the graph of the equation + + + =, is a plane having the vector n = (a, b, c) as a normal. [5] This familiar equation for a plane is called the general form of the equation of the plane or just the plane equation. [6]
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