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The scale factor is dimensionless, with counted from the birth of the universe and set to the present age of the universe: [4] giving the current value of as () or . The evolution of the scale factor is a dynamical question, determined by the equations of general relativity , which are presented in the case of a locally isotropic, locally ...
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 ...
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 }
The scale factors for the elliptic coordinates (,) are equal to = = + = . Using the double argument identities for hyperbolic functions and trigonometric functions, the scale factors can be equivalently expressed as
In an oblique pictorial drawing, the displayed angles among the axes as well as the foreshortening factors (scale) are arbitrary. The distortion created thereby is usually attenuated by aligning one plane of the imaged object to be parallel with the plane of projection thereby creating a true shape, full-size image of the chosen plane.
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A 1:1 scale construction drawing of a boat and its parts Lines plan A scaled-down version of a full-sized drawing often including the body, plan, profile, and section views Body Plan A view of the boat from both dead ahead and dead astern split in half Plan view A view looking down on the boat from above Profile view A view of the boat from the ...
The dimension of the product manifold is the sum of the dimensions of its factors. Its topology is the product topology, and a Cartesian product of charts is a chart for the product manifold. Thus, an atlas for the product manifold can be constructed using atlases for its factors.