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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
A scale factor is usually a decimal which scales, or multiplies, some quantity. In the equation y = Cx, C is the scale factor for x. C is also the coefficient of x, and may be called the constant of proportionality of y to x. For example, doubling distances corresponds to a scale factor of two for distance, while cutting a cake in half results ...
Paraboloidal coordinates are three-dimensional orthogonal coordinates (,,) that generalize two-dimensional parabolic coordinates. They possess elliptic paraboloids as one-coordinate surfaces. As such, they should be distinguished from parabolic cylindrical coordinates and parabolic rotational coordinates , both of which are also generalizations ...
Isometric projection is a method for visually representing three-dimensional objects in two dimensions in technical and engineering drawings. It is an axonometric projection in which the three coordinate axes appear equally foreshortened and the angle between any two of them is 120 degrees.
The six independent scalar products g ij =h i.h j of the natural basis vectors generalize the three scale factors defined above for orthogonal coordinates. The nine g ij are the components of the metric tensor , which has only three non zero components in orthogonal coordinates: g 11 = h 1 h 1 , g 22 = h 2 h 2 , g 33 = h 3 h 3 .
The scale factors for the parabolic coordinates (,) are equal = = + Hence, the infinitesimal element of area is = (+) and the Laplacian equals = + (+) Other differential operators such as and can be expressed in the coordinates (,) by substituting the scale factors into the general formulae found in orthogonal coordinates.
Contains the three translations along the coordinate axes; μ – scale factor, which is unitless; if it is given in ppm, it must be divided by 1,000,000 and added to 1. R – rotation matrix. Consists of three axes (small [clarification needed] rotations around each of the three coordinate axes) r x, r y, r z. The rotation matrix is an ...
The cone is unrolled, and the parallel that was touching the sphere is assigned unit scale. That parallel is called the standard parallel. By scaling the resulting map, two parallels can be assigned unit scale, with scale decreasing between the two parallels and increasing outside them. This gives the map two standard parallels.