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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 ...
at latitude 45° the scale factor is k = sec 45° ≈ 1.41, at latitude 60° the scale factor is k = sec 60° = 2, at latitude 80° the scale factor is k = sec 80° ≈ 5.76, at latitude 85° the scale factor is k = sec 85° ≈ 11.5. The area scale factor is the product of the parallel and meridian scales hk = sec 2 φ.
The scale depends on location, but not on direction. This is equivalent to preservation of angles, the defining characteristic of a conformal map. Scale is constant along any parallel in the direction of the parallel. This applies for any cylindrical or pseudocylindrical projection in normal aspect.
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 orthogonal matrix. The angles are given in either degrees or radians.
The polar angle may be called inclination angle, zenith angle, normal angle, or the colatitude. The user may choose to replace the inclination angle by its complement , the elevation angle (or altitude angle ), measured upward between the reference plane and the radial line—i.e., from the reference plane upward (towards to the positive z-axis ...
In the new coordinate system, the point P will appear to have been rotated in the opposite direction, that is, clockwise through the angle . A rotation of axes in more than two dimensions is defined similarly. [2] [3] A rotation of axes is a linear map [4] [5] and a rigid transformation.
The trend is the compass direction of the line, and the plunge is the downward angle it makes with a horizontal plane. [15] For a plane, the two angles are called its strike (angle) and its dip (angle). A strike line is the intersection of a horizontal plane with the observed planar feature (and therefore a horizontal line), and the strike ...
The geometrical definition of a projected area is: "the rectilinear parallel projection of a surface of any shape onto a plane". This translates into the equation: A projected = ∫ A cos β d A {\displaystyle A_{\text{projected}}=\int _{A}\cos {\beta }\,dA} where A is the original area, and β {\displaystyle \beta } is the angle between ...