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The first two values, Δ(1) and Δ(2), refer to the unit line segment and unit square respectively. For the three-dimensional case, the mean line segment length of a unit cube is also known as Robbins constant, named after David P. Robbins. This constant has a closed form, [6]
The distance from (x 0, y 0) to this line is measured along a vertical line segment of length |y 0 - (-c/b)| = |by 0 + c| / |b| in accordance with the formula. Similarly, for vertical lines (b = 0) the distance between the same point and the line is |ax 0 + c| / |a|, as measured along a horizontal line segment.
The European Berne Gauge has a width of 3,150 mm (10 ft 4 in). The minimal distance to structures on the side of the track is half of it but that is doubled again for double track lines. As the carriage can sway and bounce within the rail gauge, one adds 100 mm (3.9 in) and with a possible displacement of tracks over time one adds some 250 mm ...
Some amount of tolerance is necessarily allowed from the nominal gauge to allow for wear, etc.; this tolerance is typically greater for track limited to slower speeds, and tighter for track where higher speeds are expected (as an example, in the US the gauge is allowed to vary between 4 ft 8 in (1,420 mm) to 4 ft 10 in (1,470 mm) for track ...
L= Inclined length measured A= Inclined angle When distances are measured along the slope, the equivalent horizontal distance may be determined by applying a slope correction. The vertical slope angle of the length measured must be measured. (Refer to the figure on the other side) Thus, For gentle slopes, < %
For example, if we take the two-dimensional sphere S 2 as a subset of , the Euclidean metric on induces the straight-line metric on S 2 described above. Two more useful examples are the open interval (0, 1) and the closed interval [0, 1] thought of as subspaces of the real line.
A closed line segment includes both endpoints, while an open line segment excludes both endpoints; a half-open line segment includes exactly one of the endpoints. In geometry , a line segment is often denoted using an overline ( vinculum ) above the symbols for the two endpoints, such as in AB .
The horizontal alignment (or alinement in the United States) is done by using a predefined length of string line (such as 62-foot in the US and 20 meters in Australia [5]) to measure along the gauge side of the reference rail. It is the distance (in inches or millimeters) from the midpoint of the string line to the gauge of the reference rail.