Instruments for office work.
1. Drobyshev's ruler is used to construct coordinate grids on topographic map sheets.
2. Control measurements accurate to 0.02 mm, say, ones involved in the determination of the length of a working meter and one-decimeter graduations on levelling staffs use a control (Geneva) ruler.
3. The construction of rectangular kilometer grids and the plotting of points on a plan from their grid coordinates uses coordinatographs.
To do arithmetic, use is made of various calculation aids and also electronic computers.
Electronic computers, large and small, may rely for power both on dc and ac source in the range from 110 to 220 V and on storage batteries. Pocket calculators are particularly convenient in the practical work of a surveyor. They are small in size and use primary or secondary cells. The range of operations of pocket calculators is great: they can perform both simple arithmetic operations and complex computations. These calculators make it possible to make calculations without the use of any tables. A pocket calculator can calculate natural (Naperian) and decimal logarithms, trigonometric functions, perform involution and evolution and find reciprocals of numbers.
Pocket calculators are reliable, simple and easy to use. In addition, they are extremely accurate.
Practical work. Solving problems concerned with plane rectangular coordinates.
Sec. 2-1. Scales
The scale is the degree of the reduction of the horizontal distance of a line on the ground during the representation of it on a plan or map. Consequently, the scale is the ratio of a line length on a plan (map) to the corresponding horizontal projection of the line on the ground. For ease of use the scale is expressed as a common fraction 1/M, 1/500, 1/10 000, 1/200 000 etc. This scale type is called a numerical scale or a representative fraction (RF). The numerator of this fraction is invariably a unit of some measure on the plan and the denominator shows the number of such units contained in the horizontal distance of this line length on the ground. For example, an RF 1/500 indicates that 1 mm in a plan corresponds to 500 mm = 0.5 m on the ground; an RF 1/25000 indicates that 1 cm in the plan corresponds to 25 000 cm = 250 m of the horizontal projection of a line on the ground.
The larger is the denominator of a numerical scale, the smaller is the scale of the plan or map, and vice versa.
By using a numerical scale (RF) we can measure line length in a plan and plot on the plan sections of horizontal distances of lines on the ground. Suppose that in a 1/25 000 scale map the line length is 1.84 cm. Let us calculate the length of the corresponding horizontal distance of the line on the ground: 1.84 x 25 000 = 46 000cm = 460 m. Given the horizontal distance of a line on the ground 285,25 m and the map's scale 1/10 000, let us find the length of this line as shown in the map: 285.25 x 10 000 = 28,525 mm = 2.85 cm.
One can see that when we use a numerical scale we have to perform mathematical operations which is not quite convenient. It is better to refer to diagrams called a linear or a transverse scale.
To plot a linear scale, draw a straight line AB and measure out from one of its ends a segment of a uniform length a known as a scale base. The scale base is generally taken equal to 2 cm (Fig. 5). The linear scale with the scale base a =2 cm is termed normal. The first (left-hand) segment is subdivided into 10 equal parts. Each part of the scale base is called the least division. Then the linear scale divisions have their values written according to the RF of the particular
Linear scale
parallels and
are circles. The parallel whose plane passes through the centre of
the spheroid is called the equator.
The lines
EO =
a and
OP =
b are
called the major
and the minor semi axes of the spheroid (a is
the equator's radius, b
is the
semi axis of the Earth's rotation). The size of the earth spheroid is
determined by the length of these semi axes. The quantity
is
termed the flattening
of the spheroid.
The determination of the shape of the mathematical surface of the Earth reduces to the calculation of the lengths of these semi axes and of the amount of the flattening of an ellipsoid that will best fit the geoid and that will be appropriately accommodated inside the body of the Earth. This ellipsoid is said to be the ellipsoid of reference. The size of the earth's ellipsoid (spheroid) and its flattening have been determined many times by different scientists.
Since 1946 the practice has been to refer, for geodetic and mapping operations in the USSR, to the dimensions of the earth's ellipsoid as determined by F. N. Krasovskii, viz., a = 6 378 245 m, b = 6 356 863 m, α = 1 : 298.3. The flattening in F. N. Krasovskii's ellipsoid has been corroborated by satellite observations.
The flattening of the earth's spheroid is roughly 1 : 300. If one imagines a globe whose major semi axis is a = 300 mm then the difference between such a globe and the earth will be as small as 1 mm. In view of the smallness of the flattening it is usual practice, when solving a wide spectrum, of problems in engineering surveying, to assume that the Earth is a sphere with a radius of 6371.11 km.
Modern theory concerning the shape of the Earth has been developed by Soviet workers, notably M. S. Molodenskii.