Maximum distance to the horizon. The influence of the curvature of the earth on the measured distances and heights of points

Shape and dimensions of the earth

The general shape of the Earth, as a material body, is determined by the action of internal and external forces on its particles. If the Earth were a stationary homogeneous body and subject only to internal gravitational forces, it would have the shape of a sphere. The action of centrifugal force caused by the rotation of the Earth around its axis determines the oblateness of the Earth at the poles. Under the influence of internal and external forces, the physical (topographic) surface of the Earth forms an irregular, complex shape. At the same time, on the physical surface of the Earth there are a variety of irregularities: mountains, ridges, valleys, basins, etc. It is impossible to describe such a figure using any analytical dependencies. At the same time, to solve geodetic problems in the final form, it is necessary to be based on a certain mathematically strict figure - only then is it possible to obtain calculation formulas. Based on this, the task of determining the shape and size of the Earth is usually divided into two parts:

1) establishing the shape and size of some typical figure representing the Earth in general view;

2) study of deviations of the physical surface of the Earth from this typical figure.

It is known that 71% of the earth's surface is covered by seas and oceans, land – only 29%. The surface of the seas and oceans is characterized by the fact that at any point it is perpendicular to the plumb line, i.e. the direction of gravity (if the water is at rest). The direction of gravity can be set at any point and, accordingly, a surface perpendicular to the direction of this force can be constructed. A closed surface that at any point is perpendicular to the direction of gravity, i.e. perpendicular to the plumb line is called a level surface.

The level surface, which coincides with the average water level in the seas and oceans in their calm state and is mentally continued under the continents, is called the main (initial, zero) level surface. In geodesy, the general figure of the Earth is taken to be a figure limited by the main level surface, and such a figure is called a geoid (Fig. 1.1).

Due to the special complexity and geometric irregularity of the geoid, it is replaced by another figure - an ellipsoid, formed by rotating the ellipse around its minor axis RR 1 (Fig. 1.2). The dimensions of the ellipsoid were determined repeatedly by scientists from a number of countries. IN Russian Federation they were calculated under the guidance of Professor F.N. Krasovsky in 1940 and in 1946, by resolution of the Council of Ministers of the USSR, the following were approved: the semi-major axis A= 6,378,245 m, semi-minor axis b= 6,356,863 m, compression

The Earth's ellipsoid is oriented in the Earth's body so that its surface most closely matches the surface of the geoid. An ellipsoid with certain dimensions and a certain way oriented in the body of the Earth is called a reference ellipsoid (spheroid).

The largest deviations of the geoid from the spheroid are 100–150 m. In cases where, when solving practical problems, the figure of the Earth is taken to be a sphere, the radius of the sphere, equal in volume to the Krasovsky ellipsoid, is R= 6,371,110 m = 6371.11 km.

When solving practical problems, a spheroid or a sphere is taken as a typical figure of the Earth, and for small areas the curvature of the Earth is not taken into account at all. Such deviations are advisable, since geodetic work is simplified. But these deviations lead to distortions when displaying the physical surface of the Earth using the method that is commonly called in geodesy the method of projections.

The projection method in drawing up maps and plans is based on the fact that points on the physical surface of the Earth A, B and so on are projected with plumb lines onto a level surface (see Fig. 1.3, A,b). Points a, b and so on are called horizontal projections of the corresponding points of the physical surface. Then the position of these points on a level surface is determined using various coordinate systems, and then they can be plotted on a sheet of paper, i.e. a segment will be plotted on a sheet of paper ab, which is the horizontal projection of the segment AB. But, in order to determine the actual value of the segment from the horizontal projection AB, need to know the lengths aA And bB(see Fig. 1.3, b), i.e. distances from points A And IN to a level surface. These distances are called absolute heights of terrain points.

Thus, the task of drawing up maps and plans breaks down into two:

determining the position of horizontal projections of points;

determining the heights of terrain points.

When projecting points onto a plane, and not onto a level surface, distortions appear: instead of a segment ab there will be a segment a"b" instead of terrain point heights aA And bB will a"A And b"B(see Fig. 1.3, A,b).

So, the lengths of horizontal projections of segments and the heights of points will be different when projected onto a level surface, i.e. when taking into account the curvature of the Earth, and when projecting onto a plane, when the curvature of the Earth is not taken into account (Fig. 1.4). These differences will be observed in the projection lengths D S = t–S, at the heights of points D h = b"O – bO = b"O – R.

Rice. 1.3. Projection method

The problem with regard to taking into account the curvature of the Earth comes down to the following: taking the Earth as a ball with a radius R,it is necessary to determine for which highest value segment S the curvature of the Earth can be ignored, provided that at present the relative error is considered acceptable with the most accurate distance measurements (-1 cm per 10 km). The length distortion will be
D S = t – S = R tga - R a = R(tga – a). But since S small compared to the radius of the Earth R, then for a small angle we can take . Then . But even then . Respectively and km (rounded to the nearest 1 km).

Rice. 1.4. Scheme for solving the problem of the influence of the curvature of the Earth
on the amount of distortion in projections and heights

Consequently, a section of the spherical surface of the Earth with a diameter of 20 km can be taken as a plane, i.e. The curvature of the Earth within such an area, based on the error, can be ignored.

Distortion in the height of point D h = b"О – bО = R seca - R = R(seca – 1). Taking , we get
. At different meanings S we get:

S, km:	0,1;	0,2;	0,3;	1;	10;
D h, cm:	0,1;	0,3;	0,7;	7,8;	78,4.

In engineering and geodetic work, the permissible error is usually no more than 5 cm per 1 km, and therefore the curvature of the Earth should be taken into account at relatively small distances between points, about 0.8 km.

1.2. General concepts about maps, plans and profiles

The main difference between a plan and a map is that when depicting sections of the earth's surface on a plan, horizontal projections of the corresponding segments are drawn without taking into account the curvature of the Earth. When drawing maps, the curvature of the Earth must be taken into account.

Practical needs for accurate images of areas of the earth's surface are different. When drawing up projects for construction projects, they are significantly higher than during a general study of the area, geological surveys, etc.

It is known that, taking into account the permissible error when measuring distances D S= 1 cm per 10 km, a section of the spherical surface of the Earth with a diameter of 20 km can be taken as a plane, i.e. The curvature of the Earth for such a site can be ignored.

Accordingly, the creation of a plan can be schematically represented as follows. Directly on the ground (see Fig. 1.3, A) measure distances AB, BC..., horizontal angles b 1; b 2 ... and the angles of inclination of the lines to the horizon n 1, n 2 .... Then from the measured length of the terrain line, for example AB, go to the length of its orthogonal projection a"b" on a horizontal plane, i.e. determine the horizontal location of this line using the formula a"b" = AB cosn, and, decreasing by a certain number of times (scale), plot the segment a"b" on paper. Having calculated in a similar way the horizontal positions of other lines, a polygon is obtained on paper (reduced and similar to the polygon a"b"c"d"e"), which is the outline plan of the area ABCDE.

Plan – a reduced and similar image on a horizontal projection plane of a small area of ​​the earth's surface without taking into account the curvature of the earth.

Plans are usually divided according to content and scale. If only local objects are depicted on the plan, then such a plan is called contour (situational). If the plan additionally shows the relief, then such a plan is called topographical.

Standard plan scales are 1:500; 1:1000; 1:2000; 1:5000.

Maps are usually developed for a wide area of ​​the earth's surface, and the curvature of the earth must be taken into account. The image of a section of an ellipsoid or sphere cannot be transferred to paper without breaks. At the same time, the corresponding maps are intended to solve specific problems, for example, to determine distances, area areas, etc. When developing maps, the goal is not to completely eliminate distortions, which is impossible, but to reduce distortions and determine their values ​​mathematically so that real values ​​can be calculated from distorted images. For this purpose, map projections are used, which make it possible to depict the surface of a spheroid or sphere on a plane according to mathematical laws that provide measurements on a map.

Various requirements for maps have determined the presence of many map projections, which are divided into equiangular, equal-area and arbitrary. In equiangular (conformal) projections of a spheroid onto a plane, the angles of the depicted figures are preserved, but the scale changes when moving from point to point, which leads to distortion of figures of finite sizes. However, small areas of the map within which changes in scale are not significant can be considered and used as a plan.

In equal-area (equivalent) projections, the ratio of the areas of any figures on the spheroid and on the map is preserved, i.e. the scales of the areas are the same everywhere (with different scales in different directions).

In arbitrary projections, neither equiangularity nor equal area is observed. They are used for small-scale overview maps, as well as for special maps in cases where the maps have some specific useful property.

Map – constructed according to certain mathematical laws, a reduced and generalized image of the Earth’s surface on a plane.

Maps are usually divided according to content, purpose and scale.

In terms of content, maps can be general geographical and thematic, and in terms of purpose – universal and special. General geographic maps for universal purposes display the earth's surface showing all its main elements ( settlements, hydrography, etc.). The mathematical basis, content and design of special maps are subject to their intended purpose (marine, aviation and many other relatively narrow-purpose maps).

Based on scale, maps are conventionally divided into three types:

large-scale (1:100,000 and larger);

medium-scale (1:200,000 – 1:1,000,000);

small-scale (smaller than 1:1,000,000).

Maps, like plans, are contour and topographic. In the Russian Federation, state topographic maps are published on a scale of 1:1,000,000 – 1:10,000.

In cases where maps or plans are used to design engineering structures, to obtain optimal solution Of particular importance is the visibility of the physical surface of the Earth in any direction. For example, when designing linear structures (roads, canals, etc.) it is necessary: ​​a detailed assessment of the steepness of the slopes in individual sections of the route, a clear understanding of the soil, ground and hydrological conditions of the area through which the route passes. Profiles provide this visibility, allowing you to make informed engineering decisions.

Profile– an image on the plane of a vertical section of the earth’s surface in a given direction. To make the unevenness of the earth's surface more noticeable, the vertical scale should be chosen larger than the horizontal one (usually 10–20 times). Thus, as a rule, the profile is not similar, but a distorted image of a vertical section of the earth's surface.

Scale

Horizontal projections of segments (see Fig. 1.3, b segments ab or a"b") when drawing up maps and plans, they are depicted on paper in a reduced form. The degree of such reduction is characterized by scale.

Scale map (plan) - the ratio of the length of a line on a map (plan) to the length of the horizontal layout of the corresponding terrain line:

.

Scales can be numerical or graphic. The numerical scale is fixed in two ways.

1. As a simple fraction – the numerator is one, the denominator is the degree of reduction m, for example (or M = 1:2000).

2. In the form of a named ratio, for example, 1 cm 20 m. The expediency of such a ratio is determined by the fact that when studying the terrain on a map, it is convenient and customary to estimate the length of segments on the map in centimeters, and to represent the length of horizontal lines on the ground in meters or kilometers. To do this, the numerical scale is converted into different types of units of measurement: 1 cm of the map corresponds to such and such a number of meters (kilometers) of terrain.

Example 1. On the plan (1 cm 50 m) the distance between the points is 1.5 cm. Determine the horizontal distance between these same points on the ground.

Solution: 1.5 ´ 5000 = 7500 cm = 75 m (or 1.5 ´ 50 = 75 m).

Example 2. The horizontal distance between two points on the ground is 40 m. What will be the distance between these same points on the plan? M = 1:2000 (in 1 cm 20 m)?

Solution: see .

To avoid calculations and speed up work, use graphical scales. There are two such scales: linear and transverse.

For building linear scale choose an initial segment convenient for a given scale (usually 2 cm long). This initial segment is called the base of the scale (Fig. 1.5). The base is laid on a straight line the required number of times, the leftmost base is divided into parts (usually into 10 parts). Then the linear scale is signed based on the numerical scale for which it is constructed (in Fig. 1.5, A For M = 1:25,000). Such a linear scale makes it possible to estimate a segment in a certain way with an accuracy of 0.1 fraction of the base; an additional part of this fraction has to be estimated by eye.

To ensure the required measurement accuracy, the angle between the map plane and each leg of the measuring compass (Fig. 1.5, b)should not be less than 60°, and the length of the segment should be measured at least twice. Divergence D S, m between the measurement results there should be , Where T– the number of thousands in the denominator of the numerical scale. So, for example, when measuring segments on a map M and using a linear scale, which is usually placed behind the southern side of the frame of the map sheet, discrepancies in double measurements should not exceed 1.5 ´ 10 = 15 m.

Rice. 1.5. Linear scale

If the segment is longer than the constructed linear scale, then it is measured in parts. In this case, the discrepancy between the measurement results in the forward and reverse directions should not exceed , where P - number of meter settings when measuring a given segment.

For more accurate measurements use transverse scale, having an additional vertical construction on a linear scale (Fig. 1.6).

After the required number of scale bases have been set aside (also usually 2 cm long, then the scale is called normal), perpendiculars to the original line are restored and divided into equal segments (by m parts). If the base is divided into P parts and division points of the upper and lower bases are connected by inclined lines (transversals) as shown in Fig. 1.6, then the segment . Accordingly, the segment ef= 2CD;рq = 3cd etc. If m = n= 10, then cd = 0.01 base, i.e. such a transverse scale allows you to evaluate a segment in a certain way with an accuracy of 0.01 fraction of a base, an additional part of this fraction - by eye. Transverse scale, which has a base length of 2 cm and m = n = 10 is called the hundredth normal.

Rice. 1.6. Constructing a transverse scale

The transverse scale is engraved on metal rulers, which are called scales. Before using the scale ruler, you should evaluate the base and its shares according to the following diagram.

Let the numerical scale be 1:5000, the named ratio will be: 1 cm 50 m. If the transverse scale is normal (base 2 cm, Fig. 1.7), then the base will be 100 m; 0.1 base – 10 m; 0.01 bases – 1 m. The task of laying down a segment of a given length comes down to determining the number of bases, its tenths and hundredths and, if necessary, an eye-based determination of part of its smallest fraction. Let, for example, you want to set aside a segment d = 173.35 m, i.e. you need to take into the meter solution: 1 base +7 (0.1 base) +3 (0.01 base) and by eye place the legs of the meter between the horizontal lines 3 And 4 (see Fig. 1.7) so that the line AB cut off 0.35 of the space between these lines (segment DE). The inverse problem (determining the length of a segment taken into the meter solution) is accordingly solved in the reverse order. Having achieved alignment of the meter needles with the corresponding vertical and inclined lines so that both legs of the meter are on the same horizontal line, read the number of bases and its shares ( d BG = 235.3 m).

Rice. 1.7. Transverse scale

When conducting terrain surveys to obtain plans, the question inevitably arises: what is the smallest size of terrain objects that should be displayed on the plan? Obviously, the larger the shooting scale, the smaller the linear size of such objects will be. In order for a certain decision to be made in relation to a specific scale of the plan, the concept of scale accuracy is introduced. In this case, we proceed from the following. It has been experimentally established that it is impossible to measure the distance using a compass and a scale ruler more accurately than 0.1 mm. Accordingly, scale accuracy is understood as the length of a segment on the ground corresponding to 0.1 mm on a plan of a given scale. So, if M 1:2000, then the accuracy will be: , But d pl = 0.1 mm then d local = 2000 ´ 0.1 mm = 200 mm = 0.2 m. Consequently, on this scale (1:2000) the maximum graphic accuracy when drawing lines on the plan will be characterized by a value of 0.2 m, although the lines on the ground could be measured with higher accuracy.

It should be borne in mind that when measuring the relative position of contours on a plan, the accuracy is determined not by the graphical accuracy, but by the accuracy of the plan itself, where errors can average 0.5 mm due to the influence of errors other than graphic ones.

Practical part

I. Solve the following problems.

1. Determine the numerical scale if the horizontal location of a 50 m long terrain line on the plan is expressed by a segment of 5 cm.

2. The plan should display a building whose actual length is 15.6 m. Determine the length of the building on the plan in mm.

II. Construct a linear scale by drawing a line 8 cm long (see Fig. 1.5, A). Having chosen a scale base 2 cm long, set aside 4 bases, divide the leftmost base into 10 parts, digitize for three scales: ; ; .

III. Solve the following problems.

1. Lay out a segment 144 m long on paper in the three indicated scales.

2. Using the linear scale of the training map, measure the horizontal length of the three segments. Evaluate the measurement accuracy using the dependence. Here T– the number of thousands in the denominator of the numerical scale.

IV. Using a scale ruler, solve the following problems.

Put down the length of the terrain lines on paper, recording the results of the exercise in the table. 1.1.

OBJECTS FALL EXACTLY DOWN WITHOUT DISPLACEMENT

If the earth beneath us actually rotated in an easterly direction, as the heliocentric model suggests, then cannonballs fired vertically should land noticeably further west. In fact, whenever this experiment was carried out, cannonballs fired in a perfectly vertical line, illuminated by a fire cord, reached the top in an average of 14 seconds and fell back within 14 seconds no more than 2 feet (0.6 m). away from the gun, or sometimes straight back into the barrel! If the Earth actually rotated at 600-700 mph (965-1120 km/h) in the mid-latitudes of England and America, where the experiments were conducted, cannonballs should fall as much as 8,400 ft (2.6 km) or so miles and a half behind the gun!

PLANES FLY THE SAME IN ALL DIRECTIONS AND WITHOUT CORRECTION FOR THE CURVATURE AND ROTATION OF THE EARTH

If the Earth beneath our feet was spinning at several hundred miles per hour, then helicopter and hot air balloon pilots would simply have to fly straight up, hover, and wait for their destination to reach them! This has never happened in the history of aeronautics.

For example, if the Earth and its lower atmosphere were supposedly rotating together in an easterly direction at 1,038 mph (1,670 km/h) at the equator, then airplane pilots would have to accelerate an additional 1,038 mph when flying to the West! And pilots heading north and south must, of necessity, set diagonal headings to compensate! But since no compensation is required, except in the imagination of astronomers, it follows that the Earth is motionless.

CLOUDS AND WIND MOVE INDEPENDENT OF THE HIGH SPEED OF THE EARTH'S ROTATION

If the Earth and atmosphere are constantly rotating in an easterly direction at 1000 miles per hour, then how clouds, wind and weather patterns randomly and unpredictably move into different sides, often heading in opposite directions at the same time? Why can we feel a slight westerly breeze, but not the incredible supposed 1000 mph eastward rotation of the Earth!? And how is this magic Velcro-gravity strong enough to pull miles alone earth's atmosphere, but at the same time so weak that it allows small bugs, birds, clouds and airplanes to move freely at the same pace in any direction?

THE WATER IS FLAT EVERYWHERE, DESPITE THE CURVATION OF THE EARTH

If we lived on a rotating spherical Earth, then every pond, lake, swamp, canal and other places with standing water would have a small arc or semicircle expanding from the center downwards.

In Cambridge, England, there is a 20 mile canal called "Old Bedford" that runs in a straight line through the Fenlands known as Bedford Plain. The water is not interrupted by gates and sluices and remains stationary, making it ideal for determining whether curvature actually exists. In the second half of the 19th century, Dr. Samuel Rowbotham, the famous “flat-earther” and author of the wonderful book “The Earth Is Not a Globe! Experimental study of the true shape of the Earth: proof that it is a plane, without axial or orbital motion; and the only material world in the Universe!”, went to Bedford Plain and conducted a series of experiments to determine whether the surface of standing water was flat or convex.
The 6 mile (9.6 km) surface did not show any dip or curvature down from the line of sight. But if the earth is a sphere, then the surface of water 6 miles long would have to be 6 feet higher at the center than at its ends. From this experiment it follows that the surface of standing water is not convex and, therefore, the Earth is not a sphere!

WATER DOES NOT SPLIT DUE TO THE HUGE ROTATION OF THE EARTH AND CENTRIFUGAL FORCE
“If the Earth were a ball, rotating and dashingly flying in “space” at a speed of “one hundred miles in 5 seconds,” then the waters of the seas and oceans could not, according to any laws, float on the surface. To suggest that they could be held in these circumstances is an outrage upon human understanding and trust! But if the Earth - which is an inhabited landmass - were recognized as "protruding from the water and standing in the water" from the "immense depth" which is surrounded by a boundary of ice, we can throw that statement back in the teeth of those who made it and wave before them the flag of reason and common sense, with signed proof that the earth is not a sphere." - William Carpenter

THE LONGEST RIVERS IN THE WORLD HAVE NO CHANGES IN WATER LEVEL DUE TO THE CURVATION OF THE EARTH

In one part of its long route, the great Nile River flows for a thousand miles with a drop of only 1 foot (30 cm). This feat would be completely impossible if the Earth had a spherical curve. Many other rivers, including the Congo in West Africa, the Amazon in South America and Mississippi in North America, all of them floating thousands of miles in directions completely inconsistent with the supposed sphericity of the Earth

RIVERS FLOW IN ALL DIRECTIONS, NOT UP TO BOTTOM

“There are rivers that flow east, west, north and south, that is, rivers flow in all directions on the surface of the Earth at the same time. If the Earth were a ball, then some would flow uphill and others downhill, meaning what "up" and "down" actually mean in nature, no matter what shape they take. But since rivers do not flow uphill, and the theory of the sphericity of the earth requires this, this proves that the Earth is not a sphere

ALWAYS A FLAT HORIZON

Whether at sea level, at the top of Mount Everest, or flying hundreds of thousands of feet in the air, the horizontal line of the horizon rises upward to be at eye level and remains perfectly straight. You can check it out yourself at the beach or hilltop, at large field or the desert, on board a hot air balloon or helicopter; you will see that the panoramic horizon will rise with you and remain absolutely horizontal everywhere. If the Earth were actually a big ball, the horizon would have to drop as you rise, not rising to your eye level, but moving away from each end of the periphery of your vision, not remaining level along its entire length.

If the Earth were actually a large ball 25,000 miles (40,233 km) in circumference, then the horizon would be noticeably curved even at sea level, and everything on or tending towards the horizon would appear slightly tilted from our perspective. Distant buildings along the skyline would look like the Leaning Tower of Pisa falling away from the observer. A balloon, having risen and then gradually moving away from you, on a spherical Earth would appear to be slowly and constantly leaning back more and more as it recedes; the bottom of the basket gradually comes into view, while the top of the balloon disappears from view. In reality, however, buildings, balloons, trees, people, anything and everything remains at the same angle relative to the surface or horizon no matter how far away the observer is.

“Wide areas show an absolutely flat surface, from the Carpathians to the Urals, a distance of 1500 (2414 km) miles, there is only a slight rise. South of the Baltic the country is so flat that the prevailing north wind will drive water from the Szczecin Bay to the mouth of the Odra, and will reverse the river 30 or 40 miles (48-64km). The plains of Venezuela and New Granada in South America, located on the left side of the Orinoco River, are called Llanos or plain fields. Often over a distance of 270 square miles (700 sq km) the surface does not change a foot. The Amazon descends 12 feet (3.5m) only in the last 700 miles (1126km) of its course; La Plata descends only one-thirty-third of an inch per mile (0.08 cm/1.6 km),” Rev. T. Milner, “Atlas of Physical Geography”

The lighthouse at Port Nicholson, New Zealand, is 420 feet (128m) above sea level and visible from 35 miles (56km), but that means it must be 220 feet (67m) below the horizon. The Jogero Lighthouse in Norway is 154 feet (47m) above sea level and visible from 28 statute miles (46km), which means it would be 230 feet below the horizon. The lighthouse at Madras, on the Esplanade, is 132 feet (40m) high and visible from 28 miles (46km), when it should be 250 feet (76m) below the line of sight. The 207-foot (63m) high Cordonin lighthouse on the west coast of 47 France is visible from 31 miles (50km), which would be 280 feet (85m) below the line of sight. The lighthouse at Cape Bonavista, Newfoundland is 150 feet (46m) above sea level and visible from 35 miles (56km), when it should be 491 feet (150m) below the horizon. The lighthouse spire of St. Botolph's Church in Boston is 290 feet (88m) high, visible from a distance of more than 40 miles (64km), when it should be hidden as much as 800 feet (244m) below the horizon!

CHANNELS AND RAILWAYS ARE DESIGNED WITHOUT CONSIDERATION OF THE EARTH'S CURVATURE

Surveyors, engineers and architects never take into account the supposed curvature of the Earth in their projects, which is yet another proof that the world is a plane and not a planet. Canals and railways, for example, are always laid horizontally, often for hundreds of miles, without taking into account any curvature.
Engineer W. Winkler, in his “Earth Survey” of October 1893, wrote regarding the supposed curvature of the Earth: “As an engineer with 52 years of experience, I have seen that this absurd assumption is used only in school textbooks. Not a single engineer even thinks of taking into account attention to things of this kind. I have designed many miles of railroads and many more canals, and it never even occurred to me to allow for surface curvature, much less take it into account. Allowing for curvature means - 8 inches in the first mile of the canal, then increasing according to the indicator , being the square of the distance in miles; thus a small shipping canal, say 30 miles in length, will, by the above rule, have a setback for curvature of 600 feet (183m).Think about this, and please believe that the engineers not such fools. Nothing like that is taken into account. We don't think about taking into account the 600 foot curvature, for the line railway or a canal 30 miles (965 km) long, more than we spend our time trying to embrace the immensity."

AIRPLANES FLY ONLY AT EVEN, EQUAL ALTITUDES, WITHOUT CORRECTION FOR EARTH CURVATURE

If the Earth were a sphere, airplane pilots would have to constantly adjust their altitude to avoid flying straight into "outer space!" If the Earth were truly a sphere 25,000 miles (40,233 km) in circumference with a tilt of 8 inches per mile squared, then a pilot wishing to maintain the same altitude at a typical speed of 500 mph (804 km/h) would have to continually nose-down and descend at 2777 feet (846m) every minute! Otherwise, without adjustment, after an hour the pilot will be 166,666 feet (51 km) higher than expected! An airplane flying at a normal altitude of 35,000 feet (10km), wanting to maintain that altitude at the top edge of the so-called "troposphere", would in one hour find itself more than 200,000 feet (61km) 57 in the "mesosphere", and the further it will fly, the longer the trajectory will be. I've spoken to several pilots and no compensation is being made for the supposed curvature of the Earth. When pilots reach the required altitude, their artificial horizon indicator remains level, as does their heading; no required 2777 feet per minute (846 km/min) of incline is ever taken into account.

ANTARCTICA AND ARTICA HAVE DIFFERENT CLIMATES

If the Earth really were a sphere, then the polar regions of the Arctic and Antarctic at the corresponding latitudes north and south of the equator would have similar conditions and features: similar temperatures, seasonal changes, duration daylight hours, features of flora and fauna. In fact, comparable latitudes north and south of the equator in the Arctic and Antarctic regions are very different in many ways. "If the earth is a sphere, according to popular opinion, then the same amount of heat and cold, summer and winter, should be present at the corresponding latitudes north and south of the equator. The number of plants and animals would be the same, and the general conditions would be the same. Everything is as follows on the contrary, which refutes the assumption of sphericity. Large contrasts between areas at the same latitudes north and south of the equator are a strong argument against the accepted doctrine of the sphericity of the Earth

Horizon visibility range

The line observed in the sea, along which the sea seems to connect with the sky, is called the visible horizon of the observer.

If the observer's eye is at a height eat above sea level (i.e. A rice. 2.13), then the line of sight running tangentially to the earth’s surface defines a small circle on the earth’s surface ahh, radius D.

Rice. 2.13. Horizon visibility range

This would be true if the Earth were not surrounded by an atmosphere.

If we take the Earth as a sphere and exclude the influence of the atmosphere, then from a right triangle OAa follows: OA=R+e

Since the value is extremely small ( For e = 50m at R = 6371km – 0,000004 ), then we finally have:

Under the influence of earthly refraction, as a result of the refraction of the visual ray in the atmosphere, the observer sees the horizon further (in a circle bb).

(2.7)

Where X– coefficient of terrestrial refraction (» 0.16).

If we take the range of the visible horizon D e in miles, and the height of the observer's eye above sea level ( eat) in meters and substitute the value of the Earth's radius ( R=3437,7 miles = 6371 km), then we finally obtain the formula for calculating the range of the visible horizon

(2.8)

For example:1) e = 4 m D e = 4,16 miles; 2) e = 9 m D e = 6,24 miles;

3) e = 16 m D e = 8,32 miles; 4) e = 25 m D e = 10,4 miles.

Using formula (2.8), table No. 22 “MT-75” (p. 248) and table No. 2.1 “MT-2000” (p. 255) were compiled according to ( eat) from 0.25 m¸ 5100 m. (see table 2.2)

Visibility range of landmarks at sea

If an observer whose eye height is at the height eat above sea level (i.e. A rice. 2.14), observes the horizon line (i.e. IN) on distance D e(miles), then, by analogy, and from a reference point (i.e. B), whose height above sea level h M, visible horizon (i.e. IN) observed at a distance D h(miles).

Rice. 2.14. Visibility range of landmarks at sea

From Fig. 2.14 it is obvious that the visibility range of an object (landmark) having a height above sea level h M, from the height of the observer's eye above sea level eat will be expressed by the formula:

Formula (2.9) is solved using table 22 “MT-75” p. 248 or table 2.3 “MT-2000” (p. 256).

For example: e= 4 m, h= 30 m, D P = ?

Solution: For e= 4 m ® D e= 4.2 miles;

For h= 30 m® D h= 11.4 miles.

D P= D e + D h= 4,2 + 11,4 = 15.6 miles.

Rice. 2.15. Nomogram 2.4. "MT-2000"

Formula (2.9) can also be solved using Applications 6 to "MT-75" or nomogram 2.4 “MT-2000” (p. 257) ® fig. 2.15.

For example: e= 8 m, h= 30 m, D P = ?

Solution: Values e= 8 m (right scale) and h= 30 m (left scale) connect with a straight line. The point of intersection of this line with the average scale ( D P) and will give us the desired value 17.3 miles. ( see table 2.3 ).

Geographic visibility range of objects (from Table 2.3. “MT-2000”)

Note:

The height of the navigational landmark above sea level is selected from the navigational guide for navigation "Lights and Signs" ("Lights").

2.6.3. Visibility range of the landmark light shown on the map (Fig. 2.16)

Rice. 2.16. Lighthouse light visibility ranges shown

On navigation sea charts and in navigation manuals, the visibility range of the landmark light is given for the height of the observer's eye above sea level e= 5 m, i.e.:

If the actual height of the observer’s eye above sea level differs from 5 m, then to determine the visibility range of the landmark light it is necessary to add to the range shown on the map (in the manual) (if e> 5 m), or subtract (if e < 5 м) поправку к дальности видимости огня ориентира (DD K), shown on the map for the height of the eye.

(2.11)

(2.12)

For example: D K= 20 miles, e= 9 m.

D ABOUT = 20,0+1,54=21,54miles

Then: DABOUT = D K + ∆ D TO = 20.0+1.54 =21.54 miles

Answer: D O= 21.54 miles.

Problems for calculating visibility ranges

A) Visible horizon ( D e) and landmark ( D P)

B) Opening of the lighthouse fire

conclusions

1. The main ones for the observer are:

A) plane:

Plane of the observer's true horizon (PLI);

Plane of the true meridian of the observer (PL).

The plane of the first vertical of the observer;

b) lines:

The plumb line (normal) of the observer,

Observer true meridian line ® noon line N-S;

Line E-W.

2. Direction counting systems are:

Circular (0°¸360°);

Semicircular (0°¸180°);

Quarter note (0°¸90°).

3. Any direction on the Earth's surface can be measured by an angle in the plane of the true horizon, taking the observer's true meridian line as the origin.

4. True directions (IR, IP) are determined on the ship relative to the northern part of the observer’s true meridian, and CU (heading angle) - relative to the bow of the longitudinal axis of the ship.

5. Range of the observer's visible horizon ( D e) is calculated using the formula:

.

6. The visibility range of a navigation landmark (in good visibility during the day) is calculated using the formula:

7. Visibility range of the navigation landmark light, according to its range ( D K), shown on the map, is calculated using the formula:

, Where .

Rice. 4 Basic lines and planes of the observer

For orientation at sea, a system of conventional lines and planes of the observer has been adopted. In Fig. 4 shows a globe on the surface of which at a point M the observer is located. His eye is at the point A. Letter e indicates the height of the observer's eye above sea level. The line ZMn drawn through the observer's place and the center of the globe is called a plumb or vertical line. All planes drawn through this line are called vertical, and perpendicular to it - horizontal. The horizontal plane НН/ passing through the observer's eye is called true horizon plane. The vertical plane VV / passing through the observer's place M and the earth's axis is called the plane of the true meridian. At the intersection of this plane with the surface of the Earth, a big circleРnQPsQ / , called observer's true meridian. The straight line obtained from the intersection of the plane of the true horizon with the plane of the true meridian is called true meridian line or the midday line N-S. This line determines the direction to the northern and southern points of the horizon. The vertical plane FF / perpendicular to the plane of the true meridian is called plane of the first vertical. At the intersection with the plane of the true horizon, it forms line E-W, perpendicular to the N-S line and defining the directions to the eastern and western points of the horizon. Lines N-S and E-W divide the plane of the true horizon into quarters: NE, SE, SW and NW.

Fig.5. Horizon visibility range

In the open sea, the observer sees a water surface around the ship, limited by a small circle CC1 (Fig. 5). This circle is called the visible horizon. The distance De from the position of the vessel M to the visible horizon line CC 1 is called range of the visible horizon. The theoretical range of the visible horizon Dt (segment AB) is always less than its actual range De. This is explained by the fact that, due to the different density of atmospheric layers in height, a ray of light does not propagate in it rectilinearly, but along an AC curve. As a result, the observer can additionally see some part of the water surface located behind the line of the theoretical visible horizon and limited by the small circle CC 1. This circle is the line of the observer's visible horizon. The phenomenon of refraction of light rays in the atmosphere is called terrestrial refraction. Refraction depends on atmospheric pressure, temperature and humidity. In the same place on Earth, refraction can change even over the course of one day. Therefore, when calculating, the average refraction value is taken. Formula for determining the range of the visible horizon:

As a result of refraction, the observer sees the horizon line in the direction AC / (Fig. 5), tangent to the arc AC. This line is raised at an angle r above the direct ray AB. Corner r also called terrestrial refraction. Corner d between the plane of the true horizon NN / and the direction to the visible horizon is called inclination of the visible horizon.

VISIBILITY RANGE OF OBJECTS AND LIGHTS. The range of the visible horizon allows one to judge the visibility of objects located at water level. If an object has a certain height h above sea level, then an observer can detect it at a distance:

On nautical charts and in navigation manuals, the pre-calculated visibility range of lighthouse lights is given. Dk from an observer's eye height of 5 m. From such a height De equals 4.7 miles. At e, different from 5 m, an amendment should be made. Its value is equal to:

Then the visibility range of the lighthouse Dn is equal to:

The visibility range of objects calculated using this formula is called geometric or geographic. The calculated results correspond to a certain average state of the atmosphere during the daytime. When there is darkness, rain, snow or foggy weather, the visibility of objects is naturally reduced. On the contrary, under a certain state of the atmosphere, refraction can be very large, as a result of which the visibility range of objects turns out to be much greater than calculated.

Distance of the visible horizon. Table 22 MT-75:

The table is calculated using the formula:

De = 2.0809 ,

Entering the table 22 MT-75 with item height h above sea level, get the visibility range of this object from sea level. If we add to the obtained range the range of the visible horizon, found in the same table according to the height of the observer’s eye e above sea level, then the sum of these ranges will be the visibility range of the object, without taking into account the transparency of the atmosphere.

To obtain the range of the radar horizon Dp accepted selected from the table. 22 increase the range of the visible horizon by 15%, then Dp=2.3930 . This formula is valid for standard atmospheric conditions: pressure 760 mm, temperature +15°C, temperature gradient - 0.0065 degrees per meter, relative humidity, constant with altitude, 60%. Any deviation from the accepted standard state of the atmosphere will cause a partial change in the range of the radar horizon. In addition, this range, i.e. the distance from which reflected signals can be visible on the radar screen, largely depends on the individual characteristics of the radar and the reflective properties of the object. For these reasons, use the coefficient of 1.15 and the data in table. 22 should be used with caution.

The sum of the ranges of the radar horizon of the antenna Ld and the observed object of height A will represent the maximum distance from which the reflected signal can return.

Example 1. Determine the detection range of a beacon with height h=42 m from sea level from the height of the observer's eye e=15.5 m.
Solution. From the table 22 choose:
for h = 42 m..... . Dh= 13.5 miles;
For e= 15.5 m. . . . . . De= 8.2 miles,
therefore, the detection range of the beacon
Dp = Dh+De = 21.7 miles.

The visibility range of an object can also be determined by the nomogram placed on the insert (Appendix 6). MT-75

Example 2. Find the radar range of an object with height h=122 m, if the effective height of the radar antenna is Hd = 18.3 m above sea level.
Solution. From the table 22 choose the visibility range of the object and antenna from sea level, respectively, 23.0 and 8.9 miles. Summing these ranges and multiplying them by a factor of 1.15, the object is likely to be detected from a distance of 36.7 miles under standard atmospheric conditions.

When geodetic work is carried out on small areas of terrain, the level surface is taken as a horizontal plane. Such a replacement entails some distortions in the lengths of lines and heights of points.
Let us consider at what size of the area these distortions can be neglected. Let us assume that the level surface is the surface of a ball of radius R (Fig. 1.2). Let us replace the section of the ball AoBoCo with the horizontal plane ABC, tangent to the ball in the center of the section at point B. The distance between points B (Bo) and Co is equal to r, the central angle corresponding to this arc is denoted by a, the tangent segment

BC = t, then in the horizontal distance between points B (Bo) and Co there will be an error Ad = t - d. From Fig. 1.2 we find t = R tga and d = R a, where the angle a is expressed in radians a = d / R, then A d = R(tga -a) and since the value of d is insignificant compared to R, the angle is so small,
O

that approximately we can take tga -a = a /3. Applying the formula for determining angle a, we finally obtain: A d = R- a /3 = d /3R. At d = 10 km and R = 6371 km, the error in determining the distance when replacing a spherical surface with a plane will be 1 cm. Taking into account the real accuracy with which measurements are made on the ground during geodetic work, we can assume that in areas with a radius of 2025 km the error from replacing a level surface plane has no practical significance. The situation is different with the influence of the curvature of the Earth on the heights of points. From the right triangle OBC

(1.2)
where
(1.3) where p is a segment of the vertical line ССО, expressing the influence of the curvature of the Earth on the heights of point C. Since the obtained value of p is very small compared to R, this value can be neglected in the denominator of the resulting formula. Then we get

(1.4)
For various distances l, we determine corrections to the heights of terrain points, the values ​​of which are presented in Table. 1.1, from which it is clear that the influence of the curvature of the Earth on the heights of points is already felt at a distance of 0.3 km. This must be taken into account when carrying out geodetic work.
Table 1.1
Errors in measuring point heights at different distances

l, km	0,3	0,5	1,0	2,0	5,0	10,0	20,0
R, m	0,01	0,02	0,08	0,31	1,96	7,85	33,40