Portraits of the Earth

10

PORTRAITS
EARTH

Portraits, models, maps
Schemes, drawings, plans, fomo-
graphies, portraits, executed
made by artists and horses
constructors - all this is fashion-
whether the subject they depict is
mov (“oricenals”). There is a model
a kind of image substitute
his desired subject. Every fashion
l, weakening us with information about
portrayed as his “origin”
le", at the same time, a number of drawings are “origin-
la” does not reproduce, omits them, or, as in
they say, abstracts himself from them. Yes, this is understandable:
“complete” model, which is an exact copy of the “original”
zienala", would not have, from the point of view of lightness
studying has no advantages in comparison with it.
For the sake of simplifying the process of cognition, one has to sacrifice its completeness. As a rule, a flat image is
the strangeness of the object distorts the form of the original. To the number
The most important objects include the globe, ways of illustrating
the creation of “portraits” of whom, i.e. simply geodra-
physical maps, this article is devoted to. Ways
these (depending on the purpose of the card) due to
weighs a lot. But with all their diversity
all of them in one way or another include varbi-
there are two main “motives”: design-
rotation and development onto a plane.

V. N. Berezin,
M. L. Smolyansky

11

“Bust of the Earth” - globe

Before moving on to various ways of depicting the earth’s surface on a map, or, as they are called, cartographic projections, let’s say a few words about the “sculptural portrait” of the Earth, which can be considered an intermediate stage for obtaining other “portraits” - in other words, about the well-known globe.

The fact that our planet is close in shape to a sphere was quite clearly understood by ancient Greek scientists, who were based mainly on the observations of sailors and travelers. Of course, these observations themselves only allowed us to assert that the part of the Earth known to the Greeks, that is, the parts of Europe, Asia and Africa immediately adjacent to the Mediterranean and Black Seas and the eastern part of the Atlantic Ocean, is a bulge. It is not surprising that, given the highly developed culture of abstract thinking among the Greeks, these rather indirect data allowed scientists of that time to put forward a bold hypothesis about the closed surface of the Earth. The shape of the earth’s body on the surface of the Moon during an eclipse (the realization that this is the shadow of the Earth was in itself an achievement that is difficult to overestimate), and then speculative arguments,

entailing in one form or another the idea of ​​symmetry, is led to the conviction that the Earth is a sphere.

How the Greeks imagine the “face” of the Earth can be judged from the oldest geographical map that has come down to us (in later Latin copies), compiled by the famous Greek scientist Claudius Ptolemy (about 150 BC). Ptolemy, using the coordinate grid introduced even before him by Hipparchus, tries to expand the spherical surface of the Earth onto a plane (Fig. 1). The meridans are depicted as two-dimensional broken lines, and the latitudinal circles are represented by arcs of circles, the upper of which corresponds approximately to the polar circle, the lower one corresponds to the equator, and the one separating the truncated temperate cone from the cylindrical equatorial one (latitude 0°) corresponds to the Tropic of Cancer. A constant scale is maintained along the meridans (concentric arcs of parallels are separated from each other every 10° by equal intervals). Understanding that south of the equator the parallels must contract again, and therefore the lower half of the scan must be separated from the upper half depicted by him by a cut. Ptolemy is not trying to depict the Southern Hemisphere. He had to continue Africa from the south and Asia from the east with the hypothetical Terra incognita (“Unknown Land”), south of the Indian Ocean turning into the “Starry Land” (Terra australia; this name was subsequently given to the fifth continent discovered approximately in the same region). The Ptolemy map is made, as modern cartographers would say, in an “arbitrary projection.” All the more remarkable is the accuracy with which the coordinates (especially latitudes) of many cities and ports are plotted on it.

Ptolemy also gives another projection, from his point of view more beautiful and correct, but less convenient for practical constructions.

12

Fig. 2.

Let us give the floor to Polemaeus himself *): “… one could imagine in the form of a straight line only the middle meridian, with the plane of which the axis of vision coincides. The meridians, located on both sides of it, seem to be facing it with their concavity, returning as they move away from it (Fig. 2).

The last drawing conveys this while maintaining proper alignment of the curves. In addition, the mutual proportionality of each parallel conveys as accurately as possible the true relationship of one parallel to another…

Thanks to all this, the second method has, perhaps, an advantage over the first, inferior to it, however, in the ease of execution of the drawing. There it was possible to draw only one parallel, divide it into divisions and plot any point on the map by applying and moving a ruler. Here this convenience is not available, since the linear meridians approach the average meridian by bending, and, in addition, it is necessary to draw all the circles, and the position relative to the surrounding sides of the square of the points falling on the gaps of the grid must be determined from the points plotted on the map by calculation. Pos_

*) “Ancient Geography”, compiled by prof. M. S. Bodnarsky, Geography, 1953.

Fig. 3.

Therefore, although I personally in this matter and in this section prefer what is better and more difficult to what is worse and easier, it is still necessary to adhere to both established methods, bearing in mind people who are attracted by difference to the easier…”

The manufacture of the first known globe apparently dates back to approximately the same time (mid-2nd century BC). This globe, made by the philosopher Crates of Malos, has not survived and is known from later descriptions, the schematic nature of which makes it necessary to make an inevitable allowance when assessing its accuracy. The knowledge of this era was fully consistent with the image on the globe of the continent of Oecumene (“Inhabited”), consisting of Eurasia and Africa (Crates had reasons for this, of course, since the Susa crossing was cut by a canal, indeed, somewhat later). The proportions of the usual outlines of coastlines (especially in the peripheral areas) are noticeably disrupted, but the relative positions of such characteristic map elements as the Mediterranean, Black, Red and White Seas, the Persian Gulf, the Arabian Peninsula and the South European peninsulas are generally conveyed correctly.

Crates’s globe (as opposed to the much more detailed map of Pto-

13

Fig. 4.

Lemeya) played, very indo, mainly a demonstration role. The appearance on it of such fantastic details as a continent symmetrical to the Ecumene relative to the equator, supposedly inhabited by “Anteks”, and continents in the north and south of the opposite hemisphere (the habitats of the “Perizki” and “Antipodes”) can, please, be explained by an unconscious desire for “symmetry”, which, by the way, is very characteristic of modern science. (Fig.3)

A globe has survived to this day (see the intro to the article), made by a resident of Nuremberg, Martin Biheim, much later - in the same 1492, which, in honor of Columbus’s discovery of America, began to be considered the conventional border between medieval and modern times. On Biheim’s polymeter globe and especially on the map (see Fig. 4), made with the help of this globe, it is clearly visible that geographers of that time exceeded the size of Asia, and the route from Spain to its eastern shores (China and India) was therefore estimated to be almost half as small as in reality.

A mistake, of course, is a mistake, but objectively, these ideas certainly contributed to the optimism of Columbus, inspiring him to a bold plan to reach the eastern agricultural lands.

regs of Asia, moving their caravels all the time to the west.

From “model” to “portrait”
But now that globes are doing a wave of good things, what prevents you from using them in all cases when you need a map? Or are they also not free from distortions - do they not take into account, for example, whether the Earth is flattened near the stripes or its mountainous terrain? No, this is not very significant here: the difference between the polar and equatorial radii of the Earth is no more than 0.3% of either of them, and the height of the highest mountain peak - Chomolungma - is about 0.1% of the Earth’s radius (and of approximately the same order - larger than the depression at the bottom of the oceans), so within the limits of the dimensional accuracy of manufacturing, these factors are generally practically invisible; It is much more convenient to depict the relief on a map or globe using so-called “level lines”, coloring the designs between them in different colors for clarity. The essence of the matter is very simple: a globe is less convenient than a map - passports are not weakened by busts and bas-reliefs of the owners!

But, having a globe, you can think about “expanding” it - at least approximately. To make a portrait of the Earth, we need a miniature model of it.

Having understood how the globe “unfolds” into a map, we will come, by the way, to understanding an equally interesting inverse problem: how the true appearance of the original—the Earth—is recreated using flat images of pieces of the earth’s surface obtained as a result of aerial photography.

Let us now recall how surface developments are made. In a polygon, for example, you can cut almost all the edges, leaving only one uncut edge at each face so that the surface does not fall into pieces. Cylinders and cones are even easier to deploy. You will receive a model of the cylinder (more precisely, the cylindrical surface),

14

Fig. 5.

The slade is two opposite sides of a straight piece of paper, and the cone model is just a paper function. It is enough to cut any of these models according to the image and place the resulting piece of paper on the table - and the development is ready! The lengths of the lines drawn on the original surfaces are also preserved on the developments (except that the line - if it intersected the generatrix along which the cut was made - will unfold into two parts).

But our Earth is a ball. As Leonhard Euler proved in 1777, no part of a spherical surface (no matter how angularly small!) can be turned onto a plane while maintaining all distances. The idea behind the Euler proof is very simple. Whatever small part of the spherical surface of the globe we take, we can isolate a round “cap” inside it. Due to the complete equality of each point of the sphere (each diameter is an axis of symmetry), we can consider the center of this “cap” to be a strip, and the border to be a parallel (latitudinal circle). The distances between the points of the cap must, of course, be changed along the surface of the cap. In part, the distance from the shelf to any point on the cap’s border will be equal to the length $r$ of the corresponding meridian arc (Fig. 5). This means that on a development that does not distort the dimensions, this parallel should be depicted on the plane by a circle of radius $r$ and, therefore, has a length $2\pi r$. But it’s easy to prove (do it yourself!) that in fact the length of our circle is less than $2\pi r$. The resulting contradiction proves the impossibility of accurately developing onto a plane even the smallest cap in the middle of the sphere, and in general the lower part of the spherical surface.

“Microcards” - plans

From the proof of Euler’s theorem (and simply from the messages of “common sense”) it is clear that when we are talking about very small areas of the earth’s surface, replacing such small “caps” with flat circles distorts the dimensions transferred to the map, the smaller the smaller the diameter of the circle. Ultimately it comes down to what kind of image accuracy we really need. Estimating the margin of error is very simple (see problem on page 25) and allows you to answer this question in a reasonable way in each specific case.

Let’s face it, the architect planning the development of the city is not at all interested in the shape of this globe - some ravine worries him, of course, much more than such “universal” problems! The person conducting exploration of a particular deposit cares little about the roundness of the Earth and geology; all sorts of local slopes and other “incorrectities” of the decision are much more important to him, which can affect the increasing choice of well drilling sites. In short, when it comes to compiling maps of small areas, we can safely become like the ancient Egyptians, who five thousand years ago made maps (as these “micromaps” are commonly called) of sections of the earth’s surface, completely ignoring the convexity of the Earth as a whole. The only requirement for the plan is

15

Fig. 6.

is that for arbitrary points $A, B, C$ and $D$ on the surface of the depicted area of ​​the Earth and their images on the plan - points $A', B', C'$ and $D'$ with sufficient accuracy the equality $\frac{A'B'}{AB} = \frac{C'D'}{CD}$ will be fulfilled above.

The magnitude of the ratio $\frac{A'B'}{AB}$ is naturally called the image scale of the area $AB$; Thus, our condition expresses the principle of constancy of scale. From Euler’s theorem it follows, of course, that a piece of the earth’s surface cannot be depicted on a strictly constant scale. Therefore, in each specific case, an acceptable value of the permissible error is established and the plan is considered “correct” if the error does not exceed the established norm.

Very simple and visual geometric distances (see Fig. 6) convince us that all meridians (lines of constant longitude) are equal to each other, and all parallels (lines of constant latitude) decrease from the equator to the poles from $2\pi R_{земли}$ to zero (“the 90th parallel” degenerates into a point - a pole). It immediately follows that the meridian segments between adjacent parallels on the globe are equal, and the parallel segments between adjacent meridians decrease in the direction from the equator to the poles.

It is also easy to check that all angles between parallels and meridians at their points of intersection (i.e., angles between tangents to these curves at these points) are right.

A little more history

Do these well-known geographical occurrences of the (XV century) cartography, if developed, was quite spontaneous. The surviving medieval maps often even represented a significant step back from the Ptolemy map described above. An example is the so-called Hereford map (1280), a rather fantastic image of a round and flat Earth on which no special comments are required and is only of historical interest (Fig. 7).

For seafarers, however, special nautical charts (Fig. 8) were already compiled at that time - the so-called portulans (or portolans), equipped together with a coordinate grid (whether they used it and what they used at all, in addition to purely empirical techniques, compiled portulans - it’s difficult to say exactly) with the so-called “wind rose” - the directions of the “lines of rhumbs” corresponding to the divisions of the cartushka (round scale) of the magnetic compass, with

17

changing in Europe since the 12th century (Fig. 9). They made it possible to plot the ship’s course on the map without any additional calculations, using the so-called navigational ruler (which, by the way, has survived to this day) straight through the ship’s position point, parallel to the corresponding rumba line (Fig. 10).

Naturally, the sailors were interested in ensuring that the direction of the course, deviating from the meridian by a given number of degrees, was the same at any point on the map. For this it is necessary (although, of course, not enough) that the meri dianas and parallels are depicted directly on the map. Gerard Mercator, whose name is often associated with the beginning of scientific cartography, was the first to satisfy this requirement in combination with decent image accuracy. The Mercator map of the world, published on 18 separate sheets in 1569, has not lost its significance to this day. (figure on page 18 shows a fragment of this map.)

The rapid progress of geographical knowledge in general and in cartography in particular was characteristic of the 16th-17th centuries: the invention of the mathematical clock (X. Huygens, 1675),

19

Fig. 12.

and then the chronometer made it possible to determine with great accuracy the geographical longitude of a point (latitude was well defined by the ancient Greeks), various projection systems appeared, adapted to different needs and based on different principles. A map dates back to that time (Fig. 11), depicting the entire surface of the Earth in the form of two “hemispheres” (de Wit, 1700). Below we will get acquainted with several relatively simple cartographic processes. Of course, these are only individual riches of cartoff - science associated with such words as Euler, Gauss, Chebyshev, Lambert.

Square projection

Let’s imagine a globe made in the form of a regularly spherical ball, the shell of which is made of fabric that is easily stretchable in one direction and completely inextensible in the perpendicular to it. We can, for example, assume that the horizontal threads are rubber, and the vertical threads are steel, and the rubber threads are closed and pull our ball-globe along parallels, and the thin flexible steel rods perpendicular to them tightly fit the globe along the polymers, adjoining each other with ends in stripes. Since the ends of the merdians are not attached to each other, you can stretch the rubber parallels close to the strips and insert a thin hard knitting needle into the hole. However, not necessarily thin. Who is stopping us from stretching the parallels further? Let’s do just that and push into the holes, stretch to the size of the equator, a “thick spoke”, which is a straight circular cylinder of the same radius as our spherical globe (Fig. 12).

So we have “overcome” Euler’s theorem: after all, the resulting cylinder, being cut along the frontal semi-meridan (one can imagine that the rubber parallels, so that they would not tighten again after this operation, were subjected to a special heat treatment before cutting, depriving them of the ability to further compress and stretch), perfectly unfolds onto a plane! Of course, this is nothing more than a joke - the miracle of “overcoming” did not happen: after all, all the parallels (except the equator) were stretched, and the next continents and islands were stretched in the transverse direction. This stretching is especially noticeable, of course, in the subpolar region.

We received an image of the globe in the form of a rectilinear tree, the base of which (equal to the equator),

21

Fig. 14.

As you approach the pole, the rhoxodrome on the globe looks like, and the infinity of its “growing development” will probably no longer seem surprising to you.

Another stretch. Mercator projection

Let’s now imagine ourselves for a moment in the role of Mercator, looking with displeasure at the awkwardly flattened outlines of Scandinavia and Canada in a square projection (Antarctica, God knows, they didn’t yet know - I wish he could admire whose image!). But the unaesthetic quality of the map could eventually be overcome. And stretching distances along a parallel line will not be misleading, since we already know the rule for obtaining the true distance. It is not the distorting mirror that is scary, but the confidence in its correctness.

Another thing is worse: a map in a square projection is inconvenient for sailors who are interested not so much in the accuracy of measuring distances, but in the ability to easily “plot” a constant course on the map, in other words, in the straightness of the loxodrome. (Let us remember that even in the old “portulas”, despite all their handicrafts, this requirement was satisfied with sufficient accuracy for practical purposes.) So what is upsetting on figure 14 is not so much the “continuity” of the map, but the appearance of the rhyme.

Is it possible to straighten it? Why did it turn out crooked? Because the map is stretched $\frac{1}{\cos \varphi}$ times in latitude?— Great! Who can place us the stretched meridians of a square coordinate grid, the same $\frac{1}{\cos \varphi}$ times!? The squares of the former coordinate grid have now stretched out into rectangles, and the map no longer has a constant scale along the meridians (see figure 15, all the circles on which depict equal sections; the blue line depicts the rhoxodrome). But on the Mercator map the angle between two lines intersecting on the globe remains unchanged. (Try to prove it!) Quite small parts of the earth’s surface are transmitted “almost” with similarity preserved (a circle of radius 500 km, for example, is depicted on the Mercator map as a closed oval, visually indistinguishable from a circle). And loxodromes, of course, are straightforward.

As one approaches the stripes, the stretching of the vertical scale in the Mercator projection increases without limit, therefore, a complete map of the Earth in this projection would be

22

Fig. 15.

development of a cylinder with infinite height and would require an infinite sheet of paper for its image. Therefore, the circumpolar regions are not depicted on such maps (the upper and lower boundaries of the map, of course, are determined purely arbitrarily, from messages of convenience; for example, figure 15 is limited from above by the 85th latitude, and by the 30th latitude).

And two more comments. Firstly, our reconstruction of Mercator’s train of thought may not at all coincide with the actual course of his reasoning; It just seemed to us that this was the easiest way to talk about it. Further, it should be noted that the conclusion about the stretching of the map along the meridians by $\frac{1}{\cos \varphi}$ times, which we obtained “on our fingers”, for a strict (although not complicated) justification would require knowledge of the elements of mathematical analysis, including natural logarithms and the beginnings of differential and integral calculus *). By the way, Mercator himself actually did not do this - a rigorous solution to the problem of “straightening rhoxodromes” was obtained only thirty years later by Richard

*) This information can, for example, be found in the books by P. Courantan G. Robbins “What is mathematics?” (M., 1967) or N. Ya. Vilenkina S. I. Shvartsburg and “Mathematical Analysis” (M., 1969).

Wright from Cambridge (whose editors publish modern editions of what we call the “Mercator projection”).

Lambert cylindrical projection

Let’s consider another projection with a rectangular grid; it turns out very simple. Let us describe a right circular cylinder around the globe (contacting it along the equator and in stripes) and project onto its lateral surface the surface of the globe (without stripes!) so that the projection ray for each point of this surface is the passage through it of a perpendicular to the earth’s axis (Fig. 16).

Using the usual formulas of school stereometry, it is easy to show (try it!) that the areas of any sections of the surface of the globe and their projections onto the lateral surface of the cylinder turn out to be equal, and therefore the development of this cylinder (cut along any meridian) is called an equal-volume cylindrical projection (or Lambert’s projection). Correctly conveying areas, the Lambert projection greatly distorts the shape of even small sections of the projected surface (compare the ellipses on figure 17 with the corresponding circular ones on figure 15): the circumpolar regions are solidified here

23

Fig. 17.
stronger than in the square projection, from which the Lambert projection can be very much obtained by compression along the vertical axis. The straightness of the images of meridians and parallels of the perfect is obvious.

Orthodromi

Until now, we have taken it almost for granted that the simplicity of plotting a course on a map is important to sailors, but the distance that a ship has to travel is of much less interest to them. Well, during the time of sailing ships, sorry, that’s how things were. But nowadays, when sailing over very long distances, not lasting at all for months, and for years, not to mention long-distance (including transcontinental and international) aviation flights, the issue of minimality of the trajectory becomes of paramount importance.

It can be proven (which we will not try to do here) that on the surface of a ball the shortest line connecting two arbitrary points of it (or, as is commonly said in mathematics, a geodesic) is the lesser of each of the great circle passing through these points. (For diametrically opposite points on the sphere, both arcs of the great circle passing through them are equal, and

24

There may be a discrepancy between the lokompnya and the orthodrome. Of course, with long flights the difference may be greater when traveling in temperate latitudes; for example, the path along the locodrome from Moscow to New York is 8500 km, and along the orthodrome - 7500 km.

Gnomonic projection

A projection in which orthodromes are depicted as straight lines is obtained by central projection from the center of the earth’s sphere: in this case, all rays directed to the points of the orthodrome (which, as we saw, are a large circle) lie in the same plane, intersecting in a straight line with any projection plane. The grid of meridians and parallels looks easiest in this so-called gnomonic projection (Fig. 19), if the projection plane touches the globe projected onto it at the pole. (A gnomon is the rod of a sundial; if we insert it onto the projection plane and consider the earth’s radius to be a gnomon, then the projection of the projecting ray will indeed appear to us as the shadow of this gnomon.)

It is easy to prove (do it yourself) that meridians are depicted in a gnomonic projection of this type as straight lines, and parallels as concentric circles. On figure 19 (where, for obvious reasons, there is no image of the screen: it “gap into infinity”) the spiral-shaped form of locodromy in the vicinity of the pole in the gnomonic projection is very clear.

For sea and air travel over long distances

25

Gnomonic projections are very convenient. Of course, the entire earth’s surface cannot be depicted on such a map, so one has to choose different projection planes, the position of which determines the shape of the coordinate grid depicted on them. Figures 20, a and 20, b depict gnomonic projections, the picture plane of which touches the globe not at the pole.

Task. Let the terrain plan be depicted by projecting a given section, shaped like a “cap” with a radius of 100 km, onto a plane tangent to the “cap” in the center ($R_{\text{земли}} \approx 6000$ km). Find the accuracy of the image, i.e. find out by what percentage small segments of meridians and parallels drawn at any point of the “cap” are distorted:

a) in gnomonic projection (central projection from the center of the Earth);
b) in orthographic projection (parallel projection in a direction perpendicular to a given tangent plane). See figure 21, a, b.

Instead of a conclusion
That’s the whole story for now about the basic principles and specific ways of depicting the earth’s surface on a map. For anyone who would like to get acquainted with the science of “portraits of the Earth” - cartographies - more deeply, in more detail and systematically, we would advise, first of all, to carefully review some large atlas, paying special attention to the difference in the projections used in it, and try to understand independently (with the help, perhaps, of rather brief notes usually given in atlases) the reasons for this difference. True, in doing so, you may be enriched not so much with answers, but with new questions. But so much the better!

You will find a systematic (but very rich and therefore much more difficult than in this country) coverage of the subject in the article “Cartographic Projections” in the latest edition of the Great Soviet Encyclopedia. In the article “Geographical Maps,” very rich in historical material, placed there, you will find further references to the literature.