How to take a “natural photo” of the earth?

Author: Wang Su (Institute of Atmospheric Physics, Chinese Academy of Sciences)

The article comes from the Science Academy official account (ID: kexuedayuan)

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Since Aristotle first proved that the earth is round, after several centuries of development, the "spherical earth theory" has become a mainstream consciousness.

However, there are many people who do not believe in such a simple truth!

There is a growing organization on Earth - the Flat Earth International Coalition (FEIC). They firmly believe that the Earth is a large flat surface, with the Arctic Circle at the center and the continents and oceans floating around it. The outermost part is the Antarctic Circle, a 45-meter-high ice wall that surrounds the Earth in the middle. The sun and the moon are "charcoal lamps" that rotate in the sky for 24 hours without stopping.

The rotation of the earth as believed by "flat earthers" (Image source: ScienceNet)

They insisted that the governments of various countries had joined forces to deceive people, otherwise why was the UN emblem a flat earth, and why was the South Pole not placed on the projection surface? In order to convince people, they also produced their own "evidence": how to explain that the distribution of ocean currents and the size of countries are completely different on a flat surface and a spherical surface, and that they change very randomly.

(Image source: Zhihu)

Let me share a little-known fact: According to the provisions of the United Nations Charter, Antarctica does not belong to any country in the world and is a place protected by all mankind, so the projection surface is chosen to be centered on the North Pole.

Same country, different size,

Who Lied? - Map Projection

Regarding the "flat earth theory", netizens have raised the question: On the map, the areas of Greenland and South Africa are not much different, but in fact, Greenland is 1/14 of South Africa and 9 times smaller than South America. How can the size of the same country on the map be so different? This starts with map projection.

Map projection is to establish a one-to-one correspondence between points on the earth's surface and points on the map plane.

The globe is large and bulky, and people in ancient times could not hold it while working and living. If they encountered some situations that required accurate determination of position and direction, they had to make the globe fit into their trouser pockets - projecting the 3D sphere onto a 2D plane to make a map. How to make a 2D map without distortion became the biggest challenge in map drawing.

If you try to cut the globe in half and lay it out on a flat surface, you will get an "orange slice" like this. Because of the change in dimension, it is inevitable that it will be deformed and distorted. We are too "southern" to take it with us everywhere.

If the earth were forcibly flattened, it would probably look like this (Image source: Cool Play Lab)

The evolution of maps: The risks of beautifying maps

1. Mercator Projection

In order to help navigators understand the world, in the 16th century, cartographer Mercator invented the conformal cylindrical projection - the Mercator projection.

He used the direction of the earth's axis as a parallel line, unfolded the earth into a cylinder, and the meridians became a set of vertical equidistant parallel lines, and then unfolded the cylinder into a plane, realizing the displacement from sphere to plane. In this way, the length ratio of any direction at a point is equal, and what's even better is that there is no problem of angle deformation. The direction and position displayed on the map are exactly the same as the actual direction, which is really a good navigation! Therefore, it has become the most popular world map projection method.

(Image source: Zhihu)

However, as the points on the surface of the Earth "travel" farther when they are pulled into a cylinder, the distance between adjacent latitude lines increases from the equator to the poles, and the area deformation increases accordingly. This leads to the misunderstanding of the different areas mentioned above: Greenland has an overwhelming advantage even when compared with Africa (the latter's actual area is more than 10 times that of the former).

This may be the most classic example of "eyes can deceive people".

Greenland and Africa as shown in the Mercator projection map (left) and restored according to their actual size (right) (Image source: Zhihu)

2. A complete list of map projection methods

In fact, the commonly used projection methods, in addition to cylindrical projection, there are azimuthal projection (also ensures that the angle remains unchanged but the area is distorted, representative: polar stereographic projection) and conic projection (guarantees that the area remains unchanged but the angle is distorted, representative: Lambert projection). According to the different projection angles, the actual projections used can be subdivided into as many as 27 types! When the North Pole is used as the tangent point of the plane and the ellipsoid, what you see is what the United Nations logo looks like~

(Image source: Baidu Encyclopedia)

But these projection methods have their own "little problems" to a greater or lesser extent. Since the map is distorted, we have been singing "Asian Rooster" for generations, but does the shape really exist?

Don't worry. Except for 1:1 million (Lambert projection), all basic-scale topographic maps issued by my country adopt Gauss-Kruger projection.

Assume that an elliptical cylinder is tangent to the north and south poles of the earth's ellipsoid at N and S, the center of the sphere is on the central axis of the elliptical cylinder, and a meridian is tangent to the elliptical cylinder (this meridian is called the central meridian). According to the equiangular condition, the longitude and latitude lines within a certain range on both sides of the central meridian are projected onto the elliptical cylinder, and then the elliptical cylinder is cut along the elliptical cylinder generatrix passing through N and S and unfolded into a plane, which is the Gauss-Kruger projection.

This projection is very suitable for maps of small areas in mid- and low-latitudes. Compared with other projections, its length and area deformation are the smallest, which can ensure that we see a real China to the greatest extent!

Gauss-Krüger projection (Image source: CSDN)

It can be seen that it is precisely because the earth is spherical that people always fight against map deformation. On the contrary, if the earth is really flat, wouldn’t it be easy to draw a globe without distortion?

Turn off “beautification”, who will take the photo of the “bare makeup map”?

All the projections mentioned above have one thing in common - they are divided according to longitude and latitude that are always perpendicular to each other (we call it orthogonal). Therefore, you can quickly find all the countries that pass through the same latitude/longitude, but you can't do anything about how big each country is.

So, is there a more realistic way of projection?

Hajime Narukawa, a designer from Keio University in Japan, found a more accurate map projection method - AuthaGraph.

(Image source: Good Design Award official website)

At first glance, the AuthaGraph world map is a bit "weird", but it is not without its flaws. Unlike the common elliptical world map, it is square, and the longitude and latitude lines that should be clear at a glance are bent into a ball of wool.

However, it is the most accurate map showing the location and proportion of land and ocean so far! In order to avoid the projection error caused by directly unfolding the sphere, Chengchuan Zhao first divided the earth into 96 equal parts, so as to preserve the area ratio of each region. Then, each part was deformed into a regular triangular pyramid, and then the tetrahedron was unfolded by cutting, and finally a rectangle with an aspect ratio of about 1 to 1.73 was obtained. All the obtained rectangles were seamlessly spliced ​​together to finally obtain a global map.

(Image source: Good Design Award official website)

We also finally saw that Antarctica is not only close to South America, but also close to Africa and Australia.

Because it successfully turned 3D into 2D and faithfully represented all oceans and continents, including the neglected Antarctica. And this template can be moved around the world and expanded around any region to obtain a more detailed and accurate regional map. AuthaGraph World Map won the 2016 Japan Good Design Award and was recognized as an equal-area map. The map also appears in Japanese elementary school textbooks.

What’s even more interesting is that one of Chengchuan Zhao’s products, AuthaGraph Globe, can achieve the “magic” of 2D to 3D! By assembling the product, you can understand how AuthaGraph achieves the transformation from sphere to plane.

(Image source: Good Design Award official website)

This is a perfect example of the application of triangular meshes! But its application is more than that! Since any polygonal mesh can be converted into a triangular mesh, compared with general polygonal meshes, triangular meshes are simpler and easier to operate. They are widely used in graphics and modeling, and can be used to simulate the surface of complex objects, such as buildings, vehicles, and human bodies. It can be said that meshing is one of the key technologies for building finite element models!

Of course, if you want to understand in detail how it is achieved, you have to open the door to computer graphics.

(Image source: CSDN)

You never thought that a small map could contain so much knowledge. Although the earth is not flat, it can be made flat through scientific methods. It really makes people marvel at the wonders of mathematics!

Grid and weather, enemies always meet

All the maps above are divided into countless small blocks by longitude and latitude, and each block can be called a grid. However, some grids are very regular, while others are very "artistic". But it is their cooperation that reproduces an organic whole - the map. How important is the grid to meteorology?

The grid can also be cut continuously. The smaller the grid is, the smaller the area it represents, which means the higher the accuracy. With the development of technology, we have even achieved double precision improvement in horizontal and vertical scales!

WebGL-based 3D digital globe (Image source: Zhihu)

We know that weather forecasting is inseparable from numerical simulation, and this work is achieved by dividing the earth into grids and using computers to calculate the interactions between different grids using countless mathematical formulas. So, how is numerical forecasting related to earth projection?

Schematic diagram of spherical coordinate system and projected coordinate system (Image source: MOOC)

In fact, in most cases, we will not study the entire earth (we need to consider the geographic coordinate system, the coordinate unit is longitude and latitude). After all, from the perspective of the earth, if you fly from one city to another, it may be a small dot that does not move. Therefore, changes in a limited area are more concerned. At this time, in order to simplify the solution, the influence of spherical curvature will be ignored, and the study area will be regarded as a plane with the specified meridian as the Y axis and the perpendicular to it as the X axis (called a projected coordinate system, the coordinate unit is meter).

This not only maintains the characteristics of the spherical coordinates of the study area on the earth, but also simplifies the equation operation because it only involves horizontal coordinate transformation. Which projection method is better to choose depends on which part of the earth you want to study.

At this time, through the coordinate system transformation method that has been summarized, we can match the grid in the numerical calculation with the grid on the projection surface one by one. Then input the initial data into the model and wait for it to tell us the answer. In the end, the result you get is a huge amount of prediction data.

Finally, by using software to process the results into a visual plane map, we can know where it will rain and where it will be sunny tomorrow, where the wind will blow, and where the pollution will go.

You may ask, due to the projection error, aren't the forecast results we get inaccurate?

Meteorologists make sure that the region of interest is centered when simulating regional models, so that projection distortion is minimized. Moreover, the errors introduced by the projection method are much smaller than the errors in the dynamic framework and physical processes used to build the model, so we can breathe a sigh of relief.

So why not apply the latest and most accurate projection methods to the model as well?

Considering the ease of use, meteorologists have found that increasing the complexity of the projection has little effect on the correction of the results and is not so "cost-effective". Therefore, the Lambert projection is still the most commonly used projection in regional models.

However, in order to see the changes in the areas of interest more clearly, scientists are working hard to improve the performance of supercomputers, so that we can simulate smaller and smaller grids and higher and higher resolutions, thereby achieving more accurate forecasts.

The fastest supercomputer currently - China's Tianhe-2 (Image source: Sohu.com)

Therefore, it can be said that the two of them are inseparable.

Last words

Of course, we are not here to criticize the "flat earth" advocates for being unreliable. This spirit of bold questioning is indeed admirable, but we want to emphasize how important the scientific spirit of careful argumentation is, and how "clever" the pioneering spirit of looking at the world from a different perspective is. As computer technology becomes more and more powerful, meteorologists have a deeper understanding of the earth. In the near future, it is not impossible to see a higher resolution forecast!

References:

1.https://allthatsinteresting.com/authagraph-world-map

2.More Accurate World Map Wins Prestigious Design Award

3.https://www.g-mark.org/award/describe/44527)

https://zhuanlan.zhihu.com/p/33295151?from_voters_page=true