JP6049665B2 - World map showing characteristics of equirectangular hemisphere projection - Google Patents

Description

  The present invention relates to the projection of maps of the world or any other sphere or other region, and more particularly to an improved equirectangular map and globe.

  The present invention is a U.S. patent application 11 / 033,420 filed on Jan. 10, 2005 by the applicant, U.S. Pat. No. 7,331,790 and Feb. 19, 2008. US patent application 12 / 321,795 filed on Jan. 26, 2012, related to and supplemented by both U.S. Pat. No. 8,172,575 registered on May 8, 2012 It is a thing.

  The above-mentioned two US patents show a method for converting a spherical and three-dimensional globe into a flat disk shape using equirectangular projection.

  The equidistant azimuth projection is one of the azimuth projections that correctly represent the azimuth (angle or direction) and are generally drawn in a circle. The projection shows the location on the globe with the equidistant (equal distance) method, and more specifically, the distance from the center point of the map to any other point increases equally. Therefore, it is possible to draw a great circle on the earth as a straight line, which is different from the Mercator projection that can only be drawn with a waveform. A great circle is a circle having the same diameter as the earth, and the point between the two points forms the shortest distance on the earth's surface. Since the great circle is always a straight line when viewed from above on the actual Earth, the waveform curve on most maps does not reflect reality.

  A normal world map drawn with equirectangular projection has a significant distortion problem at the periphery of the projection. This is caused by trying to draw all the terrain in one big circle. In order to solve this distortion problem, in the maps of the above two US patents, one large circle was divided into two small circles with a half diameter. One of the small circles is a circle centered on a predetermined point of interest, which is a body hemisphere corresponding to the half circle inside the large equirectangular diagram. Another small circle of the same size is the opposite hemisphere centered on the antipodal point (the opposite point on the earth) and corresponds to the donut-shaped part outside the large orthographic map. In order to distinguish from normal equirectangular projection, the latter projection having two hemispheres is called equirectangular hemispheric projection.

  In U.S. Patent '790, a projection was provided in which the opposite hemisphere was divided into two and the resulting hemispherical figures were placed on the left and right sides of the main hemisphere, respectively. This is to make the land hemisphere stand out in the center.

  In U.S. Patent '575, a projection was provided that could focus on any point of interest around the world. However, in this projection, the opposite hemisphere was divided into two or more following the same process as the above-mentioned US Patent '790.

  In addition to the above-mentioned prior patents, Plier et al.'S prior art entitled “Map and Calculator” (US Pat. No. 5,902,113, registered May 11, 1999) is a two-hemisphere map of equirectangular projection. Is shown. However, it differs from the previous patent map in one important respect. The map of the above-mentioned prior patent can be seen from the world. On the other hand, Plier and other maps have half of the world on the front and the other half on the back. In other words, the two hemispheres are back-to-back. If you want to see the relationship between two points that are separated from each other, you have to flip them over and over again.

  The map of Pryor et al. Has an accessory that can point the opposite point to any point on the great circle. When one of the accessories is moved from the center point, the other accessory is also moved from the opposite point by the same amount. While using the same hemisphere map of the same center point, you can determine the opposite point of any point. If the center point and the antipodal point can be moved freely, the measurement start point need not be limited to the center point. As long as it is on the line of the great circle, the distance is always correct no matter where you go from one point to another.

  The US Geological Survey's “Map projection” poster says “The distances and directions to all locations are accurate only from the center point of the projection. The distance is correct between points on a straight line through the center. All other distances are incorrect. " Although this sentence is correct, it is easy to lead to misunderstandings. It should be emphasized more on the great circle that the distance between any two points is correct, not just from the center point.

  However, the same is not true for the direction. For example, Florida is located just east of San Francisco, but on the contrary, San Francisco is not located just west of Florida. The directions from all places on the great circle are not correct except from the center point and its opposite point. This is because the earth is round, and the north-south lines for determining the direction are not parallel. This problem will be explained in more detail in the section of Detailed Description.

  FIG. 29 of US Pat. No. '790 relates to an embodiment of another idea “two bowl-shaped hemisphere maps” in equirectangular projection. This has the advantage of being the only map projection that can be directly converted to a globe.

  However, the convertibility is not considered enough at that time. In addition, conventional equirectangular hemispherical projections require the opposite hemisphere to be split or require gears that rotate in a synchronized manner around the periphery of the other hemisphere.

US Patent 7,331,790 US Patent 8,172,575 US Patent 5,902,113

   An object of the present invention is to provide a map composed of a pair of hemispherical maps that integrates conventional knowledge and ideas and can further visually explain the characteristics of equirectangular hemispherical projections.

    The present invention is a map composed of a pair of hemisphere diagrams showing a spherical world capable of visually explaining the characteristics of equirectangular hemisphere projection. In the first hemisphere diagram of the pair of hemisphere diagrams, a map of the main body hemisphere of the sphere world is drawn by the equirectangular hemisphere projection, and has a predetermined point of the sphere at the center of the map of the main body hemisphere. The second hemisphere diagram of the pair of hemisphere diagrams is a map of the opposite hemisphere of the sphere world drawn by equirectangular hemisphere projection, and has an antipodal point of a predetermined point in the center of the opposite hemisphere map. The first and second hemisphere diagrams are aligned such that the neighboring points at the perimeter of the map drawn respectively correspond to the actual geographic points of the sphere world, thereby The direction and distance from a predetermined point to another point in the sphere world can be accurately shown by simultaneously rotating and touching the respective peripheries of the hemisphere diagram. In addition, the first and second hemisphere diagrams are printed on a stretchable material such as fabric or rubber so that a plan hemisphere diagram can be converted to a globe and vice versa.

The effects obtained by the present invention are as follows.
1. In the world map, it can show how the direction from the central point and its opposite point is different from the directions from all other points on the great circle.
2. In the equirectangular world map, the distance between any two points on the great circle can be measured.
3. It shows how easily it can be converted into a sphere in the world map of equirectangular hemisphere projection.
4). A hemispheric world map centered on any point is immediately available.
Still further, the effects of the various embodiments will become apparent from a review of the following description and figures.

According to the present invention, by drawing a hemisphere diagram on a stretchable material, the following three important characteristics of equirectangular hemisphere projection can be explained with a diagram.
1. The direction from the town on the great circle is not the opposite of the direction from the center point or the opposite point.
2. The distance between two points on the great circle is correct, not only from the center point in the figure.
3. The map projection is used to convert from a spherical body to a planar body, whereas the equirectangular hemispherical projection allows the reverse conversion from a planar body to a spherical body.

Fig. 1 is a world map based on equirectangular hemisphere projection centered on Tokyo, explaining that Tokyo is not located in the east from a city on the great circle located in the west of Tokyo. It is a figure to do. FIG. 2 is the same hemispherical map as in FIG. 1, but shows that the distance between two points on the great circle is correct not only from the center point. FIG. 3 is the same hemispherical map as in FIG. 1, but shows how a planar map can be converted to a spherical map. FIG. 4 is a diagram showing a balloon globe that can be converted into a hemispherical map of any center point.

FIG. 1 A pair of stretchable hemisphere world maps As shown in FIG. 1, the equirectangular hemisphere world map is made of a stretchable material such as fabric or rubber. The material hemisphere and the opposite hemisphere are printed on the surface of the material. A hard ring made of metal or plastic is attached to the edge of these hemispheres. The loop is attached to the edge of the stretchable material by sewing or adhering, and when used in the case as shown in FIG. 3, the map is pulled evenly in all directions.

  The first hemisphere FIG. 41, which is the main body hemisphere, shows a case where the center point 40 of interest is Tokyo by the equirectangular hemisphere projection. Similarly, the second hemisphere figure 42, the opposite hemisphere, is centered on Tokyo's antipodal point 43, which shows a point in the ocean east of Argentina and in the South Atlantic. The map is printed on a stretchable material, and a hard ring 44 is attached around the edge of each hemisphere figure 41 and 42.

  A line 45 on the great circle passing through the center point (Tokyo) 40 and its opposite point 43 is aligned in the east-west direction. Indicator pieces 46 and 47 are attached to both ends of the great circle. The indication piece 46 points to the east (E), and the indication piece 47 points to the west (W). Three points on or near the great circle-Bodh Gaya 48 in India, Lusaka 49 in Zambia, and Buenos Aires 50 in Argentina-are shown by the intersection of vertical arcs 51, 52, 53 and horizontal arcs 54, 55, 56. These horizontal arc lines indicate the east-west direction of each point. Other important points-North Pole 57 and South Pole 58-have been added.

  By the way, Bodh Gaya is a sacred place where Buddha opened enlightenment. As the Japanese have said for over a thousand years, we chose this place to prove that India is really just west of Japan. In the Mercator projection, India appears to be southwest of Tokyo. The other two cities, Lusaka in Africa and Buenos Aires in South America, were chosen for the same reason. Despite the tropical image, they are all located just west of Tokyo.

  All directions from Tokyo to these points are just west, but the reverse direction from these points to Tokyo is not true east. This is not easily understood by many who are bound by stereotypes such as Mercator. In such projections, the east-west direction is indicated by latitude lines. For example, if you travel to the east of Tokyo, you will reach 2 to 300 kilometers south of San Francisco. Conversely, the west of that point should be Tokyo on such a map. This is true in practice, but not when the direction of a place is considered by the definition of true east or true west. The problem is described below.

  If you stand somewhere in the middle and low latitudes of the Northern Hemisphere and point to the North Pole with a laser beam, the beam will not reach the North Pole itself due to the curvature of the Earth. Instead it will point to the North Star. The direction of the North Star is true north from that location, and the North Pole is below it. Similarly, if you stand somewhere on the ground and point to the east (or west) with a laser beam, many people expect to go east along the latitudes of the Mercator map. It's a mistake. If the laser beam can bend along the curvature of the earth's surface, it crosses the equator, crosses the opposite point, and returns to its original location. This virtual circle is a great circle above sea level.

  As described at the beginning of [Background Art], the great circle is drawn as a straight line in equirectangular projection. And a map drawn with equirectangular hemisphere projections can show that the reverse of the above direction in the Mercator projection is not correct.

  True east of a certain point on the ground is defined as 90 degrees from true north in the basic direction, and true west is defined as 270 degrees. This Tozai Line intersects the north and south meridians at a right angle. When the center point is on the equator, the direction between the center point and any other point on the equator is opposite in the east or west. This is because all points that intersect the meridian at right angles are the same as the equator line. However, this does not apply to other cases in other great circles.

  Take again the case of the Indian Bodh Gaya mentioned above. The north of Bodhgaya follows meridian 51 toward the North Pole. According to the above definition, the true east of this place is 90 degrees from true north. It is to the right of the arc 54 and not to Tokyo. This same rule applies to all locations, including the aforementioned African Lusaka and Buenos Aires. The exception is the equator, where all points are the opposite of east or west. Only the case of Manishi and Shinto was described here, but the same can be said for all other directions. For example, northwest (southeast), north-northwest (southeast).

  As long as there is a map of equirectangular hemisphere projections centered on Mecca in Saudi Arabia, Muslims around the world are likely to be able to use it to find the direction of Mecca. However, the direction from any place is not the opposite of the direction from Mecca. Therefore, a map centered on any place on the earth cannot be used to point the direction from another place.

  There are other notable characteristics of the great circle. The line on the great circle showing true east and west is the highest latitude point (farthest from the equator and closest to both poles) at a given point or its opposite point. The great circle also crosses the equator at two points 90 degrees to the left and 90 degrees to the left of the predetermined point.

Figure 2 – The distance between two points on the great circle is measurable Figure 2 is the same hemisphere map as in Figure 1, but both hemispheres have moved to new positions. FIG. 2 shows a map in which another town is on or near another great circle line 45 'passing through Tokyo. Examples of such towns are Beijing 59 in China, Islamabad 60 in Pakistan, Abu Dhabi 61 in the United Arab Emirates, and Kinshasa 62 in the Republic of the Congo. The distance between any two towns (not just from Tokyo) is the correct distance reflecting their actual distance. Of course, the same rule applies to the distance between any towns in FIG.

  In order to show the direction of this new great circle, a compass direction 63 is attached around both hemispheres. The compass direction 63 is engraved by printing on a base sheet (background board) 64 of a hemispherical diagram of paper, plastic, wood, metal or other printable material. The term “azimuth” is defined as representing in degrees how much it deviates from true north or magnetic north (or south) to east (or west) in the horizontal direction. However, here, the east-west line replaces the north-south reference line, which is zero.

  The indication pieces 46 'and 47' are located in a new direction and are moving from the east-west line to the top of each hemisphere. The length between the indicating pieces 46 and 46 'is the same as the length between the indicating pieces 47 and 47'. The azimuth marked frequency (0, 30, -30, 60 ... between north and south passing east or west) 65 indicates how far the indicator piece is from the east-west reference line. They are marked from 0 to 90 degrees north and from 0 to -90 degrees south. Note that the orientation of the right hemisphere is upside down. That is, the top of the hemisphere is south (S) and the bottom is north (N).

  FIG. 2 shows that the pointing piece 47 ′ has moved 18 degrees from true west to north. That means Beijing, Kinshasa, and other towns along the great circle 45 'are all 18 degrees north of Tokyo (near west northwest). It is also possible to follow the east direction of this great circle line and read from another pointing piece 46 'that it is 18 degrees east south (near east southeast).

  The ability to print on stretchable material may be a matter of compromise in map accuracy. This will be described later with reference to FIG. Thus, if a map is used only to illustrate the characteristics associated with FIGS. 1 and 2, a hard material such as paper or plastic may be more suitable for that purpose. . In addition, the map works even better when combined with a gear that meshes with the ruler of US Pat. The ruler, the gear, and the above-mentioned compass direction are the optimal combination that makes rotation of both hemispheres easier and more accurate when measuring the distance and direction from the center point.

Figure 3-Transformation from a map to a semi-terrestrial As is known, cartography is used to show a 3D sphere on a 2D plane. There are over a hundred different projections for that purpose, but there has never been a planar map that can be converted into a three-dimensional sphere. However, the equirectangular hemisphere projection can do this inverse transformation. As the term “equal distance” indicates, in this projection, the distance from the center point to the edge of the hemisphere increases at the same rate. This advantage of equirectangular hemisphere projection allows the conversion from a planar map to a half-size globe by this projection if it is stretched evenly in the horizontal and vertical directions.

  FIG. 3 shows that the hemisphere diagram of FIG. 1 is converted to a spherical shape. Two bowl bowl half spheres 66 are placed side by side. And the half sphere is made of transparent plastic or glass. The hemisphere corresponding to the main body hemisphere is already covered by the first hemisphere figure 41 which can be expanded and contracted. The hemisphere corresponding to the opposite hemisphere is about to be covered by the second hemisphere FIG. Here, the term “half sphere” is used to mean a half sphere with no map drawn, and is distinguished from the term “half globe” with a map.

  This pair of semi-terrestrial globes is superior to the ordinary globes in that you can see the entire globe at the same time, despite being spherical. When these hemisphere diagrams 41 and 42 are removed from the hemisphere 66, the pair of hemispheres returns to the original planar map.

  Two semi-globes can be attached to form a normal globe. If you put the hemisphere over the back of the hemisphere and view the map through clear plastic or glass, you will have a pair of concave hemispheres. Naturally, a pair of convex concave hemispheres in FIG. 29 of US Pat. No. 790, or a combination of convex and concave integral hemispheres formed when both hemisphere maps are backed up and one hemisphere is covered. There can be.

  You can create a temporary semi-terrestrial globe by putting a hemisphere map on a beach ball, basketball, or other normal size ball. If two balls are placed side-by-side on the main hemisphere and the opposite hemisphere, they provide an educational and fun way to learn geography.

  The above-mentioned principle of conversion from a planar map to a sphere is used as one of the globe manufacturing methods. However, it is not a well-known fact that equirectangular projection is used in globe manufacturing. Furthermore, unlike the map of the present invention, this map production method lacks the ability to return to its original form.

Figure 4-Equidistant hemisphere map that can be instantly converted to an arbitrary center Thus, the present invention provides a method for instantly centering another point of interest. This can be done using computer software with appropriate data and programming, but provides an analog method using the same characteristics of the diagram in FIG.

  FIG. 4A shows a case where printing on the surface of the rubber balloon 67 is easily possible. In this case, the world map is printed directly on the surface of the rubber balloon 67 (the map is omitted for clarity). This is accomplished by using a round ring 68 made of a shape memory alloy such as Nitinol, which is a nickel-titanium alloy with a transformation temperature. The ring 68 deforms from its original shape at a low temperature below the transformation temperature, and returns to its original shape at a room temperature above the transformation temperature. The ring 68 is deformed at a low temperature and is inserted into the balloon through the balloon inlet 69. The plasticity of Nitinol allows it to be crushed (not shown) enough to allow a round ring to be easily inserted through the inlet 69. After the collapsed ring is inserted, Nitinol is returned to room temperature and regains its shape. Due to the presence of a ring inside the balloon, it becomes a globe when inflated, and becomes an equirectangular hemisphere map with the same diameter as the ring when inflated.

  To center on another point of interest, the balloon is inflated somewhat larger than the ring, releasing the ring from the inner wall of the balloon. Then the wheel can be moved easily and the new point of interest can be centered. If you then deflate the balloon completely, it will again become a planar hemispherical map with a new center point.

  In this way, any point on the ground can be the center point of the map by the above procedure, but it may be quite difficult to print the world map directly on the surface of the rubber balloon. More accurate prints may be required to be used as a practical map or globe. For these cases, printing was made possible by using a bag 70 (FIG. 4B) made of stretchable and printable fabric. The bag is printed with both the front and back hemispheres of equirectangular hemisphere projection. In FIG. 1, a ring is attached to the edge of the elastic material, but instead of such a ring of the edge, the hemispherical view is stitched back to back along the edge line 71 to form a bag 70. The small opening 72 is left unsewed for the insertion of a balloon 67 containing a round ring 68 made of shape memory alloy. The material of the bag must be elastic enough to make it spherical when the balloon inserted in it is inflated.

Normally, the line to sew will be along the equator, but any other edge of any combination of hemispheres. The hemisphere diagram shown in FIG. 1 can be stitched together as the bag described above. However, in FIG. 4, in order to show that the center point can be selected, a hemispherical map when the center point 73 is San Francisco is used. The opposite hemisphere is sewn on the back side, but is not shown here.

  As described above, a world map has been provided to illustrate the characteristics of equirectangular hemisphere projections. As described in connection with FIG. 1, the characteristic that the reverse direction from the point on the great circle is not the exact opposite of the direction from the central point is a well-known fact among Japanese cartographers. . As mentioned in Figure 2, another characteristic that the distance between any point on the great circle is correct is also known to cartographers, as Pryer et al. Knew. It may be. The third property that this projection allows conversion to a spherical shape and return to its original shape may be quite unique.

  Generally speaking, these characteristics are not only apparent to the general public, but are not visually explained by figures. And these characteristics can be well explained by a pair of hemispherical figures that can rotate around each other's perimeter. The problems solved by these maps rarely or never have been mentioned, but can eliminate the common misunderstandings caused by Mercator and other projections.

Other characteristics and advantages of equirectangular hemisphere projection are as follows.
1. A world map with equirectangular hemisphere projection shows the shortest airway and exact direction from the center point of interest to any other point on the earth in a realistic and discernable way, even on remote continents be able to.
2. It differs from other normal hemispheric world maps in that any point of interest on the earth can be selected as the center point of the map.
3. Even though it is a flat world map, it still represents the roundness of the earth.
4). Show the opposite points clearly.
5. It is different from the wavy line in the Mercator projection and other projections in that any great circular line passing through the center point and the opposite point can be drawn with a straight line.
6). The distance between any two points on the great circle line can be measured accurately, and the correct direction from the center point and the opposite point can be known.

  Although the above description includes many specifications, these should not be construed to limit the scope of the invention, but merely illustrate some of the preferred forms for carrying out the invention at the present time. Absent. Many variations and derivations are possible.

  For example, if you make a stretchable ruler made of rubber-like material, make a thread knot on both ends of the ruler, attach it, and limit the maximum length to 1.57 (π / 2) times when stretched If so, it is possible to measure the distance between any two points on the globe and also the distance between the two points on the great circle in the plane hemisphere.

  The equirectangular hemisphere projection, as described in US Pat. No. '790, is also useful for drawing maps of other spheres such as Mars and the Moon. Now that we can understand the characteristics of this projection, it is possible to draw a map of such a sphere with accurate measurements of direction and distance.

  By using a stretchable material, hemispherical maps of other celestial bodies can be created. If you have a transparent bowl-shaped hemisphere and a map of the earth, and if you have a set of maps of Mars, Moon, Venus, etc. Can be enjoyed easily. Many are bulky and cannot afford the spheres of other planets. So with such a set of hemispheres and stacked planetary maps, the intimacy of planetary geography will increase rapidly.

  If a hemispherical bag with balloons replaces the interchangeable map described above, it would perform much the same as stacking planetary maps. This type of set does not require a half sphere, so it saves more space to get done. On top of that, the advantage of not taking this place is a set of spheres with small terrestrial planets and several satellites in different sized bags, balloons, inner rings etc. in the same ratio as the actual celestial body. Allows you to have a living room.

  If you have a set of celestial objects in this proportion, you can use the same ruler that you use for Earth maps and globes to measure the distance of other objects. For example, it is possible to directly compare the distance traveled from the Mars Rover landing site with the distance from our town.

  Accordingly, the scope of the invention should be determined not by the examples illustrated, but by the appended claims and their legal equivalents.

40: Center point 41, 42: Hemisphere diagram 43: Antipodal point 44: Wheel 45, 45 ': Great circular line 46, 47, 46', 47 ': Pointer piece 51: Meridian 54: Arc line 51, 52, 53: Vertical arc line 54, 55, 56: Horizontal arc line 63: Compass direction 64: Background board 65: Degree 66: Half sphere 67: Balloon 68: Wheel 69: Air inlet 70: Bag 71: Edge line 72: Opening

Claims (6)

As a characteristic of equirectangular hemisphere projection,
Direction to the center point from the point on the great circle, it does not indicate positive opposite direction from the center point to the point on the great circle,
The distance between the two points on the great circle is correct not only from the center point of the figure, and the conversion from a planar body to a spherical body is possible,
A map composed of a pair of hemisphere diagrams showing a spherical world that can visually explain
A first hemisphere diagram of the pair of hemisphere diagrams, wherein a map of the body hemisphere of the sphere world is drawn by equirectangular hemisphere projection and has a predetermined point of the sphere at the center of the map of the body hemisphere;
A second hemisphere diagram of the pair of hemisphere diagrams, wherein a map of the opposite hemisphere of the sphere world is drawn by equirectangular hemisphere projection and has the opposite point of the predetermined point at the center of the map of the opposite hemisphere;
Consists of
The first and second hemisphere diagrams are aligned such that adjacent points at the perimeter of the drawn map correspond to actual geographical points in the sphere world, respectively. And by simultaneously rotating and touching the respective peripheries of the second hemisphere diagram, the direction and distance from the predetermined point to the other point of the sphere world are accurately indicated,
The pair of hemispherical maps are made of a stretchable material that can be restored, and are configured to be convertible between a two-dimensional plan view and a three-dimensional spherical view while maintaining the characteristics.   Furthermore, a first indicating piece attached to the periphery of the first hemispherical view showing the east of the predetermined point and a second indicator attached to the peripheral edge of the second hemispherical view showing the west of the opposite point The map according to claim 1, wherein the map includes two indicating pieces.   In addition, a background plate for the first and second hemisphere diagrams is included, and a plurality of compass directions are inscribed on the background plate, and the predetermined point and any other point coincide with a straight line of a great circle The frequency in the direction from the predetermined point to any other point is measured by reading the numerical value of the compass direction indicated by the first and second indicating pieces. Map described in.   The compass direction on the background plate is marked with a reference value of 0 on an extension line of the east-west line passing through the predetermined point, and is recorded 90 degrees northward and -90 degrees southward. Item 4. The map according to item 3.   In order to support the pair of hemispheres when a pulling force is applied to the pair of hemispheres formed of the stretchable material, a hard round ring is provided at each periphery of the pair of hemispheres. The map according to claim 1, wherein:   The map according to claim 1, wherein the sphere is one of celestial bodies selected from the group consisting of the earth, the moon, the planets, and their satellites.

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