Creation of New Type of the World Map-application of inversion
Takushi Amemiya 741075B S1 37(the University of Tokyo) 15 July 2007
Abstract
In this research I invented the way to create a new type of the world map by using inversion (a kind of mathematical transformation) based on the fact when you invert sphere that passes the origin, you get the plane that does not pass the origin. After that I calculated the inversion of longitudes and latitudes and drew the maps; one’s center is the South pole and the another’s is Shanghai.
Introduction
From recent research on inversion, I proved that by inverting the sphere that passes the origin I can get a plane, and based on this fact, I would be able to create a new type of the world map because Earth is sphere and the map is a plane, I thought. So far there are some kinds of maps, each of which has some good points but also some bad points. This new type of the world map might solve these bad points. However, the problem is that the plane got by inversion exists on 3-dimention space slantingly, and even though computer software can draw 3D graphs, they are not accurate graphs. What I want to study here is to get the equation of the longitudes and latitudes of the world map using inversion, and draw them on plane exactly. I am going to use computer graphics software like Mathematica, GNUPLOT, or Grapes. The result will give not only the new type of the world map, but also the way to record information which is originally recorded on the surface of sphere on plane, or the information which is originally recorded on the surface of plane on sphere.
Background
So far several kinds of maps have been invented. However, when projecting the Earth on a plane distortion happens. This distortion is categorized into three types: distance, area, angle. The maps that solve distortion of distance are called equidistant maps. And the maps that dissolve distortion of area are called equal-area maps. Among them is Lambert's azimuthal equal-area map. The maps that dissolve distortion of angle are called conformal map/ orthomorphic map/ isogonal map. Among them is Mercator’s map. Some maps are not exact in all the points: area, distance angle, but they have good balance of these three, and this is the good point of these kinds of maps.
As for inversion, it has been studied since ancient Greece. Inversion itself is not studied nowadays, however, Mobius transformation, an extension of inversion is studied. The fact that inverting the circle that passes the origin gives a plane that does not pass the origin is widely known.
Method
World maps are invented by following projections each of which translates the sphere to the plane. Each map has some bad points but on the other hand it has some good points. The world map using inversion is new world map. Therefore the map should include some new characteristics and good points solve other maps’ problems.
In addition to that, studying how inversion of circles of latitude and circles of longitude are drawn on a plane will lead well understanding about inversion itself.
the map has information about sphere. Hence it can be said that my research is revealing a new way of recording information which is originally recorded on a sphere on a plane, and the information which is originally recorded on a plane on sphere.
These are the reasons why I conducted this studying.
In this research, I calculated the equation of sphere which is the inversion of a plane . Next I set the steps to produce inversion-map. After that I examine the equation of inverted circles of latitude and longitude. Because the map exists in the 3D space slantingly, I rotated the plane in order to make it parallel to xy plane with rotation-matrixes, calculating the equations of the graphs. And I drew them on xy plane by software grapes.
Def.
In the space, the inverse of a point P in respect to a circle of center O and radius R is a point P' such that P and P' are on the same ray going from O, and OP times OP' equals the radius squared,
from Wikipedia
Fig 1 illustrate the definition above.
In one dimension r→1/r (in the graph -2→-1/2)
In the two dimension blue point (P) is inverted to the red point (Q)
Fig 2 illustrate the inversion on R×R×R
Fig 2
Fig 1
The fact that when you invert a circle that goes through the origin, you get the line that does not go through the origin is known. The figure left shows that inversion respect to the circle blue of purple circle is the red line
Theory 1
From this definition, I got a result that inversion on R×R×R is,
Cor1
Thus, when a plane （）is inverted, it becomes the sphere that passes the origin whose equation is,
.
As the equation shows, it is the sphere whose center is and the length of radius is
The steps to make inversion-world map
Before making the inversion map, let assume the Earth is a sphere whose length of radius is 1 in order to make this problem simply. From the fact above, it is concluded that the sphere that passes the origin is translated to a plane that does not passes the origin by inversion. Hence, to translate Earth to the map by using inversion, it is necessary that the sphere(the Earth) passes the origin on the space R×Ｒ×Ｒ
So I define the steps for making world map whose center is point A( A is town or city) as follows.
1. Let assume that the Earth is the sphere; the center is (0,0,0) and the length of radius is 1
2. The coordinates of the point A is (a,b,c)
3. Move the center of the sphere to (a,b,c) So you can get the sphere whose center is (a,b,c)
4. Invert the sphere which is defined at 3
(by the last steps we can get the place whose center is the inverted point of A and the opposite point of A at sphere is inverted to the infinite point. )
The equation of the sphere made by the step 2 is
.
Thus the plane which is made by the step 4 is
By simplifying this equation we get,
( this is the plane.)
Examination about meridian and circle of latitude
The most important and fundamental factors of map is meridians and circle of latitude. If I am able to draw these two, I will be able to draw the map even if I don’t know about which point is inverted to which point. And these two will help the user of this map to get information. So I am going to consider about how these two are drawn on the plane.
Prop. Inverting one of circles of latitude of the sphere gives a circle on the plane, and one of meridians of the sphere gives a circle on the plane.
<Proof>
Set that the sphere is S, and the plane which is made by inversion as S is S’.
Circle of latitude is the intersection of a plane Z=Constant (set that this plane is α)and S.
α is a plane. Therefore inverting α gives a sphereα’. And inversion of the circle latitude, which is part of α, exists on S’. Thus, the union of α’ and S’ is a circle. This circle is the same circle which is gotten by inversion of the circle latitude.
In the same way, inversion of meridian is a circle that is drawn on S’.
Example of inversion-world-map
(1) Map whose center is South Pole
In this case Point A is the South Pole.
By the steps, the equation of the sphere (the Earth) is
Inverting this sphere gives the plane whose equation is
Because the circle of latitude is the intersection of Fig 4
and (θis the latitude)
Thus, inversion of the circle of latitude is the intersection of
θ=-90 means the south pole, θ=0 is equator, and θ=90 is north pole
In the following map, circles of latitude(-90°≤θ≤ 70°) are drawn. And in this case meridians are straight lines.
Map whose center is the South Pole
As another example of maps, I am going to draw the map whose center is Shanghai.
(2)Map whose center is Shanghai
(Latitude is +30°and meridian is 120°E)
Set that latitude is the degree from plane xy and meridian is the degree measured from x-axis.
In this circumstances, the coordinates of Shanghai is
So the equation of the sphere( the Earth) is …Ｓ
The circle of latitude (before inversion) is the intersection of S and …Ｂ
Inversion of S is (Fig 5)
Inversion of Ｂ is like Fig 5
Fig 5
Fig 4
Fig 6 fig 6indicates the intersection of sphere and the plane but it is not accurate graph at all.
Thus, by using rotation-matrix around axis, make the plane parallel to xy plane.
Rotation-matrix
Around x axis→
Around y axis→
Around z axis→
To make normal vector of perpendicular to xy plane, it is necessary that ,
If vector belongs to the plane is translated to the vector, becomes by transformation
＝
Therefore when rotating the plane under the condition of , the plane is produced.
On the other hand when the plane is inverted, the sphere whose equation is is generated.
When rotating this sphere under the condition of , the center of the sphere is translated to the point
In rotation the length of radius is preserved, so the radius of sphere
Therefore the equation of the sphere which is gotten by the transformation of the original sphere is
Thus the result of the transformation of circle latitude is
,which is simplified to the intersection of
Next I will examine about the inversion of meridian
Meridian at the original sphere( before step 4 but after the step 3) is the intersection of the plane θ=constant (this plane is perpendicular to xy plane) and sphere. As I proved at Prop.1, inversion of meridian is a circle. (note: θ here is different from θof latitude)
The plane θ=constant is described as the flowing equation
Therefore the intersection of inversion of this plane and (inversion of sphere) is the inversion of meridian.
Inversion of is ,which is simplified to …Ｃ
If , then C is simplified to
This is the sphere whose center is , and square of length of radius is
So when rotating the sphere under condition of ,the center of sphere above is translated to the point since ＝
So the equation of the sphere (after rotation) is
Thus the inversion of meridian is the intersection of this sphere and the plane
i.e. on the plane
If
C⇔ So rotating this plane gives the plane x=0
Therefore the inversion of the meridian( longitude is 120°) is the intersection of and x=0
The graph below is the circles of latitude and meridians. K means longitudes and I means latitude. For example I -40 is the circle of latitude 40°(south) and I50 is the circle of latitude 50°latitude north. And K0 is the Greenwich line. The center of this map is Shanghai.
Discussion and summary
In this way I invented the new type of the world map using inversion. Since Mobius transformation is a transformation that preserve angle, and angles between two circles on the sphere are preserved even on the inversion map. As for distance and area, I have not discovered the way to read the map to gain such information. Therefore I would like to solve these problems in future research.
And this time I drew the maps whose centre are the south pole and Shanghai because sin(latitude, longitude) cos(latitude, longitude) are easily calculated at these places. Thus I want to invent the function which enables to draw the map whose center is anyplace.
At the same time I want to study how this inversion map can be applied in future study. One thing that I am thinking is that by using complexities which inversion map has, I will be able to use this map as cipher for geographic information. Even if there are someone who tries to steal geographic information which is communicated between sender and receiver, they can not understand about which points they are talking due to the complexities of inversion map.
Acknowledgement
Pro. Gally and Mr. Tam gave valuable criticism and suggestions for structure and grammatical things. And Pro. Gally and Mr. Naito help me to write more clear and understandable paper.