Home Definition CMEX Mars Legal & Licensing Physics Forum Contact
NAC Geographic Products Inc. The Natural Area Coding System International NAC Society, Inc.

The Martian Area Coding System


Abstract

The Martian Area Coding SystemTM is a new system to standardize and integrate geographic coordinates, geographic area codes, map grids, and addresses on Mars. The system employs revolutionary approaches:
  1. It has unified the concepts of geodetic points, line sections, areas, and three-dimensional regions.

  2. It employs the 30 most popular characters in the world instead of ten digits and makes full use of these characters to produce the most efficient representations;

  3. It creates one standard representation for all locations, 2D areas and 3D space blocks on Mars.

These approaches make the Martian Area Coding SystemTM superior over longitude/latitude coordinates. A set of coordinates of the system is called a Martian Area Code (MAC) that can represent a point, a line section, an area or a 3D block simultaneously. When representing a geodetic point to the same resolution, it requires only half of the number of characters as required by a longitude/latitude. Using MAC to represent line sections, rectangles or three-dimensional regions can save even more in required characters compared with other systems.

Description

The Martian Area Coding System is a geodetic system with its origin at the Mars gravity center and axis extending to the infinitely distant universe. It employs a character set consisting of digits 0 to 9 and all English capital consonants since these characters are the most popular characters widely used in natural languages such as English, French, Spanish, German, Chinese, and all categories of science and engineering. Each character in the character set represents an integer ranging from 0 to 29, as shown in the following table:

  Table of the MAC Character and Integer Correspondences 
===========================================================
||Character|Integer||Character|Integer||Character|Integer|| 
||---------|-------||---------|-------||---------|-------||
||   0     |   0   ||    B    |  10   ||    N    |   20  || 
||   1     |   1   ||    C    |  11   ||    P    |   21  || 
||   2     |   2   ||    D    |  12   ||    Q    |   22  || 
||   3     |   3   ||    F    |  13   ||    R    |   23  || 
||   4     |   4   ||    G    |  14   ||    S    |   24  || 
||   5     |   5   ||    H    |  15   ||    T    |   25  || 
||   6     |   6   ||    J    |  16   ||    V    |   26  || 
||   7     |   7   ||    K    |  17   ||    W    |   27  || 
||   8     |   8   ||    L    |  18   ||    X    |   28  || 
||   9     |   9   ||    M    |  19   ||    Z    |   29  ||
===========================================================

A Martian Area Code (MAC) consists of three character strings separated by blank spaces. The first character string represents longitude, the second string represents latitude, and the third string represents altitude. The system divides the whole range of longitude (0 - 360 degrees), latitude (0 - 180 degrees) and altitude (from the Mars center to the infinite outer space) into 30 discrete divisions respectively, each of which is named by one character from the character set according to the order of the characters. And each resulting division is divided into 30 subdivisions, and each of the subdivisions is named by one character. The division process can continue to the third , fourth, and other levels. The resulting divisions in three dimensions form many regions called MAC blocks. Therefore, a first level MAC block can be represented by a MAC of three characters separated by blank spaces, each of which represents the character string for longitude, latitude and altitude respectively, for example, MAC: 5 6 7. A second level MAC block can be represented by a MAC of six characters to form three character strings: the first two characters form the longitudinal string, the third and fourth characters form the latitudinal string, and the last two characters form the altitudinal string. A blank space is placed between these strings, for example, MAC: JB KH LN represents a MAC block at the second level, in which the characters J, K and L represent coordinates of a first level MAC block which contains the second level MAC block, and the characters B, H and N are the relative coordinates of the second level MAC block in the first level MAC block. A region formed by sides at different division levels is called a MAC region and can be represented by a single MAC too. Any three MAC character strings can form a MAC which represents a completely defined region in the universe.

If the third string of a MAC is omitted, the resulting MAC represents an area on the Mars surface, called a MAC area if the number of characters in the two coordinate strings are different, and called a MAC cell if the number of characters in the two coordinate strings are the same. Any two MAC character strings can form a MAC representing a completely defined area on Mars. When the sides are very different in length, a rectangular area will turn out to be a line section automatically. When the sides are relative small, a rectangular area will become a geodetic point.

Therefore, a MAC can represent a geodetic point anywhere in the universe, a line section of constant longitude or constant latitude on Mars, an area bounded by constant longitude and constant latitude anywhere on Mars and a three-dimensional region bounded by constant longitude, constant latitude and constant altitude anywhere in the universe.

The Correlations between the Martian Area Coding System and the longitude/latitude/altitude coordinates of Mars

From (Longitude, Latitude, Altitude) to MAC

The MAC of a region that contains a geodetic point expressed by the longitude, latitude and altitude coordinates can be determined by the following algorithm:

	LONG = (Longitude + 180)/360
	x1 = Integer part of(   LONG*30)
	x2 = Integer part of((  LONG*30-x1)*30)
	x3 = Integer part of((( LONG*30-x1)*30-x2)*30)
	x4 = Integer part of((((LONG*30-x1)*30-x2)*30-x3)*30)
     ...

	LAT =  (Latitude + 90)/180
	y1 = Integer part of(   LAT*30 )
	y2 = Integer part of((  LAT*30-y1)*30)
	y3 = Integer part of((( LAT*30-y1)*30-y2)*30)
	y4 = Integer part of((((LAT*30-y1)*30-y2)*30-y3)*30)
     ...

	ALT = Arctan(Altitude/R)/90
	z1 = Integer part of(   ALT*30)
	z2 = Integer part of((  ALT*30-z1)*30)
	z3 = Integer part of((( ALT*30-z1)*30-z2)*30)
	z4 = Integer part of((((ALT*30-z1)*30-z2)*30-z3)*30)
     ...

where Longitude is positive in the eastern hemisphere but negative in the western; Latitude is positive in the northern hemisphere but negative in the southern; both Longitude and Latitude are in degrees plus decimals; Altitude is measured along the gravitational force line from the center of the geoid of Mars in kilometers; the symbol * is the multiplication sign; x1, x2, x3, x4, ..., y1, y2, y3, y4, ..., z1, z2, z3, z4, ... are integers ranging from 0 to 29 here; Arctan( ) is the arctangent function with value in degrees; R is in km the distance from the Mars center along the gravitational force line to the geoid surface and can be approximated by the Mars radius at the location:





	R = sqrt[b^2+(a^2-b^2)/(1+b^2/a^2*tan^2(Latitude))]
or more accurately the distance from the gravitation center to the geoid surface along a parabola passing the gravitation center and perpendicular to the geoid surface:
	C1 = [1 - 2*(1 - b^2/a^2)]*tan(Latitude)
	C2 = (1-b^2/a^2)*tan(Latitude)*sqrt[a^2+b^2*tan^2(Latitude)]/a^2
	C3 = 2*a*C2/sqrt[1+b^2/a^2*tan^2(Latitude)]+C1
	C4 = C3*sqrt(1+C3^2)+Asinh(C3)
	C5 = C1*sqrt(1+C1^2)+Asinh(C1)
	R  = (C4 - C5)/4/C2

where a is the semi-major Mars axis (ellipsoid equatorial radius) equal to 3397 km; b is the semi- minor Mars axis (ellipsoid polar radius) equal to 3375 km; sqrt( ) is the square root function; tan( ) is a triangular tangent function; Asinh( ) is the inverse hyperbolic sine function; the symbol / is the division sign; the symbol ^ is the exponential operator.

Once x1, x2, x3, x4, ..., y1, y2, y3, y4, ..., z1, z2, z3, z4, ... are calculated, the corresponding characters can be found from the Table of the MAC character and integer correspondences: X1, X2, X3, X4, ..., Y1, Y2, Y3, Y4, ..., Z1, Z2, Z3, Z4, .... Then, the Martian Area Code of the region is written as MAC: X1X2X3X4... Y1Y2Y3Y4... Z1Z2Z3Z4... with a blank space between any two character strings. The first character string of a MAC represents longitude, the second string represents latitude, and the third represents altitude.

If a MAC has only two character strings, then the MAC represents an area on the Mars surface and the two character strings represent the longitude and latitude respectively, as defined in the beginning of this chapter. For example, MAC: 8KD8 PGGK represents a 13 by 26 meter area, while MAC: 8KD8 PGGK H000 represents a region 13 meters wide, 26 meters long and 13 meters high measured from the geoid surface.

The number of characters to be used in a character string of a MAC representing the geodetic point is determined by the required resolution or the resolution of the original coordinates of the longitude, latitude and altitude. A MAC using more characters represents a smaller area or region. The smallest area or region containing the geodetic point is the one of the size equal to the error range of the coordinates. Therefore, when a MAC is used to represent a geodetic point, it has both the information of the location and its error range.

From MAC to (Longitude, Latitude, Altitude)

If the MAC of a region is known, then the longitude, latitude and altitude of the southwestern lower corner of the region can be calculated by the following procedure:

First, convert all characters X1, X2, X3, X4, ... Y1, Y2, Y3, Y4, ... Z1, Z2, Z3, Z4, ... into integers x1, x2, x3, x4, ... y1, y2, y3, y4, ... z1, z2, z3, z4, ... according to the Table of the MAC Character and Integer Correspondences.

Then use the following formulae to calculate coordinates:

	Longitude = (x1/30+x2/30^2+x3/30^3+x4/30^4+...)*360-180

	Latitude =  (y1/30+y2/30^2+y3/30^3+y4/30^4+...)*180-90

	f = (a - b)/a ;      e = 2*f - f^2  ;

	N = a/sqrt(1 - e^2*sin^2(Latitude))  ;

	R = N*sqrt[1 - e^2*(2-e^2)*sin^2(Latitude)]

	Altitude = R*tan((z1/30+z2/30^2+z3/30^3+z4/30^4+...)*90)-R

The northeastern upper corner of the region can be calculated by repeating the same procedure with the same integers except adding 1 to the integer corresponding to the last character of each string of the MAC. Then, the region can be completely determined by the coordinates of these two geodetic points.

MAC Algebra

In the Martian Area Coding System, several algebraic rules have been introduced to simplify the notations and operations of MACs. Some of the rules are defined in the following, where symbol = represents the equivalency and symbol + represents the sum of two MAC regions or areas.

Definition a If there are a series of neighboring MAC regions in the universe, which exactly fill a region bounded by surfaces of constant longitude, constant latitude and constant altitude, then the whole region can be represented by a single group MAC which uses a hyphen to link the relative coordinate characters of the first MAC with the relative coordinate characters of the last MAC in each direction with multiple MAC regions respectively, for example:

	MAC: NHJ-L TH KJH = MAC: NHJ TH KJH + MAC: NHK TH KJH + MAC: NHL TH KJH

	MAC: NHJ-L TH-J KJH = MAC: NHJ TH KJH + MAC: NHK TH KJH + MAC: NHL TH KJH
                     + MAC: NHJ TJ KJH + MAC: NHK TJ KJH + MAC: NHL TJ KJH

	MAC: NHJ-L TH-J KJH-J = MAC: NHJ TH KJH + MAC: NHK TH KJH + MAC: NHL TH KJH
				+ MAC: NHJ TJ KJH + MAC: NHK TJ KJH + MAC: NHL TJ KJH
				+ MAC: NHJ TH KJJ + MAC: NHK TH KJJ + MAC: NHL TH KJJ
				+ MAC: NHJ TJ KJJ + MAC: NHK TJ KJJ + MAC: NHL TJ KJJ
The number of characters after the hyphen in a character string represents the number of the characters of the relative coordinate. The characters before the hyphen in a character string represent the first MAC region coordinate in this direction. The characters before the hyphen with its last characters replaced by the characters after the hyphen in the character string represent the last MAC region coordinate in this direction. For example, MAC: NHJ-LZ TH KJH represents a three-dimensional region which starts from the region of MAC: NHJ TH KJH and ends by the region of MAC: NLZ TH KJH, that is,
	MAC: NHJ-LZ TH KJH = MAC: NHJ-Z TH KJH + MAC: NK0-Z TH KJH + MAC: NL0-Z TH KJH
It is the same for MACs with hyphens in two or three character strings, such as:
	MAC: FP-GZ TH-ZK HJK = MAC: FP-Z TH-Z HJK + MAC: G0-Z TH-Z HJK
				+ MAC: FP-Z V0-Z HJK + MAC: G0-Z V0-Z HJK
				+ MAC: FP-Z W0-Z HJK + MAC: G0-Z W0-Z HJK
				+ MAC: FP-Z X0-Z HJK + MAC: G0-Z X0-Z HJK
				+ MAC: FP-Z Z0-K HJK + MAC: G0-Z Z0-K HJK 

Definition b When a MAC with 0-Z at the end of its character string, these three characters can be omitted in the character string provided there are some characters left in the character string, for example:

	MAC: JJ0-Z KKL HG = MAC: JJ KKL HG
 
	MAC: JJ0-Z KKL0-Z HG0-Z = MAC: JJ KKL HG

Definition c An exponent has been introduced to represent the repetition of one same character in a MAC coordinate string, for example:

	MAC: RGJJJJK RDF FDS = MAC: RGJ(4)K RDF FDS

	MAC: RGGGH HFF ZZZZZ = MAC: RG(3)H HF(2) Z(5)

The exponential expressions will be useful in representing far distant objects in the universe.

Definition d Rotation rule. If the character at the right side of a hyphen of a MAC character string represents a number smaller the number represented by the character at the left side of the hyphen, then the rotation rule should be applied, e.g.

	MAC: RGJ-B WDF = MAC: RGJ-Z WDF + MAC: RH0-B WDF
With the above definitions, the concept of MAC regions has been extended to include any regions in the universe, bounded by surfaces of constant longitude, constant latitude and constant altitude, and the concept of MAC areas has been extended to include any areas on Mars, bounded by lines of constant longitude and constant latitude. Every MAC region or MAC area can be expressed by a single group MAC. Since the side ratios and size of a MAC area or region can be any values, a MAC in fact can represent any point in the universe, any line section of constant longitude or constant latitude on Mars, any area bounded by lines of constant longitude and constant latitude on Mars, any region bounded by surfaces of constant longitude, constant latitude and constant altitude in the universe.

Advantages

The Martian Area Coding System has special advantages over other systems.

First, it integrates the concepts of geodetic points, line sections, areas and regions and generates a unified form to represent all these objects.

Second, it generates short coordinates for all these objects, for examples:

For a geodetic point, the following are equivalent:  
	
	MAC: 2CHD Q87M 

	Longitude West 151.3947, Latitude North 43.6508 
For a line section, the following are equivalent:
	MAC: 2C Q87M

	Piont 1: Longitude West 151.5902, Latitude North 43.6508
	Point 2: Longitude West 151.1902, Latitude North 43.6508
For an area, the following are equivalent:
	MAC: 2C Q8

	Northwest corner: Longitude West 151.5902, Latitude North 43.8033
	Southwest corner: Longitude West 151.5902, Latitude North 43.6033
	Northeast corner: Longitude West 151.1902, Latitude North 43.8033
	Southeast corner: Longitude West 151.1902, Latitude North 43.6033
For a three-dimensional region, the following are equivalent:
	MAC: 2C Q8 H000

	The bottom surface has the height = 0 meter above the geoid 
	surface and four corners on the surface are:

	Northwest corner: Longitude West 151.5902, Latitude North 43.8033
	Southwest corner: Longitude West 151.5902, Latitude North 43.6033
	Northeast corner: Longitude West 151.1902, Latitude North 43.8033
	Southeast corner: Longitude West 151.1902, Latitude North 43.6033

	The upper surface has the height = 25 meters above the geoid
	surface and four corners on the surface are:

	Northwest corner: Longitude West 151.5902, Latitude North 43.8033
	Southwest corner: Longitude West 151.5902, Latitude North 43.6033
	Northeast corner: Longitude West 151.1902, Latitude North 43.8033
	Southeast corner: Longitude West 151.1902, Latitude North 43.6033
The Martian Area Coding System can save about 50% of characters for points, 87% for 2D MAC areas and 94% for 3D MAC regions.

Third, the simple MAC can be used to represent map both in digital and hardcopy forms. If the MAC is used for digital map then all the geodetic coordinates of the map can be saved by the relative MAC to save another 50% memory and make the database of maps extremely efficient in retrieving and storing maps. If the MAC is used to name a hardcopy map, then the maps will be very well shelved and will be very conveniently retrieved too.


Copyright 2004, NAC Geographic Products Inc. All Rights Reserved.