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Distance Calculation

Any point on the Earth’s surface can be located by its latitude and longitude coordinates.

Latitude is the angle above or below the equator in degrees. The equator is zero degrees, the North pole is North 90 degrees latitude, and the South pole is South 90 degrees latitude. The continental United States falls between 25 and 50 degrees North.

Longitude is the angle East or West of the Greenwich meridian. The continental United States is between 70 and 125 degrees West.

Approximate Solutions

One degree of latitude is equal to 69.1 miles. One degree of longitude is equal to 69.1 miles at the equator. North or South of the equator, 1 degree of longitude is a smaller distance. It’s reduced by the cosine of the latitude. Dividing the latitude number by 57.3 converts it to radians.

\[DistanceLat = 69.1\, (Lat_2 - Lat_1)\]

\[\begin{split}DistanceLong = 69.1\, (Long_2- Long_1)\, cos\, (\frac{Lat_1}{57.3})\\\end{split}\]

\[Distance = \sqrt{DistanceLat^2 + DistanceLong^2}\]

If you don’t want to use the COS function, then a good approximate solution is:

\[DistanceLat = 69.1\, (Lat_2 - Lat_1)\]

\[DistanceLong = 53\, (Long_2 - Long_1)\]

\[Distance = \sqrt{DistanceLat^2 + DistanceLong^2}\]

Exact Solution

To calculate the exact distance between points requires spherical geometry.

\[D = 3959\, cos^{-1}\, (sin\, (Lat_1)\, sin(Lat_2)\, + cos\, (Lat_1)\, cos\, (Lat_2)\, cos\, (Long_2 - Long_1))\]

The above formula has one major problem: most computer languages do not support the arccosine function. Therefore a different form is necessary, one that uses the arctangent function that most languages embrace.

\[\begin{split}D = 3959\, tan^{-1}\, (\frac{\sqrt{1-A^2}}{A})\\\end{split}\]

where A is equal to:

\[A = sin\, (Lat_1)\, sin\, (Lat_2) + cos\, (Lat_1)\, cos\, (Lat_2)\, cos\, (Long_2-Long_1)\]

Most computer languages compute the sine and cosine function with the angle given in radians. To convert degrees to radians, divide degrees by the constant:

\[\begin{split}57.3\,({\frac{180}{\pi}\\})\end{split}\]

In BASIC this would be programmed as follows:

\[C = 57.3\]

\[\begin{split}A = sin\, ({\frac{Lat_1}{C})\\}\, sin\, ({\frac{Lat_2}{C})\\} + cos\, ({\frac{Lat_1}{C}\\})\, cos\, ({\frac{Lat_2}{C}\\})\, cos\, ({\frac{Long_2}{C}\\}-{\frac{Long_1}{C}\\})\end{split}\]

\[\begin{split}D = 3959\, tan^{-1}\, (\frac{\sqrt{1-A^2}}{A})\\\end{split}\]

Where:

D = distance in statue miles from the first to the second point.

C = degrees to radians constant 57.3.

Lat₁ = latitude of the first point in degrees.

Long₁ = longitude of the first point in degrees.

Lat₂ = latitude of the second point in degrees.

Long₂ = longitude of the second point in degrees.

NOTE: Dividing the latitude number by 57.3 converts it to radians.

Since all ZIP Code points in the United States are North latitude and West longitude, there is no need to check the sines (positive and negative) of the latitudes and longitudes.

Some programming languages, such as dBASE II/III, do not have the functions of the sine, cosine and arctangent. Also, the formulas given are time-consuming to calculate. A simpler but less accurate method is given here:

\[D = 69.1\, \sqrt{(Lat_2-Lat_1)^{2} + 0.6\, (Long_2 - Long_1)^{2}}\]

Although this formula is not as accurate as the first great circle method, it will give good results for most applications.

In basic, it is written as follows:

\[D = 69.1\, \sqrt{(Lat_2-Lat_1)^{2} + .6\, (Long_2 - Long_1)^{2}}\]

In dBASE III, it is written as follows:

\[D = 69.1\, \sqrt{(Lat_2-Lat_1)^{2} + 0.6\, (Long_2 - Long_1)^{2}}\]

To make using the database easier, the latitudes and longitudes are given in the decimal format instead of the degree, minute and second format. The latter would require the additional steps of converting seconds to fractions of a minute and then minutes to a fraction of a degree.

Bearing Formula

The bearing is the direction from the first point to the second point. It is expressed as an angle from North in degrees. Due North is a bearing of zero degrees, East is a bearing of 90 degrees, South is 180, and west is 270.

The bearing from the first point to the second is calculated with:

\[B = tan^{-1}\, (\frac{sin(Lat_1)\, sin(Long_2 - Long_1)}{sin(Lat_2)\, cos(Lat_1) - cos(Lat_2)\, sin(Lat_1)\, cos(Long_2 - Long_1)})\]

Since the ARCTAN function gives the angle in radians, it is necessary to convert B to degrees by multiplying by

\[\begin{split}57.3\,({\frac{180}{\pi}\\})\end{split}\]

To use this equation requires some special considerations. This example assumes that Lat₁, Lat₂, Long₁, Long₂ have been converted to radians.

Use the following steps:

\[N = sin\, (Lat_1)\, sin\, (Long_1 - Long_2)\]

\[D = sin\, (Lat_2)\, cos\, (Lat_1)\, – \, cos\, (Lat_2)\, sin\, (Lat_1)\, cos\, (Long_1 - Long_2)\]

\[\begin{split}B = 57.3\, tan^{-1}\, ({\frac{N}{D}\\})\end{split}\]

If (D > 0) then (B = 360 + B)

If (D < 0) then (B = 180 + B)

If (B < 0) then (B = 360 + B)

Where:

N is the Numerator.

D is the Denominator.

B is the Bearing.

Examples

The following examples should give you a feel for the numbers. For these examples, we will use the points of Schenectady, NY and Los Angeles, CA.

First Point is in Schenectady, NY, 12345

Lat₁ = 42.8145

Long₁ = 73.9380

Second Point is in Los Angeles, CA, 90001

Lat₁ = 34.0515

Long₁ = 118.2420

Example 1: Approximate Formula (1)

\[DistanceLat = 69.1\, (Lat_2 - Lat_1) = 605.5\]

\[DistanceLong = 69.1\, (Long_2- Long_1)\, cos\, (\frac{42.8145}{57.3}) \ = 2245.8\]

\[Distance = 2326.0 \ miles\]

Example 2: Approximate Formula (2)

\[DistanceLat = 69.1\, (Lat_2 - Lat_1) = 605.5\]

\[DistanceLong = 53\, (Long_2 - Long_1) = \ 2348.1\]

\[Distance = 2424.9 \ miles\]

Example 3: Exact Solution Formula

\[\begin{split}A_1 = sin\, ({\frac{Lat_1}{57.3})\\}\, sin\, ({\frac{Lat_2}{57.3})\\} = \ 0.3805\end{split}\]

\[\begin{split}A_2 = cos\, ({\frac{Lat_1}{57.3}\\})\, cos\, ({\frac{Lat_2}{57.3}\\})\, cos\, ({\frac{Long_2}{57.3}\\}-{\frac{Long_1}{57.3}\\}) = \ 0.4350\end{split}\]

\[A =\ A_1 + A_2 = \ 0.8155\]

\[\begin{split}D = 3959\, tan^{-1}\, (\frac{\sqrt{1-A^2}}{A})\\\end{split}\]

\[D = 3959\, tan^{-1}\, (\frac{\sqrt{1-(0.8155)^2}}{0.8155}) \ = 2443.4\]

Example 4: Bearing Formula

\[\begin{split}N = sin\, ({\frac{Lat_1}{57.3})\\}\, sin\, ({\frac{Long_2}{57.3}\\}-{\frac{Long_1}{57.3}\\}) = \ 0.4747\end{split}\]

\[\begin{split}D_1 = sin\, ({\frac{Lat_2}{57.3})\\}\, cos\, ({\frac{Lat_1}{57.3})\\} = \ 0.4107\end{split}\]

\[\begin{split}D_2 = cos\, ({\frac{Lat_2}{57.3})\\}\, sin\, ({\frac{Lat_1}{57.3})\\}\, cos\, ({\frac{Long_2}{57.3}\\}-{\frac{Long_1}{57.3}\\}) = \ 0.4029\end{split}\]

\[\begin{split}B = 57.3\, tan^{-1}\, ({\frac{N}{D_1 - D_2}\\}) = \ -81.9 \,Degrees\end{split}\]

\[Bearing = 360 + B = 360 - 89.1\]

\[Bearing = 270.9\, degrees\, from\, North\]

Abbreviations and Codes

Military

Abbreviation	Region
AA	Atlantic
AP	Pacific
AE	Europe

State

Abbreviation	Code	State
AL	01	Alabama
AK	02	Alaska
AZ	04	Arizona
AR	05	Arkansas
CA	06	California
CO	08	Colorado
CT	09	Connecticut
DE	10	Delaware
DC	11	District of Columbia
FL	12	Florida
GA	13	Georgia
HI	15	Hawaii
ID	16	Idaho
IL	17	Illinois
IN	18	Indiana
IA	19	Iowa
KS	20	Kansas
KY	21	Kentucky
LA	22	Louisiana
ME	23	Maine
MD	24	Maryland
MA	25	Massachusetts
MI	26	Michigan
MN	27	Minnesota
MS	28	Mississippi
MO	29	Missouri
MT	30	Montana
NE	31	Nebraska
NV	32	Nevada
NH	33	New Hampshire
NJ	34	New Jersey
NM	35	New Mexico
NY	36	New York
NC	37	North Carolina
ND	38	North Dakota
OH	39	Ohio
OK	40	Oklahoma
OR	41	Oregon
PA	42	Pennsylvania
RI	44	Rhode Island
SC	45	South Carolina
SD	46	South Dakota
TN	47	Tennessee
TX	48	Texas
UT	49	Utah
VT	50	Vermont
VA	51	Virginia
WA	53	Washington
WV	54	West Virginia
WI	55	Wisconsin
WY	56	Wyoming
AS	60	American Samoa
FM	64	Micronesia
GU	66	Guam
MH	68	Marshall Islands
MP	69	North Mariana Islands
PW	70	Palua
PR	72	Puerto Rico
UM	74	Minor Islands
VI	78	Virgin Islands

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