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biologist

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- TL;DR Summary
- Any length over the surface of a sphere is an arc length. Arc length increases with radius. On Earth, this would translate to increase in distance (arc length) with increase in altitude (radius). This is not the case based on GPS results.

Consider the following example:

Point A has coordinates 45 lat, 0 long. Point B has coordinates 45 lat, 2 long. Both points are 5000 ft above sea level. The distance between them is X.

Point C has coordinates 45 lat, 100 long. Point D has coordinates 45 lat, 102 long. Both points are at sea level. The distance between them is Y.

X should be greater than Y because the difference in altitude on a sphere means a difference in radius, which means a difference in arc length. This does not seem to be the case for distances on earth, according to GPS.

Here is the equation for the difference between any distance X and distance Y on Earth assuming both distances are:

1. At the same latitude.

2. Span the same amount of degrees in longitude.

X-Y =.017 x (degrees of longitude between both distances) x (difference in altitude) x cos(latitude)

For the first example, the difference would be:

X-Y = .017(2)(5000)cos(45) = 120.208 ft

The equation has a straightforward geometric derivation that I'll provide at the end of this post. Assuming the equation is valid, however, here's an example where, using GPS, there appears to be no difference between distances X and Y:

Distance X spans across Issyk Kul, a lake in Kyrgyztan, between point A (42.446918,76.18375419) and point B (42.446918,77.86265619). The lake is 5272 ft above sea level.

Distance Y spans across the Caspian Sea between point C (42.446918,47.96593019) and point D (42.446918,49.64483219) at 92 ft below sea level.

Using GPS (Earth map online service (https://satellites.pro/)) both distances are equal to 85.60 miles. According to the above equation, there should be a difference of .02 miles, which should be within GPS accuracy. Both distances span the same amount of longitudinal degrees. The distances are both over bodies of water to rule out surface irregularities.

Looking for possible explanations for this.DERIVATION:

radius of a circle of longitude = r = (radius of earth) x cos(latitude)

radius of a circle of longitude above sea level = R = (radius of Earth + altitude) x cos(latitude)

R - r = (difference in altitude) x cos(latitude)

Arc length at sea level = l = 2pi x (r) x (change in longitude/360)

Arc length above sea level = L = 2pi x (R) x (change in longitude/360)

L - l = 2pi x (change in longitude/360) x (R-r) = .017 x (change in longitude) x (R-r) = .017 x (change in longitude) x (difference in altitude) x cos(latitude)

Point A has coordinates 45 lat, 0 long. Point B has coordinates 45 lat, 2 long. Both points are 5000 ft above sea level. The distance between them is X.

Point C has coordinates 45 lat, 100 long. Point D has coordinates 45 lat, 102 long. Both points are at sea level. The distance between them is Y.

X should be greater than Y because the difference in altitude on a sphere means a difference in radius, which means a difference in arc length. This does not seem to be the case for distances on earth, according to GPS.

Here is the equation for the difference between any distance X and distance Y on Earth assuming both distances are:

1. At the same latitude.

2. Span the same amount of degrees in longitude.

X-Y =.017 x (degrees of longitude between both distances) x (difference in altitude) x cos(latitude)

For the first example, the difference would be:

X-Y = .017(2)(5000)cos(45) = 120.208 ft

The equation has a straightforward geometric derivation that I'll provide at the end of this post. Assuming the equation is valid, however, here's an example where, using GPS, there appears to be no difference between distances X and Y:

Distance X spans across Issyk Kul, a lake in Kyrgyztan, between point A (42.446918,76.18375419) and point B (42.446918,77.86265619). The lake is 5272 ft above sea level.

Distance Y spans across the Caspian Sea between point C (42.446918,47.96593019) and point D (42.446918,49.64483219) at 92 ft below sea level.

Using GPS (Earth map online service (https://satellites.pro/)) both distances are equal to 85.60 miles. According to the above equation, there should be a difference of .02 miles, which should be within GPS accuracy. Both distances span the same amount of longitudinal degrees. The distances are both over bodies of water to rule out surface irregularities.

Looking for possible explanations for this.DERIVATION:

radius of a circle of longitude = r = (radius of earth) x cos(latitude)

radius of a circle of longitude above sea level = R = (radius of Earth + altitude) x cos(latitude)

R - r = (difference in altitude) x cos(latitude)

Arc length at sea level = l = 2pi x (r) x (change in longitude/360)

Arc length above sea level = L = 2pi x (R) x (change in longitude/360)

L - l = 2pi x (change in longitude/360) x (R-r) = .017 x (change in longitude) x (R-r) = .017 x (change in longitude) x (difference in altitude) x cos(latitude)