TITLE: Distance between two points in UTM coordinates.
QUESTION [5 upvotes]: This question is partially about geometry of the sphere and partially about Universal Transverse Mercator coordinates. I realize the latter is not completely on-topic here, but I hope the question is still close enough to be deemed relevant.
UTM coordinates are given as a zone number together with an easting-northing pair. Given two points $(E_1,N_1)$, $(E_2,N_2)$ in the same zone is it the true that the distance "over the Earth" between the two points is well approximated by the euclidean distance between the easting-northing coordinates? I.e.:
$$\sqrt{(E_1 - E_2)^2 + (N_1 - N_2)^2}$$

REPLY [2 votes]: One additional aspect not addressed above is the nature of the terrain between the two points. UTM is a $2$ dimensional Cartesian coordinate system. Points on (above or under) the Earth's surface exist in a $3$-dimensional environment with hills, valleys, etc. To illustrate the simplest case, point $B$ is at a different elevation ($e$) than point $A$ with a constant slope. After calculating the planar distance ($d$) between the two points (as described above), one must then adjust for the slope such that the actual distance ($D$) is defined by the following formula:
$$D = \sqrt{( .9996d)^2 + (e_B - e_A)^2} \cdot \left(\frac{\frac{e_B+e_A}2 + R_0}{R_0}\right)$$
Where 
$d$ is the planar distance between points A and B assuming both are at sea level;
$e_B$ is the elevation of point $B$;
$e_A$ is the elevation of point $A$; and
$R_0$ is the radius of the Earth.