**Where the Distance to the Eye Is Reckoned From the Center of the Sphere:**
Given the radius (

`R`

) of the sphere and the distance (

`D`

) of the eye from the center of the sphere, the general equation for the line-of-sight distance (

`H`

) to the sphere horizon viewed from that point is:

$$\large H ~=~ \large\sqrt{D^2 ~-~ R^2}$$
The horizon surface distance

`(S)`

, corresponding to

`(H)`

, but measured along the curved surface of the sphere directly below H and corresponding to the sailing distance is:

$$\large S ~=~ R\cdot\left[\Large\frac{\pi}{2} ~-~ cos^{-1}\left(\large\frac{\sqrt{D^2 ~-~ R^2}}{D}\right)\right]$$

Where: **D** > **R** and **R** > **0** and the inverse angle is expressed in radians

Given the radius

`(R)`

of the sphere and the distance

`(d)`

(height

) of the eye above the surface of the sphere, the general equation for the line-of-sight distance

`(H)`

to the sphere horizon viewed from that point is:

$$\large H ~=~ \large\sqrt{(R ~+~ d)^2 ~-~ R^2} ~=~ \sqrt{d^2 +~ 2Rd}$$

The horizon surface distance

`(S)`

, corresponding to line-of-sight distance

`(H)`

, but measured along the curved surface of the sphere, corresponding to the sailing distance is:

$$\large S ~=~ R\cdot\left[\frac{\pi}{2} ~-~ cos^{-1}\left(\frac{\sqrt{d^2 +~ 2Rd}}{R ~+~ d}\right)\right]$$

Where: **R** > **0** and **d** > **0** and the inverse angle is expressed in radians

**© Jay Tanner - 2019 - v2.0**