Distance From Eye or Foot to Horizon of a Sphere
PHP Program by Jay Tanner
This program computes the line-of-sight distance and the curved surface distance to the horizon of a sphere at any distance of the eye as reckoned from the surface or center of the sphere.   The radius and distance can be in any convenient linear units.  

The initial default radius is the mean lunar radius and the distance is a random distance value from 1 to 500 miles above the surface of the moon.

Radius units
Distance units
Distance To Eye Reckoned From:     Center           Surface  
Line-of-Sight Distance (H) to Horizon = 547.67755111927 units
Surface Distance (S) to Horizon = 506.82108291403 units
Distance (d) to the eye is reckoned from the surface of the sphere. This corresponds to the minimum radar distance to the nearest surface point.




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:


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:


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


Where the Distance to the Eye Is Reckoned From the Surface of the Sphere:

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:


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:


Where: R > 0 and d > 0 and the inverse angle is expressed in radians
Jay Tanner - 2024