Visible or Illuminated Fraction of Total Surface Area of a Sphere

This program computes the visible fraction or illuminated fraction of the total surface area of a sphere at any distance from the eye or point light source as reckoned from the center or surface of the sphere.

The radius and distance can be in any convenient linear units.   The default values are the mean lunar radius and the mean geocentric lunar distance in statute miles.

Radius units
Distance units
Distance To Point Reckoned From:     Center           Surface  



For a sphere of radius 1079.35 units, where the distance to the eye
or point-light source is 238856 units from the center:

Visible or illuminated fraction of surface = 49.774058 percent
Invisible or shaded fraction = 50.225942 percent
Difference = 0.451884 percent




A Sphere and a Point

Mathematically, this is a simple spatial problem involving simple 3D space containing a sphere and a point.

The tiny white point in the diagram to the left could represent different things, depending on the context.

If the point represented a point-light source, then it would simply illuminate the sphere as shown.

If the point represented the eye or camera lens, then the illuminated area could indicate the fraction of the total spherical surface visible from that point.

The distance between the point and the sphere can be reckoned either from the center or from the surface of the sphere, whichever is more convenient.

Where the Distance to the Point is Reckoned
From the Center of the Sphere:

Given the radius (R) of the sphere and the distance (D) of the point from the center, the general equation for the potential fraction (f) of the total surface area of the sphere as viewed or illuminated from that point is:

As the distance (D) from the center of the sphere increases towards infinity, we can see slightly more and more of the surface but then we have to magnify the image more and more to make out details, to the extent of the limits of our imaging technology. 

As the distance from the sphere approaches infinity, the potentially visible or illuminated fraction (f) approaches a maximum limit of 0.5 (1/2) of the total surface area. 

For the moon, at (d) kilometers or statute miles from the center:

Where the Distance to the Point is Reckoned
From the Surface of the Sphere:

Given the radius (R) of the sphere and the distance (d) of the point from the surface, the general equation for the potential fraction (f) of the total surface area of the sphere as viewed or illuminated from that point is:
As the distance from the surface of the sphere approaches infinity, the potentially visible or illuminated fraction (f) approaches a maximum limit of 0.5 (1/2) of the total surface area.

For the moon, at (d) kilometers or statute miles from the surface:



© 2017 - Jay Tanner