Visible Fraction Of Total Surface Area Of A Sphere

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 Earth radius and the mean geocentric lunar distance in kilometers.

Visible or illuminated fraction = 0.491717701047
Distance to eye or point-light is reckoned from center.

A Sphere and a Point

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

The tiny white point can represent different things, depending on the context. It can represent a point light source radiating light or an electromagnetic signal unformly in all directions and illuminating a portion the sphere. This point-light source is dimensionless, that is, the only property it has is coordinates. No particular size or brightness level is associated with it.

The point can also represent the eye or camera through which we are viewing the sphere from any given distance, the illuminated area representing the portion of the sphere visible from that point. The imaginary lens is dimensionless, that is, it only has coordinates, no actual focal length or aperture width is associated with it.

The point might also represent an orbiting spacecraft or satellite, in which case, the illuminated area would indicate the region of the planetary surface from which the object would potentially be visible or within range of its communication signals.

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 eye 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:

Where: R > 0 and d > 0

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

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 eye 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:

Where: D > R and R > 0

As the distance (D) from the center of the sphere increases towards infinity, the potentially visible fraction (f) approaches a limit of 1/2 (0.5) of the total surface area.