Consider a hollow sphere of radius R, in which a light source is placed at its centre. A square plate of side length S is held in place within the sphere by a pole of length L units. The square plate's position is then such that the displacement between the centre of the square and the light source is R-L units.

The square plate is also oriented in a way such that an imaginary line drawn perpendicular to the surface of the plate and passing through the plate's centre will pass through the light source.

Determine the surface area of the shadow formed on the spherical shell, due to the square plate.

Divide the square in two with a diagonal
and double the the area of the resulting
spherical triangle.

Area of square shadow = two times area of
                        triangle shadow

Area of spherical triangle 

    = R^2*( A' + B' + C' - pi )

Where A', B', and C' are the angles between 
arcs of great circles. Let B' be the image
of the angle of the square that was not
divided by the diagonal. Then,

    A' = C' = B'/2

Therefore, Area of spherical triangle 

    = R^2*( 2*B' - pi )

The angle B' is the angle between the faces
of the square pyramid whose apex is the
center of the sphere and whose base is the
square. It is given by the following:

    arccos[ -s^2/(s^2 + 4h^2) ]

Where h = R-L which is the height of the
pyramid. Therefore, the area of the square
shadow is

    2*R^2*[2*arccos(-s^2/[s^2 + 4*(R-L)^2]) - pi]

Posted by Bractals on 2009-08-19 18:14:21