Details of the 3D inscribing correction.

To correct by hand, use the following formulae, which have been obtained empirically:

(1) For a 3D sphere or part of a sphere, multiply radius by

1 + 0.375*omega

where omega is the average solid angle subtended by a triangular segment. An equivalent formula is

1 + 4.93/n,

where n is the total number of triangles into which the total surface area 4*pi*radius**2 of the sphere is divided -eg if you enter a part of a sphere of radius 1, between bounding circles of radius 1 and 0.5 say, then its area is 0.433 that of the whole sphere, and if there are 2 reflection planes then

n = (1/.433)*4*n1,

where n1 is the subdivision number that you have entered. (To find an area use fact that a spherical cap of half-angle theta has an area 4*pi*radius**2*sin(theta/2)**2)

The above constant (.375) is empirical and is that which gives the most accurate rays in a spherical deflection analyser. The constant is similar to the theoretical value for a right-angled isosceles triangle to make its mean distance from the centre equal to the nominal radius, namely 0.333. (For triangles that are not right-angled and isosceles, multiply the constant by (4*A**2+b**4)/(4*A*b**2), where A is the area of the triangle and b is the length of the longest side. This correction is significant for triangles that are long and thin, but remember that if such a triangle is subdivided then the correction factors for the subdivisions will be much closer to unity).

(2) For a 3D cylinder or cone, or a 2D circle (in planar or cylindrical symmetry) multiply the radius by

1 + 3.91/n**2,

where n is the total number of parts into which the 360 degrees in the azimuthal direction are divided (and remember that this number is the second 'subdivision' number that you have entered, multiplied in the case of a whole cylinder by 2 for every reflection plane that passes through the axis of the cylinder) (The constant is near to the theoretical value that makes the mean distance from the centre equal to the nominal radius, (pi**2)/3 = 3.29)

The 'iterative subdivision' option is not allowed (in CPO3D) if any of the segments have been inscribed (although it has been allowed in CPO2D).

The reason for this is that the program does not save the original radius or other original information on the segments when the electrodes are subdivided in more than one stage (that is, if 'iterative subdivision' is used, for example in order to make the charges on all the segments approximately equal).

A warning: If two neighbouring electrodes touch or are close to each other, and their radii are increased by the inscribing correction, then they might overlap (which is inadvisable, see note on 3D electrode types), so leave an appropriate gap between them.