Miscellaneous formulas.

Ellipses.

The formula that the program uses for the elliptical shape is

(x/a)**2 + (y/b)**2 = 1,

where a and b are the minor and major radii respectively, and where x and y are local orthogonal axes. The area of this ellipse is pi*a*b.

Test planes.

In CPO2D the 'test planes' are defined by the 3 numbers a, b and c, giving

a*r + b*z = c for cylindrical symmetry,

a*x + b*z = c for planar symmetry.

For example, a = 0 gives b*z = c, which is z = c/b, which is a plane at constant z, for both cylindrical and axial symmetry. For cylindrical symmetry, the 'plane' defined by non-zero values of both a and b is in fact a conical surface. When b is zero it becomes the cylindrical surface r = c/a. When b and c are both zero (but a is non-zero), the 'plane' becomes the axis of the system -a frequently used condition.

In CPO3D the 'test planes' are defined by the formula

a*x + b*y + c*z = d

which gives a unique flat surface for constant values of a, b, c and d.

For example for a = 1, b = 0, c = 0, d = 5, the plane is x = 5.

Extrapolation of number of segments to infinity.

For a power law 1/Np, and values s1 and s2 obtained at N=N1 (the highest value) and N2 respectively, the extrapolated value of s is

sinf = (s2/N1p - s1/N2p)/(1/N1p - 1/N2p)

But be careful not to use this formula when N1 and N2 are too close -they should differ preferably by a factor of 4 or more, but at least 2.

Cylindrical mirror analyzer

See test2d16.

Field at tip of a cone

See test2d19.