Surface fitting

To determine the flux on an arbitrary surface two ele­ments are required, the en­ergy striking a region on a surface and the area of that individual region. The gener­ated data cube provides us with the first of these com­ponents, the energy. What is then required, is those energy values need to be compensate for the non­horizontal-planar surface en­ergy calculation and generate a curve that best fits all of these corrected data points.

Initially, from the data cube points in space representing 400 time the solar concentra­tion for planar surfaces below the focal point were extracted

an example of which can be seen in Figure 2. This provided a list of points in space of equal energy, and from this list a surface can be generated that best represents those points. For our simulations we chose to use only a quadric surface as described by [13]
and included for the readers convenience. This method describes the fitting of a quadric surface using the least squares method which has the general form

M2 + k2y2 + k3z2 + k4xy + k5yz + k6zx + k7x + k8y + k9z = 1, (1)

where x, y,z represent relative displacements about an orthogonal basis in three- dimensions and the nine coefficents ki(i = 1,2 9) define a unique quadric surface.

The input data to the fitting procedure is a set of 3D coordinates of the sample points (a series of (x, y,z) values). If there are m sample points, there are m( x, y, z) values. The­oretically, the nine unknown coefficients can be solved from a group of nine linear equa­tions of the form of Equation 1, each of which has one data point (xi, yi, zi) assigned to its corresponding variables, x, y and z. However, due to sampling errors such a result is not robust [13].

A more practical way to solve this quadric fitting problem is to find the least squares solu­tion of a group of m linear equations,

where,

generated by substituting in each of the m, points defining regions of constant illumina­tion,

X0 = [k1 k2 k3 k4 k5 k6 k7 k8 kg],

representing the coefficients and

bmx1 [1 1 … 1] .

Normally, the least squares solution does not satisfy all equations in the group, but it min­imises the value of the residual error

er = 11 AoXo_Bo |І2,

and this solution can be considered the optimal least squares solution of the problem.

The solution X0, can then be computed by the normal equation

Xo = (ATAo)’1ATb.

Having determined the coefficients k, the surface normal at each of the points is defined by the vector [14],

[2k1 Xi + k4yi + k5Zi + k7, 2k2Yi + k4Xi + kaZi + ke, 2k3Zi + k4Xi + k5yi + kg].

The initial data cube can now be rescaled by dividing each of the corresponding points by the dot product of this surface normal vector and the z-normal vector (representing the deviation of the flux from striking a non-horizontal-planar surface). The actual surface rep­resenting areas of uniform flux can then be generated by iteratively applying these meth­ods to optimise the surface area and energy combination until it falls within reasonable tolerances.