Как выбрать гостиницу для кошек
14 декабря, 2021
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.