  We propose a generalized minimum discrepancy, which derives from Legendre's
ODE and spherical harmonic theoretics to provide a new criterion of
equidistributed pointsets on the sphere. A continuous and derivative kernel in
terms of elementary functions is established to simplify the computation of the
generalized minimum discrepancy. We consider the deterministic point generated
from Pycke's statistics to integrate a Franke function for the sphere and
investigate the discrepancies of points systems embedding with different
kernels. Quantitive experiments are conducted and the results are analyzed. Our
deduced model can explore latent point systems, that have the minimum
discrepancy without the involvement of pseudodifferential operators and
Beltrami operators, by the use of derivatives. Compared to the random point
generated from the Monte Carlo method, only a few points generated by our
method are required to approximate the target in arbitrary dimensions.
