Abstract: The insolation-duration on global slope surface can be calculated by the sunrise and sunset hour angle on the slope. According to laws of the match relations of sunrise and sunset hour angle on slope surface, there are four formulae with different forms for calculating the insolation-duration on global slope surface. These formulae are as follows:
Ts=2Eω0
Ts=2Eωx
Ts=E(ω0+ωx-|ωm|)
Ts=E(2ω0+2ωx-2π)
Where Ts is insolation-duration on slope, E is a constant 3.82 (hour/radian), which is the conversion factor from radian to hour. ω0 is the sunrise and sunset hour angle on horizontal surface. ωx is the sunrise and sunset hour angle on non-horizontal surface.ωm is a parameter of the slope. These formulae can be treated as function of the latitude, slope angle, azimuth and the declination. The influence of slope and azimuth on insolation-duration on slope surface is considerable complicated. In different latitude and declination,the influence of slope and azimuth on insolation-duration is different too. According to the partial derivative of insolation-duration to the slope, through fair and foul, it is true that the insolation-duration on slope increases or the insolation-duration holds the line with slope increasing. So, the insolation-duration on slope is always longer than or equal to that value on horizontal surface.This law can be extended to any period of time such as one month or one season. According to the partial derivative of insolation-duration to azimuth, it is quite complicated that the insolation-duration on slope varies with azimuth. Beyond the two critical azimuths, the insolation-duration on slope varying with azimuth is flat or does not vary with azimuth. When the azimuth is between the two critical azimuths, there are three different cases for the insolation-duration on slope varying with azimuth. For given declination and slope, there are two critical latitudes that are function of the slope and declination. Between the two critical azimuths and between the two critical latitudes, there is a minimum of the insolation-duration on slope varying with azimuth. Between the two critical azimuths and beyond the two critical latitudes, the insolation-duration on slope varying with azimuth is flat, the insolation-duration on slope increases or decreases always with azimuth.From all azimuths of slope surface,the two critical azimuths maybe the azimuth in which the insolation-duration on slope is extremum of all insolation-duration. Between the two critical azimuth, if the latitude is equal to declination, the insolation-duration on slope does not vary with azimuth.In north pole or south pole, the insolation-duration on slope does not change with azimuth too. The insolation-duration on slope surface is equal between opposite azimuth with the same slope angle, latitude and the declination. The law of the influence of slope and azimuth on insolation-duration on slope surface in the Northern Hemisphere is fit for the Southern Hemisphere too. But in the Southern Hemisphere, the influence of slope and azimuth on insolation-duration on slope surface is opposite to the Northern Hemisphere in azimuth of slope and the declination.