Abstract: Geometrical elements(point,line,plane,etc.)of round brilliant cut exist a functional relation,which can be expressed through analytic expressions.In this paper,quantitative relation of round brilliant cut,including cut parameters and the other geometric elements,is processed by mathematical method,visual computer software Pro/E and MATLAB. Evolution of pavilion of round brilliant cut is visualized and analyzed quantitatively by Pro/E and MATLAB, The technical standards of the former Soviet Union put forward by Tolkowsky and Bruce Harding, are unified in our mathematical framework, Positive and in-verse function relationship of β'-μ is revealed, Thus, gemstone optics and geometry are linked, Strictly speaking, round brilliant cut is an aesthetic effect of optical prism., Effect of the light and the material are embedded through the angle of components.Mathematical formula of star facet inclination (γ) which is a function of table size(M).crown angle(β') and crown girdle facets(β') are derived, Star facet inclination(γ) can not take any value freely. Therefore, it corrected Tolkowsky's biased viewpoint.The thickness and total area of waist can be quantitatively described by mathematicaformula, In the paper, b_min is minimum thickness of waist ridge which meets the structural mechanical demand, and ensures that the finished product is not easy to be broken.Three geometrical forms of crown vertical view are shown by using mathematical formulas and visualizations, and radian method which is one of the detection methods to determinethe table size is denied.