A rapidly growing computer graphics community has contributed to dramatic increase in complexity with respect to geometry as well as physical phenomena. Simulating, approximating and visualizing geometry consisting of tens of millions of polygons simultaneously tested for collision or visibility is becoming increasingly common. Further, recent technological innovations from graphics card vendors have given impetus to achieving these results at very high frame rates. Despite tremendous developments in graphics hardware, capturing the complete surrounding environment poses a significant challenge. Given the added time constraint for real-time or interactive rates, simplified representations and suitable approximations of physical effects are of key importance.<br /><br />This dissertation focuses on simplified representations and computations to achieve real-time performance for complex tasks and concentrates on a variety of topics including simplification, visibility, soft shadows and voxelization.