Real-time approximate and exact CSG of implicit surfaces on the GPU

We present a simple and powerful scheme to allow CSG of implicit surfaces on the GPU. We decompose the boolean expression of surfaces into sum-of-products form. Our algorithm presented in this paper then renders each product term, sum of products can be automatically by enabling depth test. Our Approximate CSG uses adaptive marching points algorithm for finding ray-surface intersection. Once we find an interval where root exists after root-isolation, this is used for presence of intersection. We perform root-refinement only for the uncomplemented terms in the product. Exact CSG is done by using the discriminant of the ray-surface intersection for the presence of the root. Now we can simply evaluate the product expression by checking all uncomplemented terms should be true and all complemented terms should be false. If our condition is met, we find the maximum of all the roots among uncomplemented terms to be the solution. Our algorithm is linear in the number of terms O(n). We achieve real-time rates for 4-5 terms in the product for approximate CSG. We achieve more than real-time rates for Exact CSG. Our primitives are implicit surfaces so we can achieve fairly complex results with less terms.