ABSTRACT
A specific type of implicit surfaces are those for which the value of the scalar field, then referred to as a distance field, at a given point in space corresponds to (a lower bound of) the signed distance of that point to the surface, the latter being intrinsically defined by the isovalue v = 0. Instead of resorting to a numerical root finding algorithm or iteratively36 marching with a fixed step along the ray, the intersection of the latter with the surface can instead be more efficiently computed by adaptively defining each ray-marching step based on the absolute value of the distance field at the current location, the distance-estimated surface being guaranteed not to be intersected within this radius in any direction around the point. The resulting scheme, known as sphere tracing [Hart, 1996], is illustrated in Figure 7.1. The306 iterative process may then be stopped whenever the magnitude of the distance falls below a predefined threshold, indicating that the current position along the ray is sufficiently close to the surface, while the surface normal at the intersection point is readily given by the field gradient, which can be evaluated either analytically in simple cases, or numerically by using a finite-differences scheme in the case of more complex functions.
