A method and an apparatus render on a display screen (14) an image of a three-dimensional object. The object is represented by image data arranged in a constructive solid geometry format including at least one halfspace that divides an object space (28) into an interior region that lies inside the halfspace and an exterior region that lies outside the halfspace. The boundary of the halfspace is defined by an implicit mathematical function. The method includes subdividing the object space into volume elements of either a cubical or rectangular parallelepiped shape. For each volume element, an upper bound and a lower bound are calculated to the values of each function defining the boundary of a halfspace. In a first preferred embodiment, the calculation of the upper and lower bounds of the function defining a halfspace within a volume element identifies all of the extrema of the function that lie in or on the boundary of the volume element. In a second preferred embodiment, the upper and lower bounds are calculated as approximations to the absolute maximum and minimum values, respectively, of the function. An image is rendered in accordance with whether the upper and lower bounds of the function defining the boundary of a halfspace show that all locations in a volume element are inside or outside the halfspace.