Abstract: This paper describes a new algorithm for geometric displacement mapping. Its key idea is that all occluded solutions for an eye ray lie in two-dimensional manifolds perpendicular to the underlying surface to which the height map is applied. The manifold depends only on the eye position and surface geometry, and not on the height field. A simple stepping algorithm, moving along the surface within a manifold renders a curve of pixels to the view plane, which reduces height map rendering to a set of one-dimensional computations that can be done in parallel. The curves on the view plane for two specific underlying manifolds, a plane and a sphere, are straight lines. In this paper we focus on the specific geometry of simple underlying surfaces for which the geometry is more intuitive and the sampling of the rendered image direct.