Most existing light-field rendering methods rely on images that record the color of the rays. In this paper, we present a new approach that instead uses colored point clouds, which are the source of the rays. Furthermore, we study the light field from a dual-space viewpoint, where 3D scene points appear as geometry hyperlines, and rays appear as hyperpoints located on the hyperlines. Rendering the light field reduces to finding and blending the closest hyperlines to the hyperpoints corresponding to the virtual rays. Occlusion is dealt with by clustering nearby hyperlines according to their slope. Hole filling is dealt with by increasing the range of contribution of the hyperlines by a given threshold. We also propose the concept of local ray space, and show that it improves the rendering quality and removes the restriction on virtual camera position that is associated with the global ray space scheme. Finally, we show how to deal with uncertain geometric information in our framework.