The Projection of a Triangle onto Another Triangle

We model the projection of a triangle onto another triangle when viewed from a given viewpoint in 3D space. The motivation arises from the need to calculate the viewshed of a viewpoint on a triangulated terrain. A triangulated terrain (TIN) is a representation of a real terrain, where the surface of the TIN is composed of triangles. Calculating the viewshed involves finding the invisible region on a triangle caused by the terrain surface. To this end, some studies either projected the vertices of the horizon of the terrain or projected the vertices of a triangle directly onto the supporting plane of the triangle of interest, and then connected the projections to find the invisible region on the target triangle. Such a projection involves sending a ray from the viewpoint that passes through the vertex of the horizon or the vertex of the triangle, and finding out where this ray hits on the supporting plane of the target triangle. These studies assumed that such a ray hits the supporting plane of the triangle in front of the viewpoint. Our key contribution is to show, by a counter example, that the ray may hit the plane behind the viewpoint. Taking into account this fact, we show that the projection of a triangle onto another triangle is characterized by a system of nonlinear equations, which are linearized to obtain a polyhedron. Our approach can be extended to projecting objects of general shapes.