Triangulation of Points Constrained to a Plane

We study the set of image tuples arising from fixed cameras observing varying planar 3-dimensional point configurations. We derive a formula for the number of complex critical points of the triangulation problem, which seeks to reconstruct such configurations from noisy image data. Valid for an arbitrary number of views, this formula quantifies the intrinsic algebraic complexity of planar triangulation. We validate our theoretical findings through numerical experiments on both synthetic and real data, demonstrating that incorporating the planar incidence constraints leads to faster point reconstruction and improved accuracy compared to unconstrained triangulation.