Swept volumes via spacetime numerical continuation

and Adobe Research Fig. 1. We compute the surface of the volume (orange) swept by a moving space shuttle (top left). Instead of hunting everywhere in the cold, dark, vastness of space and time, our continuation algorithm traces along a 2D submanifold of a 3-manifold embedded in 4D, reducing the problem's dimension.

Given a solid 3D shape and a trajectory of it over time, we compute its swept volume -the union of all points contained within the shape at some moment in time. We consider the representation of the input and output as implicit functions, and lift the problem to 4D spacetime, where we show the problem gains a continuous structure which avoids expensive global searches. We exploit this structure via a continuation method which marches and reconstructs the zero level set of the swept volume, using the temporal dimension to avoid erroneous solutions. We show that, compared to other methods, our approach is not restricted to a limited class of shapes or trajectories, is extremely robust, and its asymptotic complexity is an order lower than standards used in the industry, enabling its use in applications such as modeling, constructive solid geometry, and path planning.