One embodiment of the invention is a pop-up engine that generates a pop-up card from a sliced 3D graphics model. In operation, a pop-up engine processes a sliced 3D model to identify locations where each slice of the 3D model is to attach to the pop-up card or to other slices of the 3D model. The pop-up engine traverses the boundary of each slice and, at intervals along the boundary, projects a ray toward the upper portion of the card. If the ray intersects a neighboring slice, then the slice attaches to the neighboring slice at that location. If, however, the ray does not intersect a neighboring slice, then the slice attaches to the upper portion of the card at that location. The pop-up engine then modifies the slice to include a hinge portion that connects the slice to either a neighboring slice or the upper portion at that location.