Theoretical analytics of stereographic projection on 3D objects' intersection predicate
Shuqing Zhang, Junyan Zhang · International Journal of Geographical Information Systems · 2010
Based on stereographic projection (SP), a novel and systematic three‐dimensional (3D) objects' intersection predicate theory is inferred and formularized. The theory is deduced from the intersection of quasi‐convex bodies. Since composite mapping has omnidirectional one‐to‐one mapping properties for any quasi‐convex bodies, they can be accurately projected to the equatorial plane, and their intersection predicate can be simplified into two dimensions. Those predicate conclusions are further applied to common polyhedrons by the introduction of the radial dividing point concept. ‘Point inclusion’ and ‘line traversing’ are mathematically abstracted as the primary intersection predicate types for all kinds of 3D objects, even including curved line and curved surface. As a special case of ‘point inclusion’, ‘polyhedron entirely included’ is also pointed out. Meanwhile, the process of intersection predicate is heuristically optimized at object and face levels with well‐matched bounding boxes: bounding sphere and projective cone.