Detailed geometric models of the real world are in increasing demand. LiDAR data is appropriate to reconstruct such models. For urban reconstruction, it works well to reconstruct the individual surfaces of the scene and connect them to form the scene geometry. There are various methods for reconstructing the free-form shape of a point sample on a single surface. However, these methods do not ta...

Given a partition P : L → ω of the lines in R, n ≥ 2, into countably many pieces, we ask if it is possible to find a partition of the points, Q : R → ω, so that each line meets at most m points of its color. Assuming Martin’s Axiom, we show this is the case for m ≥ 3. We reduce the problem for m = 2 to a purely finitary geometry problem. Although we have established a very similar, but somewhat...

This paper presents new methods for efficient extraction of intersection curves between iso-surfaces of any pair of co-located 3D scalar fields. The first method is based on the Marching Cubes algorithm which has been enhanced to produce an additional data structure that makes it possible to reduce the complexity of the general surface intersection extraction from O(N) to O( √ N), where N denot...

Some phospholipid/dihydrocholesterol Langmuir monolayers form coexisting liquid phases. Gas domains form at the interface between the phospholipid-rich and dihydrocholesterol-rich liquid phases when these monolayers undergo expansion to low surface pressure. Analysis of the domain shapes thus formed yields the relative line tensions of the gas/phospholipid, gas/dihydrocholesterol, and phospholi...

An algorithm is demonstrated that finds an ordinary intersection in an arrangement of n lines in R, not all parallel and not all passing through a common point, in time O(n log n). The algorithm is then extended to find an ordinary intersection among an arrangement of hyperplanes in R, no d passing through a line and not all passing through the same point, again, in time O(n log n). Two additio...

Using the theory of rational elliptic fibrations, we construct and discuss a one parameter family configurations $12$ conics $9$ points in projective plane that realizes an abstract configuration $(12_6,9_8)$. This is analogous to famous Hesse lines forming $(12_3,9_4)$. We also show any Halphen fibration index $2$ with four triangular singular fibers arises from such conics.