By providing explicit definitions, we show that in both affine and projective geometry of dimension ≥ 3, considered as first-order theories axiomatized in terms of lines as the only variables, and the binary line-intersection predicate as primitive notion, non-intersection of two lines can be positively defined in terms of line-intersection. Mathematics Subject Classification (2000): 51A05, 51A...

This paper presents the 3D reconstruction of planar objects using the normal vector of plane and vanishing points from a single panoramic image. We get the geometric information of plane or line with feature points which use the constraints of coplanarity, perpendicularity and parallelism of planes or lines. We describe the 3D reconstruction about any features using the properties of vanishing ...

This paper develops a series of qualitative spatial calculi that feature topological relations. While most qualitative spatial calculi have targeted spatial relations between single-type objects, our calculi support spatial relations between heterogeneous combinations of objects. Thus, with our calculi, we can disambiguate topological arrangements of objects without regard to the types of the o...

BACKGROUND AND OBJECTIVE Spatial modulation of magnetization (SPAMM) is a valuable magnetic resonance imaging technique for studying ventricular biomechanics. In order to track the intersection points of the stripes to calculate regional wall motion and strain, the stripe spacing should be at most half the wall thickness, yet sufficiently larger than the image pixel size in order that the strip...

A proper non-empty subset C of the points of a linear space S = (P; L) is called line-closed if any two intersecting lines of S , each meeting C at least twice, have their intersection in C. We show that when every line has k points and every point lies on r lines the maximum size for such sets is r + k ? 2. In addition it is shown that this cannot happen for projective spaces PG(n; q) unless q...

The convex hull of a set of points is the smallest convex set containing the points. The convex hull is a fundamental concept in mathematics and computational geometry. Other problems can be reduced to finding the convex hull – for example, halfspace intersection, Delaunay triangulation, Voronoi diagram, etc. An algorithm known as Graham scan [6] achieves an O(n log n) running time. This algori...

In [1], Buhler et al. considered the following scenario. Given a collection U of unit magnitude complex numbers and a set S of constructed points initially containing just 0 and 1, through each constructed point draw lines whose angles with the real axis are in U . The intersections of such lines are also constructed points. Upon taking the closure we form a set R(U). They investigated which U ...

We present a method for analyzing the curvature (second derivatives) of the conical intersection hyperline at an optimized critical point. Our method uses the projected Hessians of the degenerate states after elimination of the two branching space coordinates, and is equivalent to a frequency calculation on a single Born-Oppenheimer potential-energy surface. Based on the projected Hessians, we ...