A stabbing line for a set of convex polyhedra is extremal if it passes through four edges and is tangent to the polyhedra containing those edges. In this paper we present three constructions of convex polyhedra with many extremal lines. The rst construction shows (n 2) extremal stabbing lines constrained to meet two skew lines. The second shows (n 4) extremal lines which are tangent to two poly...

QQ(R) (abbreviated as Q) is the space of unordered Q points in R introduced by Almgren [AF]. In this paper, we first show that the space Q with metric G is non-negatively curved in the sense of Alexandrov. We also discuss some issues like metric tangent cone and exponential map. In the second part, we define the signature of points in Q, give a decomposition of this metric space according to th...

Tangent curves are a powerful tool for analyzing and visualizing vector fields. In this paper two of their most important properties are examined: their curvature and torsion. Furthermore, the concept of normal surfaces is introduced to the theory of 3D vector fields, and their Gaussian and mean curvature are analyzed. It is shown that those four curvature measures tend to infinity near critica...

We propose a constructive solution to the problem of finding a cubic parametric curve in a plane if the tangent vectors (derivatives with respect to the parameter) and signed curvatures are given at its end-points but the end-points themselves are unknown. We also show how these curves can be applied to construct blending curves subject to curvature, arc length, inflection and area constraints.

We show that for m points and n lines in R2, the number of distinct distances between the points and the lines is Ω(m1/5n3/5), as long as m1/2 ≤ n ≤ m2. We also prove that for any m points in the plane, not all on a line, the number of distances between these points and the lines that they span is Ω(m4/3). The problem of bounding the number of distinct point-line distances can be reduced to the...

and is the tangent of F at x. Dually then, r and 1/k being the absolute points, the conic F the Feuerbach circle, and the conic R a rectangular hyperbola on the four given orthocentric points, and having its centre c on F, if the common diameter of F and R meet R at points dd', then these points are double foci of circular curves of class 3 on the 6 lines; the circles with centres d and d' and ...

In this paper we provide a proof of the Carleson $\varepsilon^2$-conjecture. This result yields characterization (up to exceptional sets zero length) tangent points Jordan curve in terms finiteness associated $\varepsilon^2$-square function.

A regular linear line complex is a three-parameter set of lines in space, whose Plücker vectors lie hyperplane, which not tangent to the Klein quadric. Our main result bound O(n1/2m3/4+m+n) for number incidences between n and m points F3, where F field, n≤char(F)4/3 positive characteristic. Zahl has recently observed that bichromatic pairwise coming from two distinct complexes account nonzero s...