Geometric limits to geometric optical imaging with infinite, planar, non-absorbing sheets

Geometric limits to geometric optical imaging with infinite, planar, non-absorbing sheets

New ray-optical elements allow generalized refraction of light rays, but geometry imposes limitations on possible mappings between the positions of an object and its geometric image. Here I study the case of an infinite, planar, non-absorbing sheet that images the entire three-dimensional space. The most general case of such a sheet is equivalent to a thin lens with different objectand image-si...

Infinite Random Geometric Graphs

We introduce a new class of countably infinite random geometric graphs, whose vertices V are points in a metric space, and vertices are adjacent independently with probability p ∈ (0, 1) if the metric distance between the vertices is below a given threshold. If V is a countable dense set in R equipped with the metric derived from the L∞-norm, then it is shown that with probability 1 such infini...

Drawing Planar Graphs with Few Geometric Primitives

We define the visual complexity of a plane graph drawing to be the number of basic geometric objects needed to represent all its edges. In particular, one object may represent multiple edges (e.g., one needs only one line segment to draw two collinear edges of the same vertex). Let n denote the number of vertices of a graph. We show that trees can be drawn with 3n/4 straight-line segments on a ...

Planar and Quasi Planar Simultaneous Geometric Embedding

Geometric Programming with Stochastic Parameter

Geometric programming is efficient tool for solving a variety of nonlinear optimizationproblems. Geometric programming is generalized for solving engineering design. However,Now Geometric programming is powerful tool for optimization problems where decisionvariables have exponential form.The geometric programming method has been applied with known parameters. However,the observed values of the ...