Binary space partitioning trees and their uses

Binary Space Partitioning (BSP) trees have some qualities that make them useful in solving many graphics related problems. The purpose is to describe what a BSP tree is, and how it can be used to solve the problem of hidden surface removal, and constructive solid geometry. The BSP tree is based on the idea that a plane acting as a divider subdivides space into two parts with one being on the positive side and the other on the negative. A polygonal solid is then represented as the volume defined by the collective interior half spaces of the solid's bounding surfaces. The nature of how the tree is organized lends itself well for sorting polygons relative to an arbitrary point in 3 space. The speed at which the tree can be traversed for depth sorting is fast enough to provide hidden surface removal at interactive speeds. The fact that a BSP tree actually represents a polygonal solid as a bounded volume also makes it quite useful in performing the boolean operations used in constructive solid geometry. Due to the nature of the BSP tree, polygons can be classified as they are subdivided. The ability to classify polygons as they are subdivided can enhance the simplicity of implementing constructive solid geometry

Constructive Volumetric Modeling

Abstract—In this article we intend to present a method of obtaining high complexity sinthetic scenes by using simple volumes as the building blocks. The below described method can be used to obtain both homogenous and heterogenous volumes. This is done by combining volumes of different voxel densities. Index Terms—volumetric data, voxel, constructive solid geometry, volume modelling, constructive volume geometry. I

Riemannian-geometric entropy for measuring network complexity

A central issue of the science of complex systems is the quantitative characterization of complexity. In the present work we address this issue by resorting to information geometry. Actually we propose a constructive way to associate to a - in principle any - network a differentiable object (a Riemannian manifold) whose volume is used to define an entropy. The effectiveness of the latter to measure networks complexity is successfully proved through its capability of detecting a classical phase transition occurring in both random graphs and scale--free networks, as well as of characterizing small Exponential random graphs, Configuration Models and real networks.Comment: 15 pages, 3 figure

Efficient Ray Tracing of CSG Models

Tato práce zkoumá metody sledování paprsku v kombinaci s konstruktivní geometrií těles (CSG). Dále navrhuje způsob využití Embree, vysoce optimalizované knihovny používající hierarchii obálek pro sledování paprsků v trojúhelníkových sitích, pro zobrazení CSG v kombinaci s trojúhelníkovými sítěmi.This work explores ray tracing of constructive solid geometry (CSG) and its acceleration in combination with ray tracing triangles. It proposes a way how to exploit Embree, a highly optimized library using bounding volume hierarchy for ray tracing triangle meshes, for rendering CSG with triangle meshes

Doing and Showing

The persisting gap between the formal and the informal mathematics is due to an inadequate notion of mathematical theory behind the current formalization techniques. I mean the (informal) notion of axiomatic theory according to which a mathematical theory consists of a set of axioms and further theorems deduced from these axioms according to certain rules of logical inference. Thus the usual notion of axiomatic method is inadequate and needs a replacement.Comment: 54 pages, 2 figure

On Constructive Axiomatic Method

In this last version of the paper one may find a critical overview of some recent philosophical literature on Axiomatic Method and Genetic Method.Comment: 25 pages, no figure