Shooting permanent rays among disjoint polygons in the plane

We present a data structure for ray shooting-and-insertion in the free space among disjoint polygonal obstacles with a total of $n$ vertices in the plane, where each ray starts at the boundary of some obstacle. The portion of each query ray between the starting point and the first obstacle hit is inserted permanently as a new obstacle. Our data structure uses O(n log n) space and preprocessing time, and it supports m successive ray shooting-and-insertion queries in O(n log2 n + m log m) total time. We present two applications for our data structure: (1) Our data structure supports efficient implementation of auto-partitions in the plane i.e. binary space partitions where each partition is done along the supporting line of an input segment. If n input line segments are fragmented into m pieces by an auto-partition, then it can now be implemented in O(n log2n+m log m) time. This improves the expected runtime of Patersen and Yao's classical randomized auto-partition algorithm for n disjoint line segments to O(n log2 n). (2) If we are given disjoint polygonal obstacles with a total of n vertices in the plane, a permutation of the reflex vertices, and a half-line at each reflex vertex that partitions the reflex angle into two convex angles, then the folklore convex partitioning algorithm draws a ray emanating from each reflex vertex in the prescribed order in the given direction until it hits another obstacle, a previous ray, or infinity. The previously best implementation (with a semi-dynamic ray shooting data structure) requires O(n3/2-e/2) time using O(n1+e) space. Our data structure improves the runtime to O(n log2 n)

Visual Data Representation using Context-Aware Samples

The rapid growth in the complexity of geometry models has necessisated revision of several conventional techniques in computer graphics. At the heart of this trend is the representation of geometry with locally constant approximations using independent sample primitives. This generally leads to a higher sampling rate and thus a high cost of representation, transmission, and rendering. We advocate an alternate approach involving context-aware samples that capture the local variation of the geometry. We detail two approaches; one, based on differential geometry and the other based on statistics. Our differential-geometry-based approach captures the context of the local geometry using an estimation of the local Taylor's series expansion. We render such samples using programmable Graphics Processing Unit (GPU) by fast approximation of the geometry in the screen space. The benefits of this representation can also be seen in other applications such as simulation of light transport. In our statistics-based approach we capture the context of the local geometry using Principal Component Analysis (PCA). This allows us to achieve hierarchical detail by modeling the geometry in a non-deterministic fashion as a hierarchical probability distribution. We approximate the geometry and its attributes using quasi-random sampling. Our results show a significant rendering speedup and savings in the geometric bandwidth when compared to current approaches

Inductive Pattern Formation

With the extended computational limits of algorithmic recursion, scientific investigation is transitioning away from computationally decidable problems and beginning to address computationally undecidable complexity. The analysis of deductive inference in structure-property models are yielding to the synthesis of inductive inference in process-structure simulations. Process-structure modeling has examined external order parameters of inductive pattern formation, but investigation of the internal order parameters of self-organization have been hampered by the lack of a mathematical formalism with the ability to quantitatively define a specific configuration of points. This investigation addressed this issue of quantitative synthesis. Local space was developed by the Poincare inflation of a set of points to construct neighborhood intersections, defining topological distance and introducing situated Boolean topology as a local replacement for point-set topology. Parallel development of the local semi-metric topological space, the local semi-metric probability space, and the local metric space of a set of points provides a triangulation of connectivity measures to define the quantitative architectural identity of a configuration and structure independent axes of a structural configuration space. The recursive sequence of intersections constructs a probabilistic discrete spacetime model of interacting fields to define the internal order parameters of self-organization, with order parameters external to the configuration modeled by adjusting the morphological parameters of individual neighborhoods and the interplay of excitatory and inhibitory point sets. The evolutionary trajectory of a configuration maps the development of specific hierarchical structure that is emergent from a specific set of initial conditions, with nested boundaries signaling the nonlinear properties of local causative configurations. This exploration of architectural configuration space concluded with initial process-structure-property models of deductive and inductive inference spaces. In the computationally undecidable problem of human niche construction, an adaptive-inductive pattern formation model with predictive control organized the bipartite recursion between an information structure and its physical expression as hierarchical ensembles of artificial neural network-like structures. The union of architectural identity and bipartite recursion generates a predictive structural model of an evolutionary design process, offering an alternative to the limitations of cognitive descriptive modeling. The low computational complexity of these models enable them to be embedded in physical constructions to create the artificial life forms of a real-time autonomously adaptive human habitat