Leaflets are a common technology used in web services to provide high resolution maps by providing them in tiles of incremental resolutions. While for 2D display an unique level of resolution is all that is needed, within a 3D perspective display a combination of resolution levels needs to be identified. Block-Structured Adaptive Mesh Refinement (AMR) utilizes an organisation of data similar to Leaflets, but deals with 3D or 4D volumetric blocks of data. Such a data organisation is very suitable for hierarchical organisation of big data. In this article we propose using the AMR data organisation for hierarchical structuring of arbitrary geometric primitives, i.e. point clouds, sets of lines, triangle meshes, volumes. Such data organisation allows for generic and performant data access of out-of-core data on demand, as relevant for navigation through big data. Ultimately identification of data block visibility is crucial for a fluid interactive experience. We propose an algorithm based on Conformal 5D Geometric Algebra to approximate bounding boxes via spheres and the view frustum as a cone to arrive on a visibility criteria used for priority sorting of candidate blocks.

Geometric Algebra is considered as a very intuitive tool to deal with geometric problems and it appears to be increasingly efficient and useful to deal with computer graphics solutions. For example, the Conformal Geometric Algebra includes circles, spheres, planes and lines as algebraic objects, and intersections between these objects are also algebraic objects. More complex objects such as conics, quadric surfaces can also be expressed and be manipulated using an extension of the conformal Geometric Algebra. However due to high dimension of their representations in Geometric Algebra, implementations of Geometric Algebra that are currently available do not allow efficient realizations of these objects. This paper presents a Geometric Algebra implementation dedicated for both low and high dimensions. The proposed method is a hybrid solution for precomputed code with fast execution and runtime computations with low memory requirement. More specifically, the proposed method combines a precomputed table approach with a recursive method using binary trees. Some rules are defined to select the most appropriate choice, according to the dimension of the algebra and the type of multivectors involved in the product. The resulting implementation is well suited for high dimensional spaces (e.g. algebra of dimension 15) as well as for lower dimensional space. This paper details the integration of this hybrid method as a plug-in into Gaalop, which is a very advanced optimizing code generator. This paper also presents some benchmarks to show the performances of our method, especially in high dimensional spaces.

Geometric algebra (GA) is a mathematical framework that allows the compact description of geometric relationships and algorithms in many fields of science and engineering. The execution of these algorithms, however, requires significant computational power that made the use of GA impractical for many real-world applications. We describe how a GA-based formulation of the inverse kinematics problem from computer animation and robotics can be accelerated using reconfigurable FPGA-based computing and using a graphics processing unit (GPU). The practical evaluation covers not only the sheer compute performance, but also the energy efficiency.

This paper presents an approach to deal with standard tasks of computer animations and robotics based on Conformal Geometric Algebra. We will show that this algebra is very well suitable for applications of all kind of robot manipulator kinematics, representation and visualization and object robot manipulation. Due to its geometric intuitiveness and simplicity Conformal Geometric Algebra appears to be a promising mathematical tool for building intelligent man-machine interfaces.