Dietmar Hildenbrand

Technische Universität Darmstadt | TU · Department of Mathematics (Dept.4)

We present Quantum Register Algebra (QRA) as an efficient tool for quantum computing. We show the direct link between QRA and Dirac formalism. We present Geometric Algebra Algorithms Optimizer (GAALOP) implementation of our approach. We demonstrate the ability to fully describe and compute with QRA in GAALOP using the geometric product.

We introduce Quantum Register Algebra (QRA) as an efficient tool for quantum computing. We show the direct link between QRA and Dirac formalism. We present GAALOP (Geometric Algebra Algorithms Optimizer) implementation of our approach. Using the QRA basis vectors definitions given in Section 4 and the framework based on the de Witt basis presented...

We extensively survey applications of Clifford Geometric algebra in recent years (mainly 2019–2022). This includes engineering; electric engineering; optical fibers; geographic information systems; geometry; molecular geometry; protein structure; neural networks; artificial intelligence; encryption; physics; signal, image, and video processing; and...

We describe a possibility for geometric calculation of specific conics’ intersections in Geometric Algebra for Conics (GAC) using its operations that may be expressed as sums of products. The advantage is that no solver for a system of quadratic equations is needed and thus no numerical error is involved. We also describe specific conics connected...

In this paper, we present Geometric Algebra as a powerful language to describe quantum operations using its geometric intuitiveness. Using the web-based GAALOPWeb, an online geometric algebra algorithm optimizer for computing with qubits, we describe new formulations for the NOT operation, as well as a strategy to describe the Z gate and especially...

We use Geometric Algebra for quantum computing motivated by the fact that qubits and gates can be handled as elements of the same algebra. Additionally, Geometric Algebra allows us to describe gate operations very easily, based on the geometrically intuitive description of transformations in Geometric Algebra. As the main contribution of this paper...

In this paper we present a new tool for Mathematica users, based on the new web-based geometric algebra algorithm optimizer (GAALOP). GAALOPWeb for Mathematica now supports the Mathematica user with an intuitive interface for the development, testing and visualization of geometric algebra algorithms, combining the geometric intuitiveness of geometr...

We present GAALOPWeb for Matlab, a new easy to handle solution for Geometric Algebra implementations for Matlab. We demonstrate its usability for industrial applications based on a forward kinematics algorithm of a serial robot arm and illustrate it with the help of high run-time performance.

Call for Participation for the ENGAGE workshop at the CGI 2020 conference - about geometric algebra with focus on computer graphics applications.

First we introduce briefly the frameworks of CGA and PGA for doing Euclidean geometry and we summarise basic formulas. In the next section, we show that there are actually two naturally related copies of PGA in CGA. After an identification of the two copies, the duality in PGA is obtained in terms of CGA operations. This implies directly the corres...

We introduce the transcript of direct and differential kinematics of a robotic snake in terms of Compass Ruler Algebra (CRA). We suppose that the robot moves on a planar surface. We provide the original and optimized code of elementary motions in GAALOP together with runtime comparisons.

Modern Geometric Algebra software systems tend to fall into one of two categories, either fast, difficult to use, statically typed, and syntactically different from the mathematics or slow, easy to use, dynamically typed and syntactically close to the mathematical conventions. Gajit is a system that aims to get the best of both worlds. It allows us...

This work explains how to extend standard conformal geometric algebra of the Euclidean plane in a novel way to describe cubic curves in the Euclidean plane from nine contact points or from the ten coefficients of their implicit equations. As algebraic framework serves the Clifford algebra Cl(9, 7) over the real sixteen dimensional vector space \(\m...

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 (A...

Over the last few years, recent advances in user interface and mobile computing, introduce the ability to create new experiences that enhance the way we acquire, interact and display information within the world that surrounds us with virtual characters. Virtual reality (VR) is a 3D computer simulated environment that gives to user the experience o...

Because of the high numeric complexity of Geometric Algebra, its use in engineering applications relies heavily on tools and devices for efficient implementations. In this article, we present a novel hardware design for a Geometric Algebra coprocessor, called GAPPCO, which is based on Geometric Algebra Parallelism Programs (GAPP). GAPPCO is a desig...

The usage of Geometric Algebra motors instead of Euclidean vectors for describing the position and orientation of points on a surface has promising applications in Computer Science and Engineering. Common geometric transformations, such as rotations and translations of Euclidean points, are also applicable to motors. However, encoding vertex positi...

This paper describes the geometric meaning of the inner product of 2 circles based on Compass Ruler Algebra, the Conformal Geometric Algebra in 2D. It analyzes in detail the cases of intersecting and tangent circles as well as when one circle is completely inside or completely outside the other circle. The paper reflects the systematic way of deriv...

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 alge...

Starting from the situation 15 years ago with a great gap between the low symbolic complexity on the one hand and the high numeric complexity of coding in Geometric Algebra on the other hand, this paper reviews some applications showing, that, in the meantime, this gap could be closed, especially for CPUs. Today, the use of Geometric Algebra in eng...

In the last years, Geometric Algebra with its Euclidean, Homogeneous and Conformal models attracts the research interest in many areas of Computer Science and Engineering and particularly in Computer Graphics as it is shown that they can produce more efficient and smooth results than other algebras. In this paper, we present an all-inclusive algori...

The focus of this work is a simplified integration of algorithms expressed in Geometric Algebra (GA) into modern high level computer languages, namely C++, OpenCL and CUDA. A high runtime performance in terms of GA is achieved using symbolic simplification and code generation by a precompiler that is directly integrated into CMake-based build toolc...

Geometric Algebra (GA) is a branch of mathematics that generalizes complex numbers and quaternions. One of the advantages of the framework is, that it allows intuitive description and manipulation of geometric objects. While even complex operations can be described concisely, the actual evaluation of these GA expressions is extremely compute intens...

This chapter presents the inverse kinematics application of a simple robot. All the figures in this chapter are screenshots of a CLUCalc application that can be downloaded from We present the Geometric Algebra algorithm for this application step by step. The geometrically intuitive operations of CGA make it easy to compute the joint angles of this...

Thanks to powerful GPGPU techniques (see for instance [45]), one can expect impressive results using the powerful language of Geometric Algebra. In this chapter, we present the basics of Gaalop GPC for GPUs, a precompiler for parallel programming of heterogeneous systems using OpenCL and CUDA. While CUDA is vendor-specific, OpenCL is an open indust...

A molecular dynamics simulation of the kind described here [101] models the point-pair interactions of a system of molecules, each molecule consisting of several atoms, and numerically solves Newton’s and Euler’s equations of motion for each molecule. This can be expressed in the mathematical language of Conformal Geometric Algebra.

For a long time, Geometric Algebra was known primarily as an elegant mathematical language. It could indeed be used in order to develop new algorithms, but to implement them efficiently, standard linear algebra was required. This was due to the low runtime performance of naively implemented Geometric Algebra algorithms. In 2006 [59], we presented f...

Part II of this book is written in a tutorial-like style in order to encourage the reader to gain his/her own experience in developing Geometric Algebra algorithms. The relevant tool, CLUCalc, written by Christian Perwass, the tutorial examples, and the robotics applications in this part can be downloaded free of charge. While the focus of Part I w...

In this book, we focus on 5D Conformal Geometric Algebra (CGA). The “conformal” comes from the fact that it handles conformal transformations easily. These transformations leave angles invariant.

The goal of this chapter is to identify some mathematical systems in CGA and to investigate what their geometric meaning is (Fig. 4.1).

One big advantage of CGA is its easy handling of objects such as spheres and planes. Many problems in computer graphics are related to these kinds of objects.

Part I of this book provides a theoretical introduction to Geometric Algebra. We focus on 5D Conformal Geometric Algebra because of its intuitive handling of geometric entities, geometric operations, transformations, and motions. We show how a lot of other mathematical systems can be identified within this algebra, and present the fitting of points...

In order to simplify the use of the Geometric Algebra Computing technology, we have developed Gaalop GPC [20], a precompiler, which integrates Gaalop into standard programming languages such as C++, OpenCL, and CUDA. Figure 11.1 outlines the concept for the C++ programming language. With Gaalop GPC, we are able to enhance ordinary C++ code with Geo...

The Maple-based approach of Chap. 9 was the basis for the development of our Gaalop (Geometric algebra algorithms optimizer) compiler, using the CLUCalc language of Part II as the input language (see [61]). In this chapter, we introduce Gaalop based on the horizon example and present our new compilation approaches to going from Geometric Algebra al...

In this chapter, we present a Geometric Algebra algorithm for the grasping process of the robot Geometer (Fig. 8.1) constructed at Cinvestav, Guadalajara [62, 7]. We present both the Geometric Algebra algorithm and the algorithm in standard mathematics in order to highlight the difference in the symbolic descriptions.

Geometric Algebra has the power to lead easily from the geometric intuition of solving an engineering application to its efficient implementation on current and future computing platforms. It is easy to develop new algorithms in areas such as computer graphics, robotics, computer animation and computer simulation. Owing to its geometric intuitiven...

What mathematicians often call Clifford algebra is calledgeometric algebra if the focus is on the geometric meaning of the algebraic expressions and operators. Geometric algebra is a mathematical framework to easily describe geometric concepts and oper- ations. It allows us to develop algorithms fast and in an intuitive way. It is based on the work...

The focus of the this work is on the better integration of algorithms expressed in Conformal Geometric Algebra (CGA) in modern high level computer languages, namely C++ and NVIDIA's Compute Unified Device Architecture (CUDA). A high runtime performance in terms of CGA is achieved using symbolic optimizing through the invocation of Gaalop.

Images in scientific visualization are the end-product of data processing. Starting from higher-dimensional datasets, such as scalar-, vector-, tensor- fields given on 2D, 3D, 4D domains, the objective is to reduce this complexity to two-dimensional images comprehensible to the human visual system. Various mathematical fields such as in particular...

Geometric Algebra (GA), a generalization of quaternions, is a very powerful form for intuitively expressing and manipulating complex geometric relationships common to engineering problems. The actual evaluation of GA expressions, though, is extremely compute intensive due to the high-dimensionality of data being processed. On standard desktop CPUs,...

We illustrate the suitability of geometric algebra for representing struc-tures and developing algorithms in computer graphics, especially for engineering applications. A number of example applications are reviewed. Geometric algebra unites many underpinning mathematical concepts in computer graphics such as vec-tor algebra and vector fields, quate...

We present Gaalop (Geometric algebra algorithms optimizer), our tool for high-performance computing based on conformal geometric algebra. The main goal of Gaalop is to realize implementations that are most likely faster than conventional solutions. In order to achieve this goal, our focus is on parallel target platforms like FPGA (field-programmabl...

For applications like segmentation, feature extraction and classification of point sets it is essential to know the principal curvatures and the corresponding principal directions. For the purpose of curvature estimation conformal geometric algebra promises to be a natural mathematical language: Local curvatures can be described with the help of os...

Geometric Algebra (GA) is a mathematical framework that allows a compact, geometrically intuitive description of geometric relationships and algorithms. These algorithms require significant computational power because of the intrinsically high dimensionality of geometric algebras. Algorithms in an n-dimensional GA require 2 n elements to be compute...

Geometric Algebra (GA) is a mathematical framework that allows a compact and geometrically intuitive descrip-tion of geometric relationships and algorithms. In this paper a translation, rotation and scale invariant algorithm for registration of color images and other multichannel data is introduced. The use of Geometric Algebra allows to generalize...

Geometric Algebra (GA) supports the geometrically intuitive development of an algorithm with its build-in geometric primitives such as points, lines, spheres or planes. But on the negative side GA has a huge computational footprint. In this paper we study how GA can compete with traditional methods from Linear Algebra (LA) in the field of raytracin...

The usage of Conformal Geometric Algebra leads to algo- rithms that can be formulated in a very clear and easy to grasp way. But it can also increase the performance of an implementation because of its capabilities to be computed in parallel. In this paper we show how a grasping algorithm for a robotic arm is accelerated using a Conformal Geometric...

This paper presents a solution to solve the inverse kinematics for the legs of a humanoid robot using conformal geometric algebra. We geometrically intuitively develop the algorithm with the freely available CLUCalc software and optimize it with the help of the computer algebra system Maple and the Clifford package for geometric algebras. We descri...

Geometric algebra (GA) is a mathematical framework that allows the compact description of geometric rela- tionships 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-...

Geometric algebra covers a lot of other mathematical systems like vector algebra, complex numbers, Plücker coordinates, quaternions etc. and it is geometrically intuitive to work with. Furthermore there is a lot of potential for optimization and parallelization. In this paper, we investigate computers suitable for geometric algebra algorithms. Whil...

2D-3D pose estimation is an important task for computer vision, ranging from robot navigation to medical intervention. In such applications as robot guidance, the estimation procedure should be fast and automatic, but in industrial metrology applications, the precision is typically a more important factor. In this paper, a new 3D approach for infra...

We focus on inverse kinematics applications in computer graphics and robotics based on Conformal Geometric Algebra. Here, geometric objects like spheres and circles that are often needed in inverse kinematics algorithms are simply represented by algebraic objects. We present algorithms for the inverse kinematics of a human arm like kinematic chain...

This paper presents some basics for the analysis of point clouds using the geometrically intuitive mathematical framework of conformal geometric algebra. In this framework it is easy to compute with osculating circles for the description of local curvature. Also methods for the fitting of spheres as well as bounding spheres are presented. In a nuts...

configurable hardware, runtime performance. Abstract: This paper presents a very efficient approach for algorithms developed based on conformal geometric algebra using reconfigurable hardware. We use the inverse kinematics of the arm of a virtual human as an example, but we are convinced that this approach can be used in a wide field of computer an...

Early in the development of computer graphics it was realized that projective geometry was well suited for the representation of transformations. Now, it seems that another change of paradigm is lying ahead of us based on geometric computing using conformal geometric algebra.Due to its geometric intuitiveness, elegance and simplicity, the underlyin...

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 an...

In this work we will use geometric algebra to prove a number of well known theorems central to the field of fluid dynamics, such as Kelvin's Circulation Theorem and Helmholtz' Theorem, showing that it is accessible by geometric alge-bra methods and that these methods facilitate the representation of and calculation with fluid dynamics concepts. The...

This work reviews some current engineering applications of geometric algebra and observes the potential of this mathematical language to become a basis for a wide range of computational engineering applications. Geomet-ric algebra unifies many other mathematical concepts like quaternions and projective geometry and is able to easily deal with geome...

In recent years, Geometric Algebra (GA) has become more and more popular in fields of science and engineering due to its potential for compact algorithms. However, the execution of GA algorithms and the related need for high computational power is still the limiting factor for these algorithms to be used in practice. Therefore, it would be desirabl...

Abstract Early in the development,of computer,graphics it was realized that projective geometry is suited quite well to represent points and transformations. Now, maybe another change of paradigm is lying ahead of us based on Geometric Algebra. If you already use quaternions or Lie algebra in additon to the well-known vector algebra, then you may a...

This paper presents a solution to solve the inverse kinematics for the legs of a humanoid robot using conformal geometric algebra. We geometrically intuitive develop the algorithm with the freely available software CLUCalc.

The focus of the this work is a simplified integration of algorithms expressed in Geometric Algebra (GA) in modern high level computer languages, namely C++, OpenCL and CUDA. A high runtime performance in terms of GA is achieved using symbolic simplification and code generation by a Precompiler that is directly integrated into CMake-based build too...

In this paper we present how Geometric Algebra can be used for deformation simulations. The aim is to capture the elastic behavior of simple components like rods. First we will review some other works in this field. Later we present an extended Finite Element Method, which has recently been developed and investigated. Our goal is a proof of concept...

Geometric Algebra (GA), a generalization of quaternions and complex numbers, is a very powerful framework for intuitively ex-pressing and manipulating the complex geometric relationships common to engineering problems. However, actual processing of GA expressions is very compute intensive, and acceleration is generally required for prac-tical use....

We present Gaalop (Geometric algebra algorithms optimizer), our tool for high performance computing based on Conformal Geometric Algebra (GA). The main goal of Gaalop is to realize implementations that are most likely faster than conventional solutions. We describe the concepts, the state-of-the-art as well as the future perspectives of Gaalop deal...

Nad názvem: Computer Graphics, Vision and Mathematics in co-operation with Eurographics

"The 1st international workshop on Computer Graphics, Computer Vision and Mathematics was held at the University of West Bohemia in Plzen (Pilsen), Czech Republic on September 2-4, 2009"--Úvod Nad názvem: International Workshop on Computer Graphics, Computer Vision and Mathematics in co-operation with Eurographics