GAFRO: Geometric Algebra for Robotics [Tutorial

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 already be familiar with some of the algebraic ideas that will be presented in this tutorial. In fact, quaternions can be represented by Geometric Algebra, next to a number of other algebras like complex numbers, dual-quaternions, Grassmann algebra and Grassmann-Cayley algebra. In this half day tutorial we will emphasize that Geometric Algebra • is a unified language for a lot of mathematical systems used in Computer Graphics, • can be used in an easy and geometrically intuitive way in Computer Graphics. We will focus on the (5D) Conformal Geometric Algebra. It is an extension of the 4D projective geometric algebra. For example, spheres and circles are simply represented by algebraic objects. To represent a circle you only have to intersect two spheres ( or a sphere and a plane ), which can be done with a basic algebraic operation. Alternatively you can simply combine three points (using another product in the algebra) to obtain the circle through these three points. Next to the construction of algebraic entities, kinematics can also be expressed in Geometric Algebra. For example, the inverse kinematics of a robot can be computed in an easy way. The geometrically intuitive operations of Geometric Algebra make it easy to compute the joint angles of a robot which need to be set in order for the robot to reach its goal.

We show that if Projective Geometric Algebra (PGA), i.e. the geometric algebra with degenerate signature (n, 0, 1), is understood as a subalgebra of Conformal Geometric Algebra (CGA) in a mathematically correct sense, then flat primitives share the same representation in PGA and CGA. Particularly, we treat duality in PGA in the framework of CGA. This leads to unification of PGA and CGA primitives which is important especially for software implementation and symbolic calculations.

The Hansen problem is a classic and well-known geometric challenge in geodesy and surveying involving the determination of two unknown points relative to two known reference locations using angular measurements. Traditional analytical solutions rely on cumbersome trigonometric calculations and are prone to propagation errors. This paper presents a novel framework leveraging geometric algebra (GA) to formulate and solve the Hansen problem. Our approach utilizes the representational capabilities of Vector Geometric Algebra (VGA) and Conformal Geometric Algebra (CGA) to avoid the need for tedious analytical manipulations and provide an efficient, unified solution. We develop concise geometric formulas tailored for computational implementation. The rigorous analyses and simulations that were completed as part of this work demonstrate that the precision and robustness of this new technique are equal or superior to those of conventional resection methods. The integration of classical concepts like the Hansen problem with modern GA-based spatial computing delivers more intuitive solutions while advancing the mathematical discourse. This work transforms conventional perspectives through methodological innovation, avoiding the limitations of prevailing paradigms.

Conformal geometric algebra is a powerful mathematical language for describing and manipulating geometric configurations and their conformal transformations. By providing a 5D algebraic representation of 3D geometric configurations, conformal geometric algebra proves to be very helpful in pose estimation, motion design, and neuron-based machine learning (Bayro-Corrochano et al., J. Math. Imaging Vis. 24(1):55–81, 2006; Dorst et al., Geometric Algebra for Computer Science, Morgan Kaufmann, San Mateo, 2007; Hildenbrand, Comput. Graph. 29(5):795–803, 2005; Lasenby, Computer Algebra and Geometric Algebra with Applications, LNCS, vol. 3519, pp. 298–328, Springer, Berlin, 2005; Li et al., Geometric Computing with Clifford Algebras, pp. 27–60, Springer, Heidelberg, 2001; Mourrain and Stolfi, Invariant Methods in Discrete and Computational Geometry, pp. 107–139, Reidel, Dordrecht, 1995; Rosenhahn and Sommer, J. Math. Imaging Vis. 22:27–70, 2005; Sommer et al., Computer Algebra and Geometric Algebra with Applications, pp. 278–297, Springer, Berlin, 2005). In this chapter, we present some theoretical results on conformal geometric algebra which should prove to be useful in computer applications. The focus is on parameterizing 3D conformal transformations with either quaternionic Vahlen matrices or polynomial Cayley transform from the Lie algebra to the Lie group of conformal transformations in space.

Geometric computing in computer graphics using conformal geometric algebra

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.

This paper introduces conformal geometric algebra (CGA) for applications in computer vision and robotics. The authors show that CGA deals with our intuition and insight of the geometry and it helps us to reduce considerably the computational burden of the problems. The CGA can be applied not only to describe the geometry of the space, but to handle the algebra of incidence, as well as conformal transformations, to deal with kinematics or projective geometry problems. The authors show with real and simulated applications that this system can be of great advantage in robotics and computer vision.

This paper presents a method for segmentation of medical images and the application of the so called geometric or Clifford algebras for volume representation, non-rigid registration of volumes and object tracking. Segmentation is done combining texture and boundary information in a region growing strategy obtaining good results. To model 2D surfaces and 3D volumetric data we present a new approach based on marching cubes idea however using spheres. We compare our approach with other method based on the delaunay tetrahedrization. The results show that our proposed approach reduces considerably the number of spheres. Also we show how to do non-rigid registration of two volumetric data represented as sets of spheres using 5-dimensional vectors in conformal geometric algebra. Finally we show the application of geometric algebras to track surgical devices in real time.

Geometric Algebra can be understood as a set of tools to represent, construct and transform geometric objects. Some Geometric Algebras like the well-studied Conformal Geometric Algebra constructs lines, circles, planes, and spheres from control points just by using the outer product. There exist some Geometric Algebras to handle more complex objects such as quadric surfaces; however in this case, none of them is known to build quadric surfaces from control points. This paper presents a novel Geometric Algebra framework, the Geometric Algebra of $${\mathbb {R}}^{9,6}$$ , to deal with quadric surfaces where an arbitrary quadric surface is constructed by the mere wedge of nine points. The proposed framework enables us not only to intuitively represent quadric surfaces but also to construct objects using Conformal Geometric Algebra. Our proposed framework also provides the computation of the intersection of quadric surfaces, the normal vector, and the tangent plane of a quadric surface.

Topological models in Euclidean space are difficult for spatial analysis because of the lack of direct representation of geometric three-dimensional (3D) object information. A 3D cadastral data model in the form of boundary representation based on conformal geometric algebra (CGA) is proposed to realize the integrated representation of geometric and topological information based on previous research. On the basis of the 3D cadastral data model based on CGA and basic geometric operators, self-defined geometric algebraic operators are designed using the advantages of geometric algebra in spatial topological calculation. A computation framework for cadastral parcel topological error detection is put forward based on those self-defined geometric operators. A case study is designed to verify the feasibility of topological error detection methods proposed in this paper. This study is an expansion of research for a 3D cadastral data model based on CGA.

Clustering is one of the most useful methods for understanding similarity among data. However, most conventional clustering methods do not pay sufficient attention to the geometric distributions of data. Geometric algebra (GA) is a generalization of complex numbers and quaternions able to describe spatial objects and the geometric relations between them. This paper uses conformal GA (CGA), which is a part of GA. This paper transforms data from a real Euclidean vector space into a CGA space and presents a new clustering method using conformal vectors. In particular, this paper shows that the proposed method was able to extract the geometric clusters which could not be detected by conventional methods.

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 describe our Gaalop code generator which produces executable C code with just elementary expressions leading to a very efficient implementation.

Multi-modal image registration in medical image analysis is very challenging as the appearance of body structures in images from different imaging devices can be very different. In this paper, we propose a new method [GA-speeded up robust features (SURF)], which incorporates the geometric algebra (GA) into SURF framework, to detect features from images. We model the volumetric data and register the multi-modal medical images using feature spheres formulated in conformal geometric algebra (CGA). Specifically, we first extract features from medical images using GA-SURF. Second, we construct the feature spheres using the feature points and find the correspondence of feature spheres in the two images using CGA. With that, we can register the images based on the correspondence of the feature spheres. The experimental results evaluated by RIRE have shown that our method can register the multi-modal images with high accuracy. The maximum registration error is less than $4mm$ .

This highly practical Guide to Geometric Algebra in Practice reviews algebraic techniques for geometrical problems in computer science and engineering, and the relationships between them. The topics covered range from powerful new theoretical developments, to successful applications, and the development of new software and hardware tools. Topics and features: provides hands-on review exercises throughout the book, together with helpful chapter summaries; presents a concise introductory tutorial to conformal geometric algebra (CGA) in the appendices; examines the application of CGA for the description of rigid body motion, interpolation and tracking, and image processing; reviews the employment of GA in theorem proving and combinatorics; discusses the geometric algebra of lines, lower-dimensional algebras, and other alternatives to 5-dimensional CGA; proposes applications of coordinate-free methods of GA for differential geometry.

To model Euclidean spaces in computerized geometric calculations, the Geometric Algebra framework is becoming popular in computer vision, image analysis, etc. Focusing on the Conformal Geometric Algebra, the claim of the paper is that this framework is useful in digital geometry too. To illustrate this, this paper shows how the Conformal Geometric Algebra allow to simplify the description of digital objects, such as k-dimensional circles in any n-dimensional discrete space. Moreover, the notion of duality is an inherent part of the Geometric Algebra. This is particularly useful since many algorithms are based on this notion in digital geometry. We illustrate this important aspect with the definition of k-dimensional spheres.