Geometric Algebra (GA) provides a convenient mathematical notation for representing orientations and rotations of objects in three dimensions. The Conformal model of GA gives the ability to describe algorithms in a geometrically intuitive and compact manner since basic entities, such as spheres, lines, planes and circles, are simply represented by algebraic objects. GA is also more numerically stable and more efficient than rotation matrices making it popular for applications in computer graphics, character animation, and robotics. Rotors are simpler to manipulate than Euler angles and avoid the problem of gimbal lock (Gimbal lock is a common problem associated with Euler angles and occurs because two axes become aligned during rotational operations, producing unexpected behavior since one degree of freedom is lost).
For a detailed introduction to Geometric Algebra, a brief outline of the Conformal model of Geometric Algebra (CGA), and ways to examine how the Conformal model can represent geometric primitives such as points, lines, circles, planes and spheres, please refer to the second chapter of my PhD thesis Tracking and Modelling Motion for Biomechanical Analysis. In the Appendix, we also present some trigonometric solutions using Conformal Geometric Algebra, such as how to find the Nearest Point on a Circle from a Point in Space, the Nearest Point on a Line from a Circle, the Nearest Point on a Line from a Sphere, and the Nearest Point on a Sphere from a Point in Space.
In Proceedings of the International Conference on Empowering Novel Geometric Algebra for Graphics & Engineering (ENGAGE'20), Geneva, Switzerland, September 2020.
Generating human motion is one of the most challenging problems in computational modelling: movement is continuous, highly dimensional, and fundamentally expressive. Articulated motion is usually formulated by linking the skeleton bones of a character model in a hierarchical order (rigging), and then modifying the joint angles over time. Recent developments in deep learning and neural networks have shown promise in synthesizing and controlling articulated characters using convolutional, recurrent, or generative adversarial networks. Deep neural networks, though, work better on data with continuous representations, allowing efficient mapping between the tensor and real-world spaces. As shown in the literature, for 3d rotations, all representations are discontinuous in the real Euclidean spaces of four or fewer dimensions, making difficult for neural networks to learn; this discontinuity in the representation space generates motions with absurd and erroneous rotations. Currently, most of the deep neural network researchers turn to represent motion using positional data, but such a representation have ambiguity problems (roll axis is missing), and cannot ensure bone length violations, requiring the use of an additional Inverse Kinematics layer. In this work, we investigate the use of conformal geometric algebra (CGA), to ensure the continuity in motion representation when deep neural networks are used. We demonstrate the use of CGA and other mathematical frameworks (e.g., Euler angles, unit quaternions, 5d, 6d), comparing the smoothness in their synthesized motion. We also present an easy way to convert motion files (in .bvh format) in CGA representations, and vice versa, for an easy adaptation to neural networks.
Margarita Papaefthimiou, George Papagiannakis, Andreas Aristidou, Marinos Ioannides
In Proceedings of the 6th Conference on Applied Geometric Algebra in Computer Science and Engineering (AGACSE'15), Barcelona, Spain, July 2015.
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 the user the experience of being physically present in real or computer-generated worlds; on the other hand, Augmented Reality (AR) is a live direct or indirect view of a physical environment whose elements are augmented (or supplemented) by computer-generated sensory inputs. Both technologies use interactive devices to achieve the optimum adaptation of the user in the immersive world achieving enhanced presence , harnessing latest advances in computer vision, glasses or headmounted-displays featuring embedded mobile devices. A common issue in all of them is interpolation errors while using different linear and quaternion algebraic methods when a) tracking the user’s position and orientation (translation and rotation) using computer vision b) tracking using mobile sensors c) using gesture input methods to allow the user to interactively edit the augmented scene (translation, rotation and scale) d) animation blending of the virtual characters that augmented the mixed reality scenes (translation and rotation).
In this proposed talk, we aim to enhance the conformal model of Geometric Algebra (CGA) as the mathematical background for camera, display and character animation control in immersive and virtual technology, such as head-mounted displays (e.g. Google Cardboard™) or modern smartphones; a framework that offers a smooth and stable calibration/control can be used in real-time mobile mixed reality systems that featured realistic, animated virtual human actors who augmented real environments. The conformal model of Geometric Algebra is a mathematical framework that provides a convenient mathematical notation for representing orientations and rotations of objects in three dimensions, a compact and geometrically intuitive formulation of algorithms, and an easy and immediate computation of rotors; CGA extends the usefulness of the 3D GA by expanding the class of rotors to include translations, dilations and inversions. Rotors are simpler to manipulate than Euler angles; they are more numerically stable and more efficient than rotation matrices, avoiding the problem of gimbal lock. The results of this work allow us to a) unify and improve the performance of previously separated linear and quaternion algebra camera transformations b) fully replace quaternions for rotation interpolation with faster CGA rotors, c) blend rotations and translations between character animations using CGA, under a single geometric algebraic framework using CGA for Mixed Reality applications.
Andreas Aristidou, Joan Lasenby
Guide to Geometric Algebra in Practice, L. Dorst and J. Lasenby (Eds), pages 47-62, Springer Verlag, 2011.
An iterative Inverse Kinematics solver is implemented using Conformal Geometric Algebra. We use a human hand as an example of implementation where a constrained version of the IK solver is employed for pose tracking. The hand is modelled using CGA, taking advantage of CGA's compact and geometrically intuitive framework, that basic entities in CGA, such as spheres, lines, planes and circles, are simply represented by algebraic objects.
In this paper, CGA was chosen to describe a fast iterative Inverse Kinematics solver (FABRIK) since it simpifies the mathematical model (basic entities, such as spheres, lines, planes and circles, are simply represented by algebraic objects). A simple and efficient methodology for tracking and reconstructing 3d hand poses using a markered optical motion capture system is also presented. It is assumed that the hand geometry, meaning the initial joint configuration of the hand, is known. Also, it is assumed that the hand palm is flat and that the orientation of the hand is given by the hand palm. The model is highly constrained with rotational and orientational constraints implemented using CGA. FABRIK was then applied to fit the rest of the joints; the end effector positions are markers which are attached at the end of each finger and they are captured using an optical motion capture system (Phasespace). The proposed IK solver and hand model are real-time implementable and the resulting motion is smooth and natural.
© 2017 Andreas Aristidou