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The Double Conformal Space-Time Algebra (DCSTA) is a high-dimensional 12D Geometric Algebra G4,8that extends the concepts introduced with the Double Conformal / Darboux Cyclide Geometric Algebra (DCGA) G8,2 with entities for Darboux cyclides (incl. parabolic and Dupin cyclides, general quadrics, and ring torus) in spacetime with a new boost operator. The base algebra in which spacetime geometry is modeled is the Space-Time Algebra (STA) G1,3. Two Conformal Space-Time subalgebras (CSTA) G2,4 provide spacetime entities for points, flats (incl. worldlines), and hyperbolics, and a complete set of versors for their spacetime transformations that includes rotation, translation, isotropic dilation, hyperbolic rotation (boost), planar reflection, and (pseudo)spherical inversion in rounds or hyperbolics. The DCSTA G4,8 is a doubling product of two G2,4 CSTA subalgebras that inherits doubled CSTA entities and versors from CSTA and adds new bivector entities for (pseudo)quadrics and Darboux (pseudo)cyclides in space...

We show how Fourier transformations can be extended to Hamilton's algebra of quaternions. This was initially motivated by applications in nuclear magnetic resonance and electric engineering. Followed by an ever wider range of applications in color image and signal processing. Hamilton's algebra of quaternions is only one example of the larger class of Clifford's geometric algebras, complete algebras encoding a vector space and all its subspace elements. We introduce how Fourier transformations are extended to Clifford algebras and applied in electromagnetism, and in the processing of images, color images, vector field and climate data.

Conformal geometric algebra is preferred in many applications. Clifford Fourier transforms (CFT) allow holistic signal processing of (multi) vector fields, different from marginal (channel wise) processing: Flow fields, color fields, electro-magnetic fields, ... The Clifford algebra sets (manifolds) of −1 lead to continuous manifolds of CFTs. A frequently asked question is: What does a Clifford Fourier transform of conformal geometric algebra look like? We try to give a first answer.

In this paper geometric algebra of space-time is used to state Maxwell's equations in an ellegant differential and integral form. The proposed integral form involves no derivative with respect to time and is coordinate-free in contrast to what is usually refered as the integral form. The Maxwell's equations are discretized in a very natural manner using their integral form. The discretization of the material equations is discussed in detail. It is directly related to the efficiency of the resulting scheme. Finally, the relation to traditional approaches is shown.

We investigate the most general warped compactification of eleven-dimensional supergravity on eight-dimensional manifolds to AdS3 spaces (in the presence of non-vanishing four-form flux) which preserves N = 2 supersymmetry in three dimensions. Without imposing any restrictions on the chirality of the internal part of the supersymmetry generators, we use geometric algebra techniques to study some implications of the supersymmetry constraints. In particular, we discuss the Lie bracket of certain vector fields constructed as pinor bilinears on the compactification manifold.

Geometric algebra and Clifford algebra are important tools to describe and analyze the physics of the world we live in. Although there is enormous empirical evidence that we are living in four dimensional spacetime, mathematical worlds of higher dimensions can be used to present the physical laws of our world in an aesthetical and didactical more appealing way. In physics and mathematics education we are therefore confronted with the question how these high dimensional spaces should be taught. But as an immediate confrontation of students with high dimensional compactified spacetimes would expect too much from them at the beginning of their university studies, it seems reasonable to approach the mathematics and physics of higher dimensions step by step. The first step naturally is the step from four dimensional spacetime of special relativity to a five dimensional spacetime world. As a toy model for this artificial world cosmological special relativity, invented by Moshe Carmeli, can be used. This five dimensional non-compactified approach describes a spacetime which consists not only of one time dimension and three space dimensions. In addition velocity is regarded as a fifth dimension. This model very probably will not represent physics correctly. But it can be used to discuss and analyze the consequences of an additional dimension in a clear and simple way. Unfortunately Carmeli has formulated cosmological special relativity in standard vector notation. Therefore a translation of cosmological special relativity into the mathematical language of Grassmann and Clifford (Geometric algebra) is given and the physics of cosmological special relativity is discussed.

We will present a fundamental solution of the constructive interference‐problem of waves in two dimensions (also known as fringe patterns). Such problems are known from quantum physics and optics. In case of two one‐dimensional waves with same frequency but different phases and different amplitudes, the solution of their resulting superposition or interference is well known. We will generalize this solution to two dimensions. In case of two dimensions the waves can not only be described by their phases, amplitudes and frequencies, also geometric properties pop up since in two dimensions an infinite number of additional degrees of freedom exists. The wave equations will be given in a traditional Clifford‐valued tensor form. We will solve this problem in a hybrid matrix geometric algebra setting by mapping the traditional tensor expressions to Clifford numbers in conformal space. This Clifford number representation of two‐dimensional waves can be used to solve the interference‐problem linearly. Future work will consist of the generalization from 2D to multi‐dimensional waves.

A novel method which entirely resides inside conformal geometric algebra (CGA) is presented estimating the pose of a camera from one image of a known object. At first, subproblems covering only three feature points are solved and globally assessed. The object model is accordingly pruned and rigidly fitted to corresponding projection rays by evaluating a succinct CGA expression which emerged from a purely geometric approach. It results a set of 3‐point poses each given by a motor. These spinor elements of CGA embody rigid body motions from the manifold SE(3). The poses are then to be averaged according to their quality. This is the second aspect of this work as the respective motors do not come from a linear space and averaging must be carried out appropriately. For this purpose, a technique called weighted intrinsic mean is used.

After identification of multivector functions and differential forms, methods from flat Kahlerian geometry are used to gain deeper insights in the structure of the Hermitean monogenic systems.

A faithful matrix realization of the Clifford Algebra Cl(1,4) is presented in a form which shows its immediate relevance for Fundamental Theoretical Physics. We show that quadratic spaces are a fundamental clue to understand the structure of theoretical physics (see, for example, Keller in ICNAAM 2005,2006 and 2007).

The main idea is to construct a basis for the space L2([0,1]) that can be wrapped isomorphically onto a Clifford algebra Rm of dimension 2m (m going to infinity). The endomorphism algebra End(Rm), itself a Clifford algebra, is then used to encode bounded linear operators on L2([0,1]) such as the Haar wavelet transform.