Citation Context

...r Euclidean space with all sorts of applications to computer-aided geometry, robotics, computer vision and the like. A detailed description and analysis of the model is soon to be published elsewhere =-=[1]-=-, so I can concentrate on highlights here, although with a slightly different formulation that I find more convenient for applications. Also, I can assume that this audience is familiar with Geometric...

Citation Context

...e scenario. But since the stratification hierarchy is based on pure point concepts a new algebraic embedding is required when dealing with higher order entities. The conformal geometric algebra (CGA) =-=[24]-=- is well suited to solve this problem, since it subsumes the involved mathematical spaces. Operators are defined to switch entities between the algebras of the conformal space and its Euclidean and pr...

Citation Context

... expressed by special products, called inner., outer A, commutator _x and anticommutatorsproduct. The most powerful and only recently introduced algebra is the conformal geometric algebra 4, (ConfGA) =-=[8]-=-. Because it is suited to describe conformal geometry, it contains spheres as entities and a rich set of geometric manipulations. The point at infinity, e, and the origin, e0, are special elements and...

Citation Context

...ations of various geometric objects Line – L = X1 ∧X2 ∧ n Circle – C = X1 ∧X2 ∧X3 Plane – Φ = X1 ∧X2 ∧X3 ∧ n Sphere – Σ = X1 ∧X2 ∧X3 ∧X4 Inversions. Finally, inversions (x → x2/x) may be represented =-=[17, 18]-=- as F (x) → eF (x)e. Although this will not be discussed here, this becomes particularly important when considering non-Euclidean geometry. This mapping has provided a similar advantage to that of ho...

Citation Context

..., and they too can be multiplied 6via the geometric product. The two algebras which concern us in this paper are the algebras of conformal vectors for the Euclidean plane and three-dimensional space =-=[7, 8, 9]-=-. For the Euclidean plane, let {e1, e2} denote an orthonormal basis set, and write e0 = ē, e3 = e. The full basis set is then {ei}, i = 0 . . .3. These generators satisfy where eiej + ejei = 2ηij, i, ...

Citation Context

...of Hyperspheres There is not enough space here to give a full treatment of the mathematics involved. Therefore, only the most important aspects will be discussed. For a more detailed introduction see =-=[5, 6]-=-. Consider the Minkowski space R 1,1 with basis {e+,e−}, where e 2 + =+1 and e 2 − = −1. The following two null-vectors can be constructed from thissESANN'2003 proceedings - European Symposium on Arti...

Citation Context

...of both properties of the constructed embedding which are the representation of points as null-vectors and the spinor representation of the conformal group. The conformal geometric algebra G4,1 (CGA) =-=[8, 13]-=- is suited to describe conformal geometry. The point at infinity, e ≃ n, and the origin, e0 ≃ s, are special elements of the representation which are used as basis vectors instead of e+ and e− because...

Citation Context

...ircles, spheres and conics, which play a role in many computer vision applications. In particular it will be shown how within the Clifford algebra of conformal space, as introduced by Hestenes et al. =-=[11, 16]-=-, circles can be constructed from three uncertain points in 3D-Euclidean space, while propagating the covariance matrices of the points. This then enables us to obtain and visualize the uncertainty of...