Geometric Algebra for Conics

Foundations of Conic Conformal Geometric Algebra and Compact Versors for Rotation, Translation and Scaling

“…We acknowledge inspiration from the related formalism in [10]. Note that the conic conformal point definition (2) of [10] differs from our definition (43), which partly accounts for the differences and simplifications in the versor expressions for rotations and especially translations below.…”

“…We acknowledge inspiration from the related formalism in [10]. Note that the conic conformal point definition (2) of [10] differs from our definition (43), which partly accounts for the differences and simplifications in the versor expressions for rotations and especially translations below. For the successful implementation of rotations together with a simplification of the translation versors, we found it essential to define the null vector pair {e ∞3 , e o3 } in the symmetric fashion of (2) e ∞3 = 1 √ 2 (e +3 + e −3 ), e o3 = 1 √ 2 (e −3 − e +3 ).…”

“…The parameters 3 λ i , i = 1, 2, 3, are usually in conformal geometric algebra of the plane, or in the modeling of conics in Cl(5, 3) by Perwass [13] or by Hrdina et al [10], simply all set to λ 1 = λ 2 = λ 3 = √ 2. In contrast to that, e.g., El Mir et al [5] have exploited this degree of freedom in conformal geometry for an elegant new formulation of viewpoint change representation.…”

“…Most recently double CGA (DCGA) [6] and quadric CGA (QCGA) [1] have been proposed 1 in this context. We also compare with a recent modified approach provided in [10]. We first focus on the rational algebraic core structure of conic conformal geometric algebra (CCGA) to be presented in some detail.…”

“…While doing that, we introduce a small model preserving modification in the basic algebraic definitions, that is key to preserving the translation formulation of [13], but allows to consistently keep the full system of versors for rotations, translations and scaling. This formulation of the translation versors is simpler than the formulation given in [10].…”