Reflections in conics, quadrics and hyperquadrics via Clifford algebra

Abstract

Abstract.In this article we present a new and not fully employed geometric algebra model. With this model a generalization of the conformal geometric algebra model is achieved. We discuss the geometric objects that can be represented. Furthermore, we show that the Pin group of this geometric algebra corresponds to the group of inversions with respect to quadrics in principal position. We discuss the construction for the two-and three-dimensional case in detail and give the construction for arbitrary dimension.

Introduction to Cartan numbers and some geometric applications

“…Using Sylvester's law of inertia one can find a basis {e 1 , e 2 , … , e n } of  such that e 2 i is 1, 0, − 1. 5 Definition 3. The number of basis vectors that square to (1, −1, 0) is called signature (p, q, r).…”

“…If r ≠ 0, the Clifford algebra is called the degenerate Clifford algebra. 5 The quaternion algebra H has the basis 1, i, j, k ∈ R 4 , and the multiplication of quaternions is associative and non commutative multiplication. Since there is an isomorphism between the quaternion algebra and the ring M 2×2 (R) of all 2 × 2 real matrices induced by…”

“…Definition The number of basis vectors that square to $$$ \left(1,-1,0\right) $$$ is called signature $$$ \left(p,q,r\right) $$$. If $$$ r\ne 0 $$$, the Clifford algebra is called the degenerate Clifford algebra 5 …”

“…Let  be a real-valued vector space of dimension n. Furthermore, let 𝔔 ∶  → R be a quadratic form 𝔔 on  The pair (, 𝔔) is called quadratic space. 5 Definition 2. The Clifford algebra is defined by relations e i e 𝑗 + e 𝑗 e i = 2 ⟨ e i , e 𝑗 ⟩ , 1 ≤ i, 𝑗 ≤ n…”

“…Definition Let $$$ \mathcal{V} $$$ be a real‐valued vector space of dimension $$$ n $$$. Furthermore, let $$$ \mathfrak{Q}:\mathcal{V}\to \mathbb{R} $$$ be a quadratic form $$$ \mathfrak{Q} $$$ on $$$ \mathcal{V} $$$ The pair $$$ \left(\mathcal{V},\mathfrak{Q}\right) $$$ is called quadratic space 5 …”