We discuss an alternative approach to the conformal geometric algebra (CGA) in which just a single extra dimension is necessary, as compared to the two normally used. This is made possible by working in a constant curvature background space, rather than the usual Euclidean space. A possible benefit, which is explored here, is that it is possible to define cost functions for geometric object matching in computer vision that are fully covariant, in particular invariant under both rotations and translations, unlike the cost functions which have been used in CGA so far. An algorithm is given for application of this method to the problem of matching sets of lines, which replaces the standard matrix singular value decomposition, by computations wholly in Geometric Algebra terms, and which may itself be of interest in more general settings. Secondly, we consider a further perhaps surprising application of the 1d up approach, which is to the context of a recent paper by Joy Christian published by the Royal Society, which has made strong claims about Bell’s Theorem in quantum mechanics, and its relation to the sphere \(S^7\) and the exceptional group \(E_8\), and proposed a new associative version of the division algebra normally thought to require the octonians. We show that what is being discussed by Christian is mathematically the same as our 1d up approach to 3d geometry, but that after the removal of some incorrect mathematical assertions, the results he proves in the first part of the paper, and bases the application to Bell’s Theorem on, amount to no more than the statement that the combination of two rotors from the Clifford Algebra Cl(4, 0) is also a rotor.

中文翻译：

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保形几何代数的一维向上方法：在线拟合和量子力学中的应用

我们讨论了保形几何代数（CGA）的另一种方法，与通常使用的两种方法相比，仅需一个额外的维即可。通过在恒定曲率的背景空间（而不是通常的欧几里得空间）中工作，可以实现这一点。这里探讨的一个可能的好处是，可以定义完全视觉协变的成本函数，用于计算机视觉中的几何对象匹配，尤其是在旋转和平移下都是不变的，这与CGA中使用的成本函数不同。远。给出了一种将该方法应用到匹配线组问题的算法，该算法通过完全以几何代数形式进行计算来代替标准矩阵奇异值分解，并且它本身可能在更通用的设置中也会引起人们的兴趣。其次，\（S ^ 7 \）和例外群\（E_8 \），并提出了通常被认为需要奥托克数的除法代数的新关联版本。我们证明了Christian讨论的内容在数学上与我们对3d几何的1d up方法相同，但是在删除了一些错误的数学断言之后，他在论文的第一部分证明了这一点，并将其应用到了贝尔定理不等于说来自克利福德代数Cl（4，0）的两个转子的组合也是转子。