We discuss how transformations in a three dimensional euclidean space can be described in terms of the Clifford algebra \({\mathcal {C}}\ell _{3,3}\) of the quadratic space \({\mathbb {R}}^{3,3}\). We show that this algebra describes in a unified way the operations of reflection, rotation (circular and hyperbolic), translation, shear and non-uniform scale. Moreover, using Hodge duality, we define an operation called cotranslation, and show that perspective projection can be written in this Clifford algebra as a composition of translation and cotranslation. We also show that pseudo-perspective can be implemented using the cotranslation operation. In addition, we discuss how a general transformation of points can be described using this formalism. An important point is that the expressions for reflection and rotation in \({\mathcal {C}}\ell _{3,3}\) preserve the subspaces that can be associated with the algebras \({\mathcal {C}}\ell _{3,0}\) and \({\mathcal {C}}\ell _{0,3}\), so that reflection and rotation can be expressed in terms of \({\mathcal {C}}\ell _{3,0}\) or \({\mathcal {C}}\ell _{0,3}\), as is well-known. However, all the other operations mix these subspaces in such a way that these transformations need to be expressed in terms of the full Clifford algebra \({\mathcal {C}}\ell _{3,3}\). An essential aspect of our formulation is the representation of points in terms of objects called paravectors. Paravectors have been used previously to represents points in terms of an algebra closely related to the Clifford algebra \({\mathcal {C}}\ell _{3,3}\). We compare these different approaches.

中文翻译：

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关于3D欧式空间中变换的Clifford代数描述

我们讨论如何用二次空间\（{\ mathbb {R}}的Clifford代数\ {{\ mathcal {C}} \ ell _ {3,3} \）来描述三维欧几里德空间中的变换} ^ {3,3} \）。我们证明了该代数以统一的方式描述了反射，旋转（圆形和双曲线），平移，剪切和不均匀比例的运算。此外，使用Hodge对偶，我们定义了一个称为共翻译的操作，并表明透视投影可以用此Clifford代数写成翻译和共译的组合。我们还显示可以使用共翻译操作来实现伪透视。此外，我们讨论了如何使用这种形式主义来描述点的一般变换。重要的一点是\（{\ mathcal {C}} \ ell _ {3,3} \）中的反射和旋转表达式保留了可与代数\（{\ mathcal {C}}相关的子空间\ ell _ {3,0} \）和\（{\ mathcal {C}} \ ell _ {0,3} \），因此反射和旋转可以用\（{\ mathcal {C} } \ ell _ {3,0} \）或\（{\ mathcal {C}} \ ell _ {0,3} \），众所周知。但是，所有其他运算都将这些子空间混合在一起，以使这些转换需要根据完整的Clifford代数\ {{\ mathcal {C}} \ ell _ {3,3} \}来表达。我们公式化的一个重要方面是用称为对向量的对象表示点。先前已经使用超向量来表示与Clifford代数\（{\ mathcal {C}} \ ell _ {3,3} \）密切相关的代数。我们比较了这些不同的方法。