Linear Geometry in Euclidean 4-space |
Common terms and phrases
algebra angle angle-vector bijection Cartan's theorem Clifford parallel Clifford parallel lines Clifford translation collinear points complex numbers consequently cross-ratio defined distance function e₁ e₂ equation equivalent Euclidean exists following theorem geometry gonal Hence homogeneous coordinates identity transformation induces inner product intersects invariant 2-planes L₂ left Clifford parallel left resp left translation Lemma linear transformation linearly independent linearly independent vectors maximal set mutually isoclinic 2-planes n-space non-singular nonzero vector orientation associated orientation-reversing orthonormal basis plane points in ph projective line Proof Proposition prove the following prove THEOREM quadric real numbers reflection right translation satisfies the condition scalar semi-orthogonal sends set of mutually set of points Suppose trans u₁ u₂ unit pure quaternions unit quaternion unit vector V₁ V₂ vector space vn+1 vn+l zero Λ Λ