This means that if we define for a \in \mathbb{R} and (x,y,z,w) \in \mathbb{R}^4 the scalar by vector product a(x,y,z,w)=(ax,ay,az,aw), the quaternion q=(x,y,z,w) may be written unique...This means that if we define for a \in \mathbb{R} and (x,y,z,w) \in \mathbb{R}^4 the scalar by vector product a(x,y,z,w)=(ax,ay,az,aw), the quaternion q=(x,y,z,w) may be written uniquely in the form q=x1+yi+zj+wk. Now if we abbreviate x=x1, the quaternion takes the form q= x+yi+zj+wk. Addition now becomes (x+yi+zj+wk) + (a+bi+cj+dk) = (x+a) +(y+b)i+(z+c)j+(w+d)k. Products of the basis elements 1,i,j,k are defined as follows: \[1q=q1=q \mbox{ for all } q \in \…
Elliptic geometry is the geometry of the sphere (the 2-dimensional surface of a 3-dimensional solid ball), where congruence transformations are the rotations of the sphere about its center.
