10.9 The Einstein Gyroparallelogram

An Einstein gyroparallelogram is a gyroparallelogram, Def. 6.40, in an Einstein gyrovector space.

Let a, b, c be any three nongyrocollinear points in an Einstein gyrovector space ( G, ?, ?), and let d = ( b ? c)? a. Then the four points a, b, c, d are the vertices of the Einstein gyroparallelogram abdc, Def. 6.40, with gyrocenter m _abdc, Fig. 10.6. The two diagonals, ad and bc, of the gyroparallelogram intersect at their gyromidpoints m _ad and m _{bc ,} Fig. 10.6,

(10.46)

By CM velocity considerations similar to those shown in Fig. 10.3, the gyrocenter m _abcd of the Einstein gyroparallelogram abdc is given by each of the following three expressions, Fig. 10.6.

(10.47)

Figure 10.6: The Einstein gyroparallelogram. Let a, b, c be any three nongyrocollinear points in an Einstein gyrovector space ( , ?, ?), being the s-ball of the real inner product space ( , +, ), and let d = ( b ? c)? a. Then the four points a, b, c, d are the vertices of the Einstein gyroparallelogram abdc, Def. 6.40, and, by Theorem 6.45, opposite sides are equal modulo gyrations. Shown are three expressions for the gyrocenter m _abdc = m _ad = m _bc of the...