In order to perform these calculations the GRRT equation is integrated in parallel with the geodesic equations of motion of each ray.
Let S p ( ρ ( q ) ) be the forward geodesic sphere of radius ρ ( q ) centered at p. Choosing the local g ∇ ρ -orthonormal frame E 1, …, E n − 1 of S p ( ρ ( q ) ) near q, we get local vector fields E 1, …, E n − 1, E n = ∇ ρ by parallel transport along geodesic rays.
In particular, we require a formula for the parallel transport and the geodesics of the COP manifold.
The formulas for the tangent spaces, the geodesics, the parallel transport, and the projection onto the tangent spaces of Q k ( m ) follow directly from the product manifold structure.
Table 1 Flow processes and major centrality measures Duplication Transfer Parallel duplication Serial duplication Geodesics Closeness Closeness Betweenness Closeness Paths Degree Closeness Trails Degree Closeness Walks Degree Eigenvector PageRank *Adapted from Borgatti [36], p.63.
This is done via parallel transport along the geodesic γ, which we denote by τ : T x i M → T x i + 1 M. (37).
Accordingly, the tangent spaces, the geodesics, and the parallel transport for Q ( m ) and ( C P m − 1 ) m coincide locally and thus are easily derived from the geometry of C P m − 1.
Firstly it must be said that very few surfaces possess patterns of geodesics which run parallel at constant distance: They exist precisely on the intrinsically flat surfaces with vanishing Gaussian curvature.
The curvature of the earth's surface, for example, is revealed in the fact that geodesics that begin in parallel directions can begin to approach one another for example, two lines of longitude can both be perpendicular to the equator, but converge on one another as they approach the poles.
