7 On Geodesic Triangles with Right Angles in a Dually Flat Space
181
η
θ
η
θ
Fig. 7.13 Two examples of ∇-triangles with two interior right angles (the opposite angles to the
thicker vertex r )
α q ( p, r ) = 90.0,
α r ( p, q) = 12.82764159141668.
• θ( p) = (1.7128340504770114, 1.2510418358297621),
θ(q) = (1.446857135939727, 1.7930125176801988),
θ(r ) = (1.1177842396781703, 1.5922051785236535),
α p (q, r ) = 90.00000000000001,
α q ( p, r ) = 90.00000000000001,
α r ( p, q) = 6.595093466701274.
Define the following two θ -flat submanifolds:
H
θ
q : θ
r ∇
2 F(θ q )θ pq − θ
q ∇
2 F(θ q )θ pq = 0,
(7.118)
H
θ
p : θ
r ∇
2 F(θ p )θ pq − θ
p ∇
2 F(θ p )θ pq = 0.
(7.119)
We end up with the following proposition:
Proposition 2 In a non-Mahalanobis Bregman manifold M, the locii of points r
that form a double right angle with the geodesic arc γ pq is H
θ
q ∩ H
θ
q .