7 On Geodesic Triangles with Right Angles in a Dually Flat Space
185
S( p, q) = H
η
( p, q) ∩ H
θ
( p, q).
(7.133)
We should make sure S( p, q) belongs to the manifold M when solving the equations
using either the θ - or η-coordinate system.
Proposition 3 The locii of points r such that γ pq ⊥ q γ
∗
qr and γ
∗
pq ⊥ q γ qr is the
intersection of a θ -flat with a η-flat restricted to the manifold: S( p, q) = H
η
( p, q) ∩
H
θ
( p, q).
In general the intersection of a θ -flat with a η-flat is neither a θ -flat nor a η-flat. In
2D, the submanifolds H
η
( p, q) and H
θ
( p, q) can be interpreted as a dual geodesic
and a primal geodesic, respectively. Thus in 2D, S pq may not be simply connected,
and the maximum number of points of the intersection of two geodesics upper bounds
the number of solutions for r . The next section illustrates how to build such triples
of points for the Itakura–Saito manifold.
7.4.2 Explicit Construction in the Itakura–Saito Manifold
Consider the Itakura–Saito manifold described in Sect. 7.2.4.3. Let us exhibit some
triple of points ( p, q, r ) such that γ pq ⊥ q γ
∗
qr and γ
∗
pq ⊥ q γ qr . To avoid confusion,
let us write the 2D θ - and η-coordinates of p by θ
x
p and θ
y
p , and η
x
p and η
y
p .
The second orthogonality constraint equation yields the equation η
pq (θ r − θ q ) =
η
x
pq θ
x
r + η
y
pq θ
y
r − η
pq θ q = 0 which can be rewritten as:
θ
y
r = −
η
x
pq
η
y
pq
θ
x
r +
η
pq θ q
η
y
pq
= aθ
x
r + b,
(7.134)
with a = −
η
x
pq
η
y
pq
and b =
η
pq θ q
η
y
pq
.
The first orthogonality constraint equation yields (θ pq )
(η r − η q ) with η
x
r = −
1
θ x
r
and η
y
r = −
1
θ
y
r
. After multiplying both sides of the equation by θ
x
r θ
y
r , we find
− θ
x
pq θ
y
r − θ
y
pq θ
x
r − θ
pq η q θ
x
r θ
y
r = 0
(7.135)
Letting θ
y
r = aθ
x
r + b, we get the following quadratic equation to solve:
− θ
x
pq (aθ
x
r + b) − θ
y
pq θ
x
r − θ
pq η q θ
x
r (aθ
x
r + b) = 0.
(7.136)
We write the former equation into the following canonical form of quadratic equations:
185
S( p, q) = H
η
( p, q) ∩ H
θ
( p, q).
(7.133)
We should make sure S( p, q) belongs to the manifold M when solving the equations
using either the θ - or η-coordinate system.
Proposition 3 The locii of points r such that γ pq ⊥ q γ
∗
qr and γ
∗
pq ⊥ q γ qr is the
intersection of a θ -flat with a η-flat restricted to the manifold: S( p, q) = H
η
( p, q) ∩
H
θ
( p, q).
In general the intersection of a θ -flat with a η-flat is neither a θ -flat nor a η-flat. In
2D, the submanifolds H
η
( p, q) and H
θ
( p, q) can be interpreted as a dual geodesic
and a primal geodesic, respectively. Thus in 2D, S pq may not be simply connected,
and the maximum number of points of the intersection of two geodesics upper bounds
the number of solutions for r . The next section illustrates how to build such triples
of points for the Itakura–Saito manifold.
7.4.2 Explicit Construction in the Itakura–Saito Manifold
Consider the Itakura–Saito manifold described in Sect. 7.2.4.3. Let us exhibit some
triple of points ( p, q, r ) such that γ pq ⊥ q γ
∗
qr and γ
∗
pq ⊥ q γ qr . To avoid confusion,
let us write the 2D θ - and η-coordinates of p by θ
x
p and θ
y
p , and η
x
p and η
y
p .
The second orthogonality constraint equation yields the equation η
pq (θ r − θ q ) =
η
x
pq θ
x
r + η
y
pq θ
y
r − η
pq θ q = 0 which can be rewritten as:
θ
y
r = −
η
x
pq
η
y
pq
θ
x
r +
η
pq θ q
η
y
pq
= aθ
x
r + b,
(7.134)
with a = −
η
x
pq
η
y
pq
and b =
η
pq θ q
η
y
pq
.
The first orthogonality constraint equation yields (θ pq )
(η r − η q ) with η
x
r = −
1
θ x
r
and η
y
r = −
1
θ
y
r
. After multiplying both sides of the equation by θ
x
r θ
y
r , we find
− θ
x
pq θ
y
r − θ
y
pq θ
x
r − θ
pq η q θ
x
r θ
y
r = 0
(7.135)
Letting θ
y
r = aθ
x
r + b, we get the following quadratic equation to solve:
− θ
x
pq (aθ
x
r + b) − θ
y
pq θ
x
r − θ
pq η q θ
x
r (aθ
x
r + b) = 0.
(7.136)
We write the former equation into the following canonical form of quadratic equations: