Chapter 7
On Geodesic Triangles with Right Angles
in a Dually Flat Space
Frank Nielsen
Abstract The dualistic structure of statistical manifolds in information geometry
yields eight types of (possibly mixed type) geodesic triangles passing through three
given points, the triangle vertices. The interior angles of geodesic triangles can sum
up to π like in Euclidean/Mahalanobis flat geometry, or exhibit otherwise angle
excesses or angle defects. In this work, we initiate the study of geodesic triangles
in dually flat spaces, termed Bregman manifolds, where a generalized Pythagorean
theorem holds. We consider non-self dual Bregman manifolds since Mahalanobis
self-dual manifolds amount to Euclidean geometry. First, we show how to construct
geodesic triangles with either one, two, or three interior right angles, whenever it
is possible. Second, we report a construction of triples of points for which the dual
Pythagorean theorems hold simultaneously at a point, yielding two dual pairs of
dual-type geodesics with right angles at that point.
Keywords Dually flat space · Bregman divergence · Geodesic triangle · Right
angle triangle · Pythagorean theorem · Angle excess/defect · Mahalanobis
manifold · Itakura–Saito manifold · (Extended) Kullback–Leibler manifold ·
Multinoulli manifold
7.1 Introduction and Motivation
In Euclidean geometry, it is well-known that the sum of the three interior angles of any
triangle always sum up to π (Fig. 7.1), and that Euclidean triangles can have at most
one right angle (
π
2
radians or 90
◦ ). These facts are not true anymore in hyperbolic
geometry nor in spherical geometry [27], where the total sum of the interior angles
of a triangle may vary [31]:
F. Nielsen (B)
Sony Computer Science Laboratories, Inc., Tokyo, Japan
e-mail: Frank.Nielsen@acm.org
URL: https://FrankNielsen.github.io/
© Springer Nature Switzerland AG 2021
F. Nielsen (ed.), Progress in Information Geometry,
Signals and Communication Technology,
https://doi.org/10.1007/978-3-030-65459-7_7
153
On Geodesic Triangles with Right Angles
in a Dually Flat Space
Frank Nielsen
Abstract The dualistic structure of statistical manifolds in information geometry
yields eight types of (possibly mixed type) geodesic triangles passing through three
given points, the triangle vertices. The interior angles of geodesic triangles can sum
up to π like in Euclidean/Mahalanobis flat geometry, or exhibit otherwise angle
excesses or angle defects. In this work, we initiate the study of geodesic triangles
in dually flat spaces, termed Bregman manifolds, where a generalized Pythagorean
theorem holds. We consider non-self dual Bregman manifolds since Mahalanobis
self-dual manifolds amount to Euclidean geometry. First, we show how to construct
geodesic triangles with either one, two, or three interior right angles, whenever it
is possible. Second, we report a construction of triples of points for which the dual
Pythagorean theorems hold simultaneously at a point, yielding two dual pairs of
dual-type geodesics with right angles at that point.
Keywords Dually flat space · Bregman divergence · Geodesic triangle · Right
angle triangle · Pythagorean theorem · Angle excess/defect · Mahalanobis
manifold · Itakura–Saito manifold · (Extended) Kullback–Leibler manifold ·
Multinoulli manifold
7.1 Introduction and Motivation
In Euclidean geometry, it is well-known that the sum of the three interior angles of any
triangle always sum up to π (Fig. 7.1), and that Euclidean triangles can have at most
one right angle (
π
2
radians or 90
◦ ). These facts are not true anymore in hyperbolic
geometry nor in spherical geometry [27], where the total sum of the interior angles
of a triangle may vary [31]:
F. Nielsen (B)
Sony Computer Science Laboratories, Inc., Tokyo, Japan
e-mail: Frank.Nielsen@acm.org
URL: https://FrankNielsen.github.io/
© Springer Nature Switzerland AG 2021
F. Nielsen (ed.), Progress in Information Geometry,
Signals and Communication Technology,
https://doi.org/10.1007/978-3-030-65459-7_7
153