156
F. Nielsen
Table 7.1 Eight types of geodesic triangles paired by duality
Geodesic triangle T
Dual geodesic triangle T ∗
T = γ pq γ qr γ r p (type ••• ppp : a ∇-triangle)
T ∗ = γ ∗
pq γ ∗
qr γ ∗
r p (type ••• ddd: a ∇ ∗ -triangle)
T = γ pq γ qr γ ∗
r p (type ••• ppd)
T ∗ = γ ∗
pq γ ∗
qr γ r p (type ••• ddp)
T = γ pq γ ∗
qr γ ∗
r p (type ••• pdd)
T ∗ = γ ∗
pq γ qr γ r p (type ••• dpp)
T = γ pq γ ∗
qr γ r p (type ••• pdp)
T ∗ = γ ∗
pq γ qr γ ∗
r p (type ••• dpd)
r . Let “p” or • stands for primal and “d” or • stands for dual. A primal geodesic arc
γ pq and a dual geodesic arc γ
∗
pq may also be written γ •
pq and γ •
pq , respectively. The
8 types of geodesic triangles are: ••• ppp, ••• ppd, ••• pdd, ••• pdp, ••• ddd, •••
ddp, ••• dpp, and ••• dpd. We can group these eight types of geodesic triangles into
two groups of four geodesic triangles each as shown in Table 7.1.
A geodesic triangle T
∗ is dual to another geodesic triangle T iff. each geodesic
arc of T
∗ is corresponding to a dual geodesic arc of T . For example, the triangle
T
∗
= γ
∗
pq γ qr γ r p (type ••• dpp) is dual to the triangle T = γ pq γ
∗
qr γ
∗
r p (type ••• pdd).
At each triangle vertex, we have 4 geodesic arcs defining
4
2
= 6 angles. A triple
of points defines 8 geodesic triangles with 4 × 3 = 12 interior angles.
2 The interior
angle α p (c pq , c pr ) of a geodesic triangle T = c pq c qr c r p (where c ab stands either for
γ ab or for γ
∗
ab ) at p is measured according to the metric tensor g as follows:
α p (c pq , c pr ) = arccos
g p ( ˙
c pq (0), ˙
c pr (0))
˙
c pq (0) p ˙
c pr (0) p
= α p (c pr , c pq ),
(7.2)
where v p =
g p (v, v) measures the length of vector v ∈ T p . The total sum α(T )
of the three interior angles of a geodesic triangle T = c pq c qr c r p is defined by
α(T ) := α p (c pq , c pr ) + α q (c qp , c qr ) + α r (c r p , c rq ).
(7.3)
We may specify a geodesic triangle as follows: T E ( p, q, r ) where p, q, r ∈ M are the
three triangle vertices (points on M), and E ∈ (t 1 ∈ {p, d}, t 2 ∈ {p, d}, t 3 ∈ {p, d})
is the type of primal/dual geodesic edges of the triangle so that c
t i
ab = γ ab if t i = p and
c
t i
ab = γ
∗
ab if t i = d. A geodesic triangle T = T E ( p, q, r ) with all primal geodesics
arcs (E = ( p, p, p)) is called a ∇-triangle, and its dual geodesic triangle T
∗
=
T E ∗ ( p, q, r ) (with all dual geodesic arcs, i.e., E
∗
= (d, d, d)) is called a ∇
∗ -triangle.
By considering duality of geodesic triangles and permutations of the triple of
points p, q and r , we may reduce the study of interior angles between a pair of
geodesics at a vertex to three types of primal geodesic triangles (type ppp, pdp,
and pdd) with their corresponding dual geodesic triangles (type ddd, dpd and dpp,
respectively).
2 We do consider at a triangle vertex only pairs of geodesics with interior angles linking the two
other triangle vertices.
F. Nielsen
Table 7.1 Eight types of geodesic triangles paired by duality
Geodesic triangle T
Dual geodesic triangle T ∗
T = γ pq γ qr γ r p (type ••• ppp : a ∇-triangle)
T ∗ = γ ∗
pq γ ∗
qr γ ∗
r p (type ••• ddd: a ∇ ∗ -triangle)
T = γ pq γ qr γ ∗
r p (type ••• ppd)
T ∗ = γ ∗
pq γ ∗
qr γ r p (type ••• ddp)
T = γ pq γ ∗
qr γ ∗
r p (type ••• pdd)
T ∗ = γ ∗
pq γ qr γ r p (type ••• dpp)
T = γ pq γ ∗
qr γ r p (type ••• pdp)
T ∗ = γ ∗
pq γ qr γ ∗
r p (type ••• dpd)
r . Let “p” or • stands for primal and “d” or • stands for dual. A primal geodesic arc
γ pq and a dual geodesic arc γ
∗
pq may also be written γ •
pq and γ •
pq , respectively. The
8 types of geodesic triangles are: ••• ppp, ••• ppd, ••• pdd, ••• pdp, ••• ddd, •••
ddp, ••• dpp, and ••• dpd. We can group these eight types of geodesic triangles into
two groups of four geodesic triangles each as shown in Table 7.1.
A geodesic triangle T
∗ is dual to another geodesic triangle T iff. each geodesic
arc of T
∗ is corresponding to a dual geodesic arc of T . For example, the triangle
T
∗
= γ
∗
pq γ qr γ r p (type ••• dpp) is dual to the triangle T = γ pq γ
∗
qr γ
∗
r p (type ••• pdd).
At each triangle vertex, we have 4 geodesic arcs defining
4
2
= 6 angles. A triple
of points defines 8 geodesic triangles with 4 × 3 = 12 interior angles.
2 The interior
angle α p (c pq , c pr ) of a geodesic triangle T = c pq c qr c r p (where c ab stands either for
γ ab or for γ
∗
ab ) at p is measured according to the metric tensor g as follows:
α p (c pq , c pr ) = arccos
g p ( ˙
c pq (0), ˙
c pr (0))
˙
c pq (0) p ˙
c pr (0) p
= α p (c pr , c pq ),
(7.2)
where v p =
g p (v, v) measures the length of vector v ∈ T p . The total sum α(T )
of the three interior angles of a geodesic triangle T = c pq c qr c r p is defined by
α(T ) := α p (c pq , c pr ) + α q (c qp , c qr ) + α r (c r p , c rq ).
(7.3)
We may specify a geodesic triangle as follows: T E ( p, q, r ) where p, q, r ∈ M are the
three triangle vertices (points on M), and E ∈ (t 1 ∈ {p, d}, t 2 ∈ {p, d}, t 3 ∈ {p, d})
is the type of primal/dual geodesic edges of the triangle so that c
t i
ab = γ ab if t i = p and
c
t i
ab = γ
∗
ab if t i = d. A geodesic triangle T = T E ( p, q, r ) with all primal geodesics
arcs (E = ( p, p, p)) is called a ∇-triangle, and its dual geodesic triangle T
∗
=
T E ∗ ( p, q, r ) (with all dual geodesic arcs, i.e., E
∗
= (d, d, d)) is called a ∇
∗ -triangle.
By considering duality of geodesic triangles and permutations of the triple of
points p, q and r , we may reduce the study of interior angles between a pair of
geodesics at a vertex to three types of primal geodesic triangles (type ppp, pdp,
and pdd) with their corresponding dual geodesic triangles (type ddd, dpd and dpp,
respectively).
2 We do consider at a triangle vertex only pairs of geodesics with interior angles linking the two
other triangle vertices.