intrinsic simplices on spaces of nearly constant curvature › slides › mwintraecken.pdf · here:...
TRANSCRIPT
![Page 1: Intrinsic simplices on spaces of nearly constant curvature › slides › MWintraecken.pdf · Here: Model simplices using space of constant curvature I Map to space of constant curvature](https://reader033.vdocuments.us/reader033/viewer/2022052722/5f0d18db7e708231d438a9ea/html5/thumbnails/1.jpg)
Intrinsic simplices on spaces of nearly constant curvature
Ramsay Dyer, Gert Vegter and Mathijs Wintraecken
Johann Bernoulli Institute
Workshop on computational geometry in non-Euclidean spaces
Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 1 / 48
![Page 2: Intrinsic simplices on spaces of nearly constant curvature › slides › MWintraecken.pdf · Here: Model simplices using space of constant curvature I Map to space of constant curvature](https://reader033.vdocuments.us/reader033/viewer/2022052722/5f0d18db7e708231d438a9ea/html5/thumbnails/2.jpg)
Intrinsic simplices on Riemannian manifolds
Motivation: Generic triangulationcriteria
intrinsic setting
explicit quality requirements
Arbitrary dimension
Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 2 / 48
![Page 3: Intrinsic simplices on spaces of nearly constant curvature › slides › MWintraecken.pdf · Here: Model simplices using space of constant curvature I Map to space of constant curvature](https://reader033.vdocuments.us/reader033/viewer/2022052722/5f0d18db7e708231d438a9ea/html5/thumbnails/3.jpg)
Modeling simplices: quality
Non-degeneracy by quality requirements depending on curvature manifold
SOCG: Model simplices on Euclidean spaceI To tangent space via exponential mapI Almost flat
Here: Model simplices using space of constant curvatureI Map to space of constant curvatureI Curvature is (locally) nearly constantI Towards adaptive sampling
New quality measures for spaces of constant curvature
Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 3 / 48
![Page 4: Intrinsic simplices on spaces of nearly constant curvature › slides › MWintraecken.pdf · Here: Model simplices using space of constant curvature I Map to space of constant curvature](https://reader033.vdocuments.us/reader033/viewer/2022052722/5f0d18db7e708231d438a9ea/html5/thumbnails/4.jpg)
Outline
1 Intrinsic simplices and non-degeneracy
2 Topogonov comparison theorem
3 Non-degeneracy criteria for simplices modeled on simplices in Euclideanspace
4 Non-degeneracy criteria for simplices modeled on simplices in spaces ofconstant curvature
5 Quality measures for simplices in spaces of constant curvature
6 Questions about quality
7 Results
Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 4 / 48
![Page 5: Intrinsic simplices on spaces of nearly constant curvature › slides › MWintraecken.pdf · Here: Model simplices using space of constant curvature I Map to space of constant curvature](https://reader033.vdocuments.us/reader033/viewer/2022052722/5f0d18db7e708231d438a9ea/html5/thumbnails/5.jpg)
Intrinsic simplices andnon-degeneracy
Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 5 / 48
![Page 6: Intrinsic simplices on spaces of nearly constant curvature › slides › MWintraecken.pdf · Here: Model simplices using space of constant curvature I Map to space of constant curvature](https://reader033.vdocuments.us/reader033/viewer/2022052722/5f0d18db7e708231d438a9ea/html5/thumbnails/6.jpg)
Convex hulls
Natural way to “fill in” a simplex?
Convex hull bad:-Generally convex hull three points not two di-mensional-Stronger conjectured not closed
Berger 2001 (panoramic overview):‘It appears as if there is no canonical method tofill up a triangle, or more general simplex, in ageneric Riemannian manifold. But this is not true-the problem is solved by the notion of center ofmass, modeled on Euclidean geometry.’
Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 6 / 48
![Page 7: Intrinsic simplices on spaces of nearly constant curvature › slides › MWintraecken.pdf · Here: Model simplices using space of constant curvature I Map to space of constant curvature](https://reader033.vdocuments.us/reader033/viewer/2022052722/5f0d18db7e708231d438a9ea/html5/thumbnails/7.jpg)
Centres of mass in Euclidean space
Weighted average of points ∑µivi
assume∑µi = 1.
Generalizes to ∫p dµ(p)
is where the minimum of
PRn(x) =1
2
∫‖x− p‖2dµ(p),
is attained
Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 7 / 48
![Page 8: Intrinsic simplices on spaces of nearly constant curvature › slides › MWintraecken.pdf · Here: Model simplices using space of constant curvature I Map to space of constant curvature](https://reader033.vdocuments.us/reader033/viewer/2022052722/5f0d18db7e708231d438a9ea/html5/thumbnails/8.jpg)
Riemannian centres of mass
Centre of mass
Eλ(x) =1
2
∑i
λidM (x, vi)2
barycentric coordinates: λi ≥ 0,∑λi = 1
Bσj :∆j →M
λ 7→ argminx∈Bρ
Eλ(x)
∆j the standard Euclidean j-simplex, σM image
Point where minimum Eλ(x) is attained is characterized by∑λi exp−1
x (vi) = 0. (generalization of∑λi(vi − x) = 0)
Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 8 / 48
![Page 9: Intrinsic simplices on spaces of nearly constant curvature › slides › MWintraecken.pdf · Here: Model simplices using space of constant curvature I Map to space of constant curvature](https://reader033.vdocuments.us/reader033/viewer/2022052722/5f0d18db7e708231d438a9ea/html5/thumbnails/9.jpg)
Smooth map
Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 9 / 48
![Page 10: Intrinsic simplices on spaces of nearly constant curvature › slides › MWintraecken.pdf · Here: Model simplices using space of constant curvature I Map to space of constant curvature](https://reader033.vdocuments.us/reader033/viewer/2022052722/5f0d18db7e708231d438a9ea/html5/thumbnails/10.jpg)
Exponential map
Notation
vi(x) = exp−1x (vi),
σ(x) = {v0(x), . . . , vj(x)}⊂ TxM ,injectivity radius ιM
vi(x) = exp−1x (vi) is smooth
Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 10 / 48
![Page 11: Intrinsic simplices on spaces of nearly constant curvature › slides › MWintraecken.pdf · Here: Model simplices using space of constant curvature I Map to space of constant curvature](https://reader033.vdocuments.us/reader033/viewer/2022052722/5f0d18db7e708231d438a9ea/html5/thumbnails/11.jpg)
Non-degeneracy; a consequence of linear independence
Definition
A Riemannian simplex σM is non-degenerate if σM is diffeomorphic to thestandard simplex ∆n
Lemma (Consequences of linear independence)
If tangents to geodesics connecting any n (in neighbourhood) to somesubset v0, . . . , vj−1, vj+1, . . . vn (may depend on x) are linearlyindependent then
the map ∆n → σM is bijective
The inverse of ∆n → σM is smooth
Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 11 / 48
![Page 12: Intrinsic simplices on spaces of nearly constant curvature › slides › MWintraecken.pdf · Here: Model simplices using space of constant curvature I Map to space of constant curvature](https://reader033.vdocuments.us/reader033/viewer/2022052722/5f0d18db7e708231d438a9ea/html5/thumbnails/12.jpg)
∆n → σM is bijective
Proof by contradiction∑λi exp−1
x (vi) =∑
λ̃i exp−1x (vi) =
∑λ̃ivi(x) = 0
with∑λi =
∑λ̃i = 1. Because v0(x), . . . , v̂j(x), . . . , vn(x) linearly
independent λj 6= 0, λ̃j 6= 0. So
λ0
λjv0(x) + . . .+
λj−1
λjvj−1(x) +
λj+1
λjvj+1(x) + . . .+
λnλjvn(x) =
λ̃0
λ̃jv0(x) + . . .+
λ̃j−1
λ̃jvj−1(x) +
λ̃j+1
λ̃jvj+1(x) + . . .+
λ̃n
λ̃jvn(x).
Contradiction
Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 12 / 48
![Page 13: Intrinsic simplices on spaces of nearly constant curvature › slides › MWintraecken.pdf · Here: Model simplices using space of constant curvature I Map to space of constant curvature](https://reader033.vdocuments.us/reader033/viewer/2022052722/5f0d18db7e708231d438a9ea/html5/thumbnails/13.jpg)
Inverse of ∆n → σM is smooth
Proof ∑λi exp−1
x (vi) =∑
λivi(x) = 0,
λj 6= 0 because if λj = 0 then∑i 6=j
λivi(x) = 0,
contradicting linear independence. So
(v0(x), . . . , vj−1(x), vj+1(x), . . . vn(x))−1vj(x) =
(λiλj
).
Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 13 / 48
![Page 14: Intrinsic simplices on spaces of nearly constant curvature › slides › MWintraecken.pdf · Here: Model simplices using space of constant curvature I Map to space of constant curvature](https://reader033.vdocuments.us/reader033/viewer/2022052722/5f0d18db7e708231d438a9ea/html5/thumbnails/14.jpg)
Linear independence 2D
In two dimensions linear indepen-dence easy (Rustamov 2010)
If exp−1x (v0) = v0(x), v1(x),
v2(x) do not span TxMthen they are co-linear.
Equivalent to v0, v1 and v2
lying on geodesic.
Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 14 / 48
![Page 15: Intrinsic simplices on spaces of nearly constant curvature › slides › MWintraecken.pdf · Here: Model simplices using space of constant curvature I Map to space of constant curvature](https://reader033.vdocuments.us/reader033/viewer/2022052722/5f0d18db7e708231d438a9ea/html5/thumbnails/15.jpg)
Friedland’s bounds
Stability of determinants
|det(A+ E)− det(A)| ≤ nmax{‖A‖p, ‖A+ E‖p}n−1‖E‖p
with A and E, n× n-matrices‖ · ‖p p-norm on matrices 1 ≤ p ≤ ∞:
‖A‖p = maxx∈Rn
|Ax|p|x|p
,
p-norm on vectors:
|w|p = ((w1)p + . . .+ (wn)p)1/p
Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 15 / 48
![Page 16: Intrinsic simplices on spaces of nearly constant curvature › slides › MWintraecken.pdf · Here: Model simplices using space of constant curvature I Map to space of constant curvature](https://reader033.vdocuments.us/reader033/viewer/2022052722/5f0d18db7e708231d438a9ea/html5/thumbnails/16.jpg)
Topogonov comparison theorem
Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 16 / 48
![Page 17: Intrinsic simplices on spaces of nearly constant curvature › slides › MWintraecken.pdf · Here: Model simplices using space of constant curvature I Map to space of constant curvature](https://reader033.vdocuments.us/reader033/viewer/2022052722/5f0d18db7e708231d438a9ea/html5/thumbnails/17.jpg)
Triangles and cosine rules
A
C
Bb
c
a
α
β
γ
Cosine rules:
cosa
k= cos
b
kcos
c
k+ sin
b
ksin
c
kcosα,
in a space H(1/k2) of sectional curvature1/k2
a2 = b2 + c2 − 2bc cosα
in Euclidean space Rn
cosha
k= cosh
b
kcosh
c
k− sinh
b
ksinh
c
kcosα,
in a space H(−1/k2) of sectional curvature−1/k2
Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 17 / 48
![Page 18: Intrinsic simplices on spaces of nearly constant curvature › slides › MWintraecken.pdf · Here: Model simplices using space of constant curvature I Map to space of constant curvature](https://reader033.vdocuments.us/reader033/viewer/2022052722/5f0d18db7e708231d438a9ea/html5/thumbnails/18.jpg)
Geodesic triangles
Geodesic triangle T : Three minimizing geodesics connecting three pointson a arbitrary manifold (no interior)
A
C
Bb
c
a
Alexandrov triangle: A geodesic triangle with same edge lengths on spaceof constant curvature H(Λ∗).
Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 18 / 48
![Page 19: Intrinsic simplices on spaces of nearly constant curvature › slides › MWintraecken.pdf · Here: Model simplices using space of constant curvature I Map to space of constant curvature](https://reader033.vdocuments.us/reader033/viewer/2022052722/5f0d18db7e708231d438a9ea/html5/thumbnails/19.jpg)
Hinges
Hinge: Two minimizing geodesics connecting three points and enclosedangle on a arbitrary manifold
A
C
Bb
cα
Rauch hinge: A hinge with the same edge lengths and enclosed angle on aspace of constant curvature H(Λ∗).
Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 19 / 48
![Page 20: Intrinsic simplices on spaces of nearly constant curvature › slides › MWintraecken.pdf · Here: Model simplices using space of constant curvature I Map to space of constant curvature](https://reader033.vdocuments.us/reader033/viewer/2022052722/5f0d18db7e708231d438a9ea/html5/thumbnails/20.jpg)
Topogonov comparison theorem
Manifold M , sectional curvatures Λ− ≤ K ≤ Λ+.
Given: Geodesic triangle T on M then exist TΛ− , TΛ+ on H(Λ−), H(Λ+)and
αΛ− ≤ α ≤ αΛ+ ,
Given: Hinge on M then exist Rauch hinges on H(Λ−), H(Λ+) and thelength of the closing geodesics satisfy
cΛ− ≥ c ≥ cΛ+ .
Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 20 / 48
![Page 21: Intrinsic simplices on spaces of nearly constant curvature › slides › MWintraecken.pdf · Here: Model simplices using space of constant curvature I Map to space of constant curvature](https://reader033.vdocuments.us/reader033/viewer/2022052722/5f0d18db7e708231d438a9ea/html5/thumbnails/21.jpg)
Non-degeneracy criteria forsimplices modeled on simplices in
Euclidean space
Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 21 / 48
![Page 22: Intrinsic simplices on spaces of nearly constant curvature › slides › MWintraecken.pdf · Here: Model simplices using space of constant curvature I Map to space of constant curvature](https://reader033.vdocuments.us/reader033/viewer/2022052722/5f0d18db7e708231d438a9ea/html5/thumbnails/22.jpg)
Setting
Setting:
Manifold M with bounded curvature |K| ≤ Λ.
Points {v0, . . . , vn} in a small ball in M : vertices.
Choose vertex vr.
σE(vr) convex hull of the exp−1vr (vi) = vi(vr).
Goal:
Give conditions on σE(vr) that imply non-degeneracy
Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 22 / 48
![Page 23: Intrinsic simplices on spaces of nearly constant curvature › slides › MWintraecken.pdf · Here: Model simplices using space of constant curvature I Map to space of constant curvature](https://reader033.vdocuments.us/reader033/viewer/2022052722/5f0d18db7e708231d438a9ea/html5/thumbnails/23.jpg)
Step 1
We know all the geodesics emanating from vr. Topogonov for Hingesbounds the length of blue geodesics in the middle by those on the side.
Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 23 / 48
![Page 24: Intrinsic simplices on spaces of nearly constant curvature › slides › MWintraecken.pdf · Here: Model simplices using space of constant curvature I Map to space of constant curvature](https://reader033.vdocuments.us/reader033/viewer/2022052722/5f0d18db7e708231d438a9ea/html5/thumbnails/24.jpg)
Step 2
In small neighbourhood the lengths of geodesics on the left and right areclose to the lengths in the tangent space (via exp−1
H(Λ∗))
This implies that the same holds for M in the middle.
Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 24 / 48
![Page 25: Intrinsic simplices on spaces of nearly constant curvature › slides › MWintraecken.pdf · Here: Model simplices using space of constant curvature I Map to space of constant curvature](https://reader033.vdocuments.us/reader033/viewer/2022052722/5f0d18db7e708231d438a9ea/html5/thumbnails/25.jpg)
Step 2
In small neighbourhood the lengths of geodesics on the left and right areclose to the lengths in the tangent space (via exp−1
H(Λ∗))
This implies that the same holds for M in the middle.
Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 25 / 48
![Page 26: Intrinsic simplices on spaces of nearly constant curvature › slides › MWintraecken.pdf · Here: Model simplices using space of constant curvature I Map to space of constant curvature](https://reader033.vdocuments.us/reader033/viewer/2022052722/5f0d18db7e708231d438a9ea/html5/thumbnails/26.jpg)
Step 3
Use the Toponogov comparison theorem for geodesic triangles to concludethat the angles (and inner product) are close to those in the tangent space
Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 26 / 48
![Page 27: Intrinsic simplices on spaces of nearly constant curvature › slides › MWintraecken.pdf · Here: Model simplices using space of constant curvature I Map to space of constant curvature](https://reader033.vdocuments.us/reader033/viewer/2022052722/5f0d18db7e708231d438a9ea/html5/thumbnails/27.jpg)
Step 4
Gram matrix (in n directions)
(〈exp−1x vi, exp−1
x vl〉)i,l 6=j = (〈vi(x), vl(x)〉)i,l 6=j
is close to
(〈vi(vr)− x(vr), vl(vr)− x(vr)〉)i,l 6=j
Determinants:
Friedlands result on stability of determinants gives that determinantsare close
determinants zero iff n tangent vectors linearly independent
determinants are like volume squared, gives a quality measure
Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 27 / 48
![Page 28: Intrinsic simplices on spaces of nearly constant curvature › slides › MWintraecken.pdf · Here: Model simplices using space of constant curvature I Map to space of constant curvature](https://reader033.vdocuments.us/reader033/viewer/2022052722/5f0d18db7e708231d438a9ea/html5/thumbnails/28.jpg)
Step 5
(before) linear independence implies diffeomorphism
(here) If normalized volume simplex is large enough then linearindependent
Gives non-degeneracy criteria
Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 28 / 48
![Page 29: Intrinsic simplices on spaces of nearly constant curvature › slides › MWintraecken.pdf · Here: Model simplices using space of constant curvature I Map to space of constant curvature](https://reader033.vdocuments.us/reader033/viewer/2022052722/5f0d18db7e708231d438a9ea/html5/thumbnails/29.jpg)
Non-degeneracy criteria forsimplices modeled on simplices inspaces of constant curvature
Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 29 / 48
![Page 30: Intrinsic simplices on spaces of nearly constant curvature › slides › MWintraecken.pdf · Here: Model simplices using space of constant curvature I Map to space of constant curvature](https://reader033.vdocuments.us/reader033/viewer/2022052722/5f0d18db7e708231d438a9ea/html5/thumbnails/30.jpg)
Setting
Setting:
Manifold M with bounded curvature Λ− ≤ K ≤ Λ+, with 0 < Λ− orΛ+ < 0.
Points {v0, . . . , vn} in a small ball in M : vertices.
Choose vertex vr.
Model on σH(Λmid)(vr) convex hull ofexpH(Λmid) ◦ exp−1
vr (vi) = vi(vr), with Λmid ∈ [Λ−,Λ+].If Λ−,Λ+ > 0
Λmid =1
2(Λ− + Λ+).
Goal:
Give conditions on σH(Λmid)(vr) that imply non-degeneracy
Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 30 / 48
![Page 31: Intrinsic simplices on spaces of nearly constant curvature › slides › MWintraecken.pdf · Here: Model simplices using space of constant curvature I Map to space of constant curvature](https://reader033.vdocuments.us/reader033/viewer/2022052722/5f0d18db7e708231d438a9ea/html5/thumbnails/31.jpg)
Step 1
We know all the geodesics emanating from vr. Topogonov for Hingesbounds the length of blue geodesics in the middle by those on the side.
Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 31 / 48
![Page 32: Intrinsic simplices on spaces of nearly constant curvature › slides › MWintraecken.pdf · Here: Model simplices using space of constant curvature I Map to space of constant curvature](https://reader033.vdocuments.us/reader033/viewer/2022052722/5f0d18db7e708231d438a9ea/html5/thumbnails/32.jpg)
Step 2
In small neighbourhood the lengths of geodesics on the left and right areclose to the lengths on H(Λmid) (via expH(Λmid) ◦ exp−1
vr,M).
This implies that the same holds for M in the middle.Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 32 / 48
![Page 33: Intrinsic simplices on spaces of nearly constant curvature › slides › MWintraecken.pdf · Here: Model simplices using space of constant curvature I Map to space of constant curvature](https://reader033.vdocuments.us/reader033/viewer/2022052722/5f0d18db7e708231d438a9ea/html5/thumbnails/33.jpg)
Step 3
Use the Toponogov comparison theorem for geodesic triangles to concludethat the angles (and inner product) are close to those on H(Λmid)
Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 33 / 48
![Page 34: Intrinsic simplices on spaces of nearly constant curvature › slides › MWintraecken.pdf · Here: Model simplices using space of constant curvature I Map to space of constant curvature](https://reader033.vdocuments.us/reader033/viewer/2022052722/5f0d18db7e708231d438a9ea/html5/thumbnails/34.jpg)
Step 3 (continued)
These inner products are:
1
Λmidsin (
√ΛmiddM (x, vi)) sin (
√ΛmiddM (x, vj)) cos θij,M (elliptic)
1
Λmidsinh (
√ΛmiddM (x, vi)) sinh (
√ΛmiddM (x, vj)) cos θij,M (hyperbolic)
from the cosine rules:
cosa
k= cos
b
kcos
c
k+ sin
b
ksin
c
kcosα,
a2 = b2 + c2 − 2bc cosα
cosha
k= cosh
b
kcosh
c
k− sinh
b
ksinh
c
kcosα
Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 34 / 48
![Page 35: Intrinsic simplices on spaces of nearly constant curvature › slides › MWintraecken.pdf · Here: Model simplices using space of constant curvature I Map to space of constant curvature](https://reader033.vdocuments.us/reader033/viewer/2022052722/5f0d18db7e708231d438a9ea/html5/thumbnails/35.jpg)
Step 4
Gram matrix for n directions (elliptic case, hyperbolic similar)(1
Λmidsin(
√ΛmiddM (x, vi)) sin(
√ΛmiddM (x, vj)) cos θij,M
)i,l 6=j
is close to(1
Λmidsin(√
Λmid dH(Λmid)(x(vr), vi(vr)))
sin(√
ΛmiddH(Λmid)(x(vr), vl(vr)))
cos θil,H(Λmid)
)i,l 6=j
with x(w) = expH(Λmid) ◦ exp−1w x
Determinants:
Friedlands result on stability of determinants
determinants zero iff n tangent vectors linearly independent
determinants gives a new quality measure
Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 35 / 48
![Page 36: Intrinsic simplices on spaces of nearly constant curvature › slides › MWintraecken.pdf · Here: Model simplices using space of constant curvature I Map to space of constant curvature](https://reader033.vdocuments.us/reader033/viewer/2022052722/5f0d18db7e708231d438a9ea/html5/thumbnails/36.jpg)
Step 5
(before) linear independence implies diffeomorphism
(here) If quality is large enough then linear independent
Gives non-degeneracy criteria
Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 36 / 48
![Page 37: Intrinsic simplices on spaces of nearly constant curvature › slides › MWintraecken.pdf · Here: Model simplices using space of constant curvature I Map to space of constant curvature](https://reader033.vdocuments.us/reader033/viewer/2022052722/5f0d18db7e708231d438a9ea/html5/thumbnails/37.jpg)
Quality measures for simplices inspaces of constant curvature
Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 37 / 48
![Page 38: Intrinsic simplices on spaces of nearly constant curvature › slides › MWintraecken.pdf · Here: Model simplices using space of constant curvature I Map to space of constant curvature](https://reader033.vdocuments.us/reader033/viewer/2022052722/5f0d18db7e708231d438a9ea/html5/thumbnails/38.jpg)
Barycentre and division of volume
v1
v3
v2barycentre
The barycentre is the point where the maximum subvolume is minimized:
QRn(σ) = miny∈Rn
maxj
{det ((y − vi) · (y − vl))i,l 6=j
}=
(n · volume(σ)
n+ 1
)2
Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 38 / 48
![Page 39: Intrinsic simplices on spaces of nearly constant curvature › slides › MWintraecken.pdf · Here: Model simplices using space of constant curvature I Map to space of constant curvature](https://reader033.vdocuments.us/reader033/viewer/2022052722/5f0d18db7e708231d438a9ea/html5/thumbnails/39.jpg)
Quality for positive curvature spaces
Replacement for inner product
1
Λsin√
ΛdM (x, vi) sin√
ΛdM (x, vj) cos θij,M
New quality measure
QH(Λ)(σH(Λ)(vr))
= miny∈H(Λ)
maxj
{det
(1
Λsin(√
Λ dH(Λ)(y, vi(vr)))·
sin(√
ΛdH(Λ)(y, vl(vr)))
cos θil
)i,l 6=j
},
Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 39 / 48
![Page 40: Intrinsic simplices on spaces of nearly constant curvature › slides › MWintraecken.pdf · Here: Model simplices using space of constant curvature I Map to space of constant curvature](https://reader033.vdocuments.us/reader033/viewer/2022052722/5f0d18db7e708231d438a9ea/html5/thumbnails/40.jpg)
Geometric interpretation for positive curvature
0
x
vi
1√Λ
1√Λ
sin(√
Λd(x, vi))
d(x, vi)
For standard embedded sphere
1
Λsin(√
Λd(x, vi)) sin(√
Λd(x, vj))
cos θij,M
is inner product after projection ontothe tangent space.
QH(Λ) is of the form volume squaredin tangent space via projection.
Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 40 / 48
![Page 41: Intrinsic simplices on spaces of nearly constant curvature › slides › MWintraecken.pdf · Here: Model simplices using space of constant curvature I Map to space of constant curvature](https://reader033.vdocuments.us/reader033/viewer/2022052722/5f0d18db7e708231d438a9ea/html5/thumbnails/41.jpg)
Quality for spaces of negative curvature
QH(Λ) = minx∈N
maxj
{det
(1
|Λ|sinh
(√|Λ|dHn(ΛH
mid)(x, vi(vr)))
sinh(√|Λ|dHn(Λ)(x, vl(vr))
)cos θil
)i,l 6=j
}
Similar geometric interpretation using the hyperboloid model in Minkowskispace.
Both inner products have the right Euclidean limit
limΛ→0
1
Λsin(√
Λb) sin(√
Λc) cos θ = bc cos θ
limΛ→0
1
Λsinh(
√Λb) sinh(
√Λc) cos θ = bc cos θ
Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 41 / 48
![Page 42: Intrinsic simplices on spaces of nearly constant curvature › slides › MWintraecken.pdf · Here: Model simplices using space of constant curvature I Map to space of constant curvature](https://reader033.vdocuments.us/reader033/viewer/2022052722/5f0d18db7e708231d438a9ea/html5/thumbnails/42.jpg)
My problems: 1. Quality measures on space form withpositive curvature
Is our quality measure the natural one?Easier ones can be imagined, but must satisfy:
Small triangles with large volume compared to edge length are good
Large triangles with vertices near the equator are bad, even if thevertices are evenly spaced
To put it differently, quality of an equilateral triangle should decrease withsize.
Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 42 / 48
![Page 43: Intrinsic simplices on spaces of nearly constant curvature › slides › MWintraecken.pdf · Here: Model simplices using space of constant curvature I Map to space of constant curvature](https://reader033.vdocuments.us/reader033/viewer/2022052722/5f0d18db7e708231d438a9ea/html5/thumbnails/43.jpg)
My problems: 2. Quality measures on space form withnegative curvature
Is our quality measure the natural one?
We use the Minkowski model, is this the most natural way to look atthings?Expressions bounds (as we shall see) are complicated so maybe not.
Is there a natural scaling requirement? In the elliptic case the qualitydecreases with size, in Euclidean it stays the same, does it increase ordecrease here?
To put it differently, should quality of an equilateral triangle increase ordecrease with size? We do not know.
Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 43 / 48
![Page 44: Intrinsic simplices on spaces of nearly constant curvature › slides › MWintraecken.pdf · Here: Model simplices using space of constant curvature I Map to space of constant curvature](https://reader033.vdocuments.us/reader033/viewer/2022052722/5f0d18db7e708231d438a9ea/html5/thumbnails/44.jpg)
Results
Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 44 / 48
![Page 45: Intrinsic simplices on spaces of nearly constant curvature › slides › MWintraecken.pdf · Here: Model simplices using space of constant curvature I Map to space of constant curvature](https://reader033.vdocuments.us/reader033/viewer/2022052722/5f0d18db7e708231d438a9ea/html5/thumbnails/45.jpg)
Non-degeneracy result
Theorem (Non-degeneracy criteria for positive curvature)
Let M manifold with 0 < Λ− ≤ K ≤ Λ+. v0, . . . , vn vertices on M withina geodesic ball of radius 1
2D̃ with centre vr, where D̃ ≤ 1/(2√
Λ+).Simplex is non-degenerate if
QH(Λmid)(σH(Λmid)(vr))
(2D̃)2n≥ n |Λ− − Λ+| D̃2
σH(Λmid)(vr) the simplex on H(Λmid) with vertices vi(vr) defined by
vi(vr) = expH(Λmid) ◦ exp−1vr,M
(vi) and
Λmid =1
2(Λ− + Λ+).
Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 45 / 48
![Page 46: Intrinsic simplices on spaces of nearly constant curvature › slides › MWintraecken.pdf · Here: Model simplices using space of constant curvature I Map to space of constant curvature](https://reader033.vdocuments.us/reader033/viewer/2022052722/5f0d18db7e708231d438a9ea/html5/thumbnails/46.jpg)
Theorem (Non-degeneracy criteria for negative curvature)
Let M be a manifold with Λ− ≤ K ≤ Λ+ < 0. Given vertices v0, . . . , vnon M within geodesic ball of radius 1
2D̃ with centre vr. Simplex isnon-degenerate if
|Λmid|nQH(Λmid)(σH(Λmid)(vr)) > n(sinh√|Λ−|D̃)2(n−1)
·
(2 + 2 cosh
(√|Λ−|D̃
)+ |Λ−|2 cosh2
(√|Λ−|D̃
) 11D̃4
4!
)
·∣∣∣|Λ−| cosh2
√|Λ−|D̃ − |Λ+| cosh2
√|Λ+|D̃
∣∣∣ |Λmid|11D̃4
2 · 4!
σH(Λmid)(vr) simplex on H(Λmid), vertices expH(Λmid) ◦ exp−1vr,M
(vi) and∣∣∣|Λmid| cosh2√
ΛmidD̃ − |Λ−| cosh2√|Λ−|D̃
∣∣∣=∣∣∣|Λmid| cosh2
√ΛmidD̃ − |Λ+| cosh2
√|Λ+|D̃
∣∣∣ .Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 46 / 48
![Page 47: Intrinsic simplices on spaces of nearly constant curvature › slides › MWintraecken.pdf · Here: Model simplices using space of constant curvature I Map to space of constant curvature](https://reader033.vdocuments.us/reader033/viewer/2022052722/5f0d18db7e708231d438a9ea/html5/thumbnails/47.jpg)
Comment on the quality bound
The quality bound looks awful∣∣∣|Λ−| cosh2√|Λ−|D̃ − |Λ+| cosh2
√|Λ+|D̃
∣∣∣but it is roughly speaking proportional to |Λ− − Λ+|. Implies that thebounds on the quality go to zero as |Λ− − Λ+| tends to zero.
Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 47 / 48
![Page 48: Intrinsic simplices on spaces of nearly constant curvature › slides › MWintraecken.pdf · Here: Model simplices using space of constant curvature I Map to space of constant curvature](https://reader033.vdocuments.us/reader033/viewer/2022052722/5f0d18db7e708231d438a9ea/html5/thumbnails/48.jpg)
Questions?
Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 48 / 48