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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

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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

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Intrinsic simplices on Riemannian manifolds

Motivation: Generic triangulation criteria intrinsic setting explicit quality requirements Arbitrary dimension

Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 2 / 48

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Modeling simplices: quality

Non-degeneracy by quality requirements depending on curvature manifold SOCG: Model simplices on Euclidean space

◮ To tangent space via exponential map ◮ Almost flat

Here: Model simplices using space of constant curvature

◮ Map to space of constant curvature ◮ Curvature is (locally) nearly constant ◮ Towards adaptive sampling

New quality measures for spaces of constant curvature

Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 3 / 48

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Outline

1

Intrinsic simplices and non-degeneracy

2

Topogonov comparison theorem

3

Non-degeneracy criteria for simplices modeled on simplices in Euclidean space

4

Non-degeneracy criteria for simplices modeled on simplices in spaces of constant 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

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Intrinsic simplices and non-degeneracy

Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 5 / 48

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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 to fill up a triangle, or more general simplex, in a generic Riemannian manifold. But this is not true

  • the problem is solved by the notion of center of

mass, modeled on Euclidean geometry.’

Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 6 / 48

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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 − p2dµ(p),

is attained

Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 7 / 48

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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 → argmin

x∈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

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Smooth map

Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 9 / 48

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Exponential map

Notation

vi(x) = exp−1

x (vi),

σ(x) = {v0(x), . . . , vj(x)} ⊂ TxM, injectivity radius ιM vi(x) = exp−1

x (vi) is smooth

Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 10 / 48

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Non-degeneracy; a consequence of linear independence

Definition

A Riemannian simplex σM is non-degenerate if σM is diffeomorphic to the standard simplex ∆n

Lemma (Consequences of linear independence)

If tangents to geodesics connecting any n (in neighbourhood) to some subset v0, . . . , vj−1, vj+1, . . . vn (may depend on x) are linearly independent then the map ∆n → σM is bijective The inverse of ∆n → σM is smooth

Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 11 / 48

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∆n → σM is bijective

Proof by contradiction

  • λi exp−1

x (vi) =

˜ λi exp−1

x (vi) =

˜ λivi(x) = 0 with λi = ˜ λi = 1. Because v0(x), . . . , ˆ vj(x), . . . , vn(x) linearly independent λj = 0, ˜ λj = 0. So λ0 λj v0(x) + . . . + λj−1 λj vj−1(x) + λj+1 λj vj+1(x) + . . . + λn λj vn(x) = ˜ λ0 ˜ λj v0(x) + . . . + ˜ λj−1 ˜ λj vj−1(x) + ˜ λj+1 ˜ λj vj+1(x) + . . . + ˜ λn ˜ λj vn(x). Contradiction

Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 12 / 48

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Inverse of ∆n → σM is smooth

Proof

  • λi exp−1

x (vi) =

  • λivi(x) = 0,

λj = 0 because if λj = 0 then

  • i=j

λivi(x) = 0, contradicting linear independence. So (v0(x), . . . , vj−1(x), vj+1(x), . . . vn(x))−1vj(x) = λi λj

  • .

Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 13 / 48

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Linear independence 2D

In two dimensions linear indepen- dence easy (Rustamov 2010) If exp−1

x (v0) = v0(x), v1(x),

v2(x) do not span TxM then they are co-linear. Equivalent to v0, v1 and v2 lying on geodesic.

Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 14 / 48

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Friedland’s bounds

Stability of determinants | det(A + E) − det(A)| ≤ n max{Ap, A + Ep}n−1Ep with A and E, n × n-matrices · p p-norm on matrices 1 ≤ p ≤ ∞: Ap = max

x∈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

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Topogonov comparison theorem

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Triangles and cosine rules

A C B b c a α β γ Cosine rules: cos a k = cos b k cos c k + sin b k sin c k cos α, in a space H(1/k2) of sectional curvature 1/k2 a2 = b2 + c2 − 2bc cos α in Euclidean space Rn cosh a k = cosh b k cosh c k − sinh b k sinh c k cos α, in a space H(−1/k2) of sectional curvature −1/k2

Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 17 / 48

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Geodesic triangles

Geodesic triangle T: Three minimizing geodesics connecting three points

  • n a arbitrary manifold (no interior)

A C B b c a Alexandrov triangle: A geodesic triangle with same edge lengths on space

  • f constant curvature H(Λ∗).

Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 18 / 48

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Hinges

Hinge: Two minimizing geodesics connecting three points and enclosed angle on a arbitrary manifold A C B b c α Rauch hinge: A hinge with the same edge lengths and enclosed angle on a space of constant curvature H(Λ∗).

Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 19 / 48

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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 the length of the closing geodesics satisfy cΛ− ≥ c ≥ cΛ+.

Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 20 / 48

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Non-degeneracy criteria for simplices modeled on simplices in Euclidean space

Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 21 / 48

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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−1

vr (vi) = vi(vr).

Goal: Give conditions on σE(vr) that imply non-degeneracy

Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 22 / 48

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Step 1

We know all the geodesics emanating from vr. Topogonov for Hinges bounds the length of blue geodesics in the middle by those on the side.

Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 23 / 48

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Step 2

In small neighbourhood the lengths of geodesics on the left and right are close 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

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Step 2

In small neighbourhood the lengths of geodesics on the left and right are close 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

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Step 3

Use the Toponogov comparison theorem for geodesic triangles to conclude that the angles (and inner product) are close to those in the tangent space

Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 26 / 48

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Step 4

Gram matrix (in n directions) (exp−1

x vi, exp−1 x vl)i,l=j = (vi(x), vl(x))i,l=j

is close to (vi(vr) − x(vr), vl(vr) − x(vr))i,l=j Determinants: Friedlands result on stability of determinants gives that determinants are 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

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Step 5

(before) linear independence implies diffeomorphism (here) If normalized volume simplex is large enough then linear independent Gives non-degeneracy criteria

Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 28 / 48

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Non-degeneracy criteria for simplices modeled on simplices in spaces of constant curvature

Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 29 / 48

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Setting

Setting: Manifold M with bounded curvature Λ− ≤ K ≤ Λ+, with 0 < Λ− or Λ+ < 0. Points {v0, . . . , vn} in a small ball in M: vertices. Choose vertex vr. Model on σH(Λmid)(vr) convex hull of expH(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

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Step 1

We know all the geodesics emanating from vr. Topogonov for Hinges bounds the length of blue geodesics in the middle by those on the side.

Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 31 / 48

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Step 2

In small neighbourhood the lengths of geodesics on the left and right are close 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

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Step 3

Use the Toponogov comparison theorem for geodesic triangles to conclude that the angles (and inner product) are close to those on H(Λmid)

Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 33 / 48

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Step 3 (continued)

These inner products are: 1 Λmid sin (

  • ΛmiddM(x, vi)) sin (
  • ΛmiddM(x, vj)) cos θij,M

(elliptic) 1 Λmid sinh (

  • ΛmiddM(x, vi)) sinh (
  • ΛmiddM(x, vj)) cos θij,M

(hyperbolic) from the cosine rules: cos a k = cos b k cos c k + sin b k sin c k cos α, a2 = b2 + c2 − 2bc cos α cosh a k = cosh b k cosh c k − sinh b k sinh c k cos α

Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 34 / 48

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Step 4

Gram matrix for n directions (elliptic case, hyperbolic similar)

  • 1

Λmid sin(

  • ΛmiddM(x, vi)) sin(
  • ΛmiddM(x, vj)) cos θij,M
  • i,l=j

is close to

  • 1

Λmid sin

  • Λmid dH(Λmid)(x(vr), vi(vr))
  • sin
  • ΛmiddH(Λmid)(x(vr), vl(vr))
  • cos θil,H(Λmid)
  • i,l=j

with x(w) = expH(Λmid) ◦ exp−1

w 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

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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

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Quality measures for simplices in spaces of constant curvature

Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 37 / 48

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Barycentre and division of volume

v1 v3 v2 barycentre The barycentre is the point where the maximum subvolume is minimized: QRn(σ) = min

y∈Rn max j

  • det ((y − vi) · (y − vl))i,l=j
  • =

n · volume(σ) n + 1 2

Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 38 / 48

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Quality for positive curvature spaces

Replacement for inner product 1 Λ sin √ ΛdM(x, vi) sin √ ΛdM(x, vj) cos θij,M New quality measure QH(Λ)(σH(Λ)(vr)) = min

y∈H(Λ) max j

  • det

1 Λ sin √ Λ dH(Λ)(y, vi(vr))

  • ·

sin √ ΛdH(Λ)(y, vl(vr))

  • cos θil
  • i,l=j
  • ,

Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 39 / 48

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Geometric interpretation for positive curvature

x vi

1 √ Λ 1 √ Λ sin(

√ Λd(x, vi)) d(x, vi) For standard embedded sphere 1 Λ sin( √ Λd(x, vi)) sin( √ Λd(x, vj)) cos θij,M is inner product after projection onto the tangent space. QH(Λ) is of the form volume squared in tangent space via projection.

Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 40 / 48

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Quality for spaces of negative curvature

QH(Λ) = min

x∈N max j

  • det

1 |Λ| sinh

  • |Λ|dHn(ΛH

mid)(x, vi(vr))

  • sinh
  • |Λ|dHn(Λ)(x, vl(vr))
  • cos θil
  • i,l=j
  • Similar geometric interpretation using the hyperboloid model in Minkowski

space. Both inner products have the right Euclidean limit lim

Λ→0

1 Λ sin( √ Λb) sin( √ Λc) cos θ = bc cos θ lim

Λ→0

1 Λ sinh( √ Λb) sinh( √ Λc) cos θ = bc cos θ

Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 41 / 48

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My problems: 1. Quality measures on space form with positive 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 the vertices are evenly spaced To put it differently, quality of an equilateral triangle should decrease with size.

Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 42 / 48

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My problems: 2. Quality measures on space form with negative curvature

Is our quality measure the natural one? We use the Minkowski model, is this the most natural way to look at things? Expressions bounds (as we shall see) are complicated so maybe not. Is there a natural scaling requirement? In the elliptic case the quality decreases with size, in Euclidean it stays the same, does it increase or decrease here? To put it differently, should quality of an equilateral triangle increase or decrease with size? We do not know.

Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 43 / 48

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Results

Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 44 / 48

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Non-degeneracy result

Theorem (Non-degeneracy criteria for positive curvature)

Let M manifold with 0 < Λ− ≤ K ≤ Λ+. v0, . . . , vn vertices on M within a geodesic ball of radius 1

2 ˜

D with centre vr, where ˜ D ≤ 1/(2√Λ+). Simplex is non-degenerate if QH(Λmid)(σH(Λmid)(vr)) (2 ˜ D)2n ≥ n |Λ− − Λ+| ˜ D2 σH(Λmid)(vr) the simplex on H(Λmid) with vertices vi(vr) defined by vi(vr) = expH(Λmid) ◦ exp−1

vr,M(vi) and

Λmid = 1 2(Λ− + Λ+).

Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 45 / 48

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Theorem (Non-degeneracy criteria for negative curvature)

Let M be a manifold with Λ− ≤ K ≤ Λ+ < 0. Given vertices v0, . . . , vn

  • n M within geodesic ball of radius 1

2 ˜

D with centre vr. Simplex is non-degenerate if |Λmid|nQH(Λmid)(σH(Λmid)(vr)) > n(sinh

  • |Λ−| ˜

D)2(n−1) ·

  • 2 + 2 cosh
  • |Λ−| ˜

D

  • + |Λ−|2 cosh2

|Λ−| ˜ D 11 ˜ D4 4!

  • ·
  • |Λ−| cosh2

|Λ−| ˜ D − |Λ+| cosh2 |Λ+| ˜ D

  • |Λmid|11 ˜

D4 2 · 4! σH(Λmid)(vr) simplex on H(Λmid), vertices expH(Λmid) ◦ exp−1

vr,M(vi) and

  • |Λmid| cosh2

Λmid ˜ D − |Λ−| cosh2 |Λ−| ˜ D

  • =
  • |Λmid| cosh2

Λmid ˜ D − |Λ+| cosh2 |Λ+| ˜ D

  • .

Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 46 / 48

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Comment on the quality bound

The quality bound looks awful

  • |Λ−| cosh2

|Λ−| ˜ D − |Λ+| cosh2 |Λ+| ˜ D

  • but it is roughly speaking proportional to |Λ− − Λ+|. Implies that the

bounds on the quality go to zero as |Λ− − Λ+| tends to zero.

Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 47 / 48

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Questions?

Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 48 / 48