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
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
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
Outline Intrinsic simplices and non-degeneracy 1 Topogonov comparison theorem 2 Non-degeneracy criteria for simplices modeled on simplices in Euclidean 3 space Non-degeneracy criteria for simplices modeled on simplices in spaces of 4 constant curvature Quality measures for simplices in spaces of constant curvature 5 Questions about quality 6 Results 7 Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 4 / 48
Intrinsic simplices and non-degeneracy Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 5 / 48
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
Centres of mass in Euclidean space Weighted average of points � µ i v i assume � µ i = 1 . Generalizes to � p d µ ( p ) is where the minimum of P R n ( x ) = 1 � � x − p � 2 d µ ( p ) , 2 is attained Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 7 / 48
Riemannian centres of mass Centre of mass E λ ( x ) = 1 � λ i d M ( x, v i ) 2 2 i barycentric coordinates: λ i ≥ 0 , � λ i = 1 B σ j : ∆ j → M λ �→ argmin E λ ( x ) x ∈ B ρ ∆ j the standard Euclidean j -simplex, σ M image Point where minimum E λ ( x ) is attained is characterized by � λ i exp − 1 x ( v i ) = 0 . (generalization of � λ i ( v i − x ) = 0 ) Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 8 / 48
Smooth map Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 9 / 48
Exponential map Notation v i ( x ) = exp − 1 x ( v i ) , σ ( x ) = { v 0 ( x ) , . . . , v j ( x ) } ⊂ T x M , injectivity radius ι M v i ( x ) = exp − 1 x ( v i ) is smooth Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 10 / 48
Non-degeneracy; a consequence of linear independence Definition A Riemannian simplex σ M is non-degenerate if σ M is diffeomorphic to the standard simplex ∆ n Lemma (Consequences of linear independence) If tangents to geodesics connecting any n (in neighbourhood) to some subset v 0 , . . . , v j − 1 , v j +1 , . . . v n (may depend on x ) are linearly independent then the map ∆ n → σ M is bijective The inverse of ∆ n → σ M is smooth Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 11 / 48
∆ n → σ M is bijective Proof by contradiction � ˜ � ˜ � λ i exp − 1 λ i exp − 1 x ( v i ) = x ( v i ) = λ i v i ( x ) = 0 with � λ i = � ˜ λ i = 1 . Because v 0 ( x ) , . . . , ˆ v j ( x ) , . . . , v n ( x ) linearly independent λ j � = 0 , ˜ λ j � = 0 . So λ 0 v 0 ( x ) + . . . + λ j − 1 v j − 1 ( x ) + λ j +1 v j +1 ( x ) + . . . + λ n v n ( x ) = λ j λ j λ j λ j ˜ ˜ ˜ ˜ λ 0 λ j − 1 λ j +1 λ n v 0 ( x ) + . . . + v j − 1 ( x ) + v j +1 ( x ) + . . . + v n ( x ) . ˜ ˜ ˜ ˜ λ j λ j λ j λ j Contradiction Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 12 / 48
Inverse of ∆ n → σ M is smooth Proof � λ i exp − 1 � x ( v i ) = λ i v i ( x ) = 0 , λ j � = 0 because if λ j = 0 then � λ i v i ( x ) = 0 , i � = j contradicting linear independence. So � λ i � ( v 0 ( x ) , . . . , v j − 1 ( x ) , v j +1 ( x ) , . . . v n ( x )) − 1 v j ( x ) = . λ j Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 13 / 48
Linear independence 2D In two dimensions linear indepen- dence easy (Rustamov 2010) If exp − 1 x ( v 0 ) = v 0 ( x ) , v 1 ( x ) , v 2 ( x ) do not span T x M then they are co-linear. Equivalent to v 0 , v 1 and v 2 lying on geodesic. Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 14 / 48
Friedland’s bounds Stability of determinants | det( A + E ) − det( A ) | ≤ n max {� A � p , � A + E � p } n − 1 � E � p with A and E , n × n -matrices � · � p p -norm on matrices 1 ≤ p ≤ ∞ : | Ax | p � A � p = max , | x | p x ∈ R n p -norm on vectors: | w | p = (( w 1 ) p + . . . + ( w n ) p ) 1 /p Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 15 / 48
Topogonov comparison theorem Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 16 / 48
Triangles and cosine rules Cosine rules: cos a k = cos b k cos c k + sin b k sin c k cos α, C in a space H (1 /k 2 ) of sectional curvature γ 1 /k 2 a a 2 = b 2 + c 2 − 2 bc cos α b β B in Euclidean space R n c α cosh a k = cosh b k cosh c k − sinh b k sinh c k cos α, A in a space H ( − 1 /k 2 ) of sectional curvature − 1 /k 2 Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 17 / 48
Geodesic triangles Geodesic triangle T : Three minimizing geodesics connecting three points on a arbitrary manifold (no interior) C a b B c A Alexandrov triangle: A geodesic triangle with same edge lengths on space of constant curvature H (Λ ∗ ) . Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 18 / 48
Hinges Hinge: Two minimizing geodesics connecting three points and enclosed angle on a arbitrary manifold C b B c α A 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
Topogonov comparison theorem Manifold M , sectional curvatures Λ − ≤ K ≤ Λ + . Given: Geodesic triangle T on M then exist T Λ − , T Λ + on H (Λ − ) , H (Λ + ) and α Λ − ≤ α ≤ α Λ + , Given: Hinge on M then exist Rauch hinges on H (Λ − ) , H (Λ + ) and the length of the closing geodesics satisfy c Λ − ≥ c ≥ c Λ + . Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 20 / 48
Non-degeneracy criteria for simplices modeled on simplices in Euclidean space Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 21 / 48
Setting Setting: Manifold M with bounded curvature | K | ≤ Λ . Points { v 0 , . . . , v n } in a small ball in M : vertices. Choose vertex v r . σ E ( v r ) convex hull of the exp − 1 v r ( v i ) = v i ( v r ) . Goal: Give conditions on σ E ( v r ) that imply non-degeneracy Dyer, Vegter, Wintraecken (JBI) Intrinsic simplices August 2015 22 / 48
Step 1 We know all the geodesics emanating from v r . 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
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
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
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
Step 4 Gram matrix (in n directions) ( � exp − 1 x v i , exp − 1 x v l � ) i,l � = j = ( � v i ( x ) , v l ( x ) � ) i,l � = j is close to ( � v i ( v r ) − x ( v r ) , v l ( v r ) − x ( v r ) � ) 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
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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