Simplicial Complexes: Theory and Implementation · Simplicial Complexes: Theory and Implementation...
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Simplicial Complexes:Theory and Implementation
Marek K. Misztal
Informatics and Mathematical Modelling, Technical University of [email protected]
DSC 2011 WorkshopKgs. Lyngby, 25th August 2011
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Affine independence
Definition (affine independce)
Let v1, . . . ,vp+1 be points in an n-dimensional Euclidean space En.We call them affinely dependent if
(∃µ1, . . . ,µp+1 ∈ R)p+1
∑i=1
µi = 1∧p+1
∑i=1
µivi = 0.
Otherwise, we call them affinely independent.
Examples:• three non-colinear points in E2 are affinely independent;• four non-coplanar points in E3 are affinely independent;
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Simplex
Definition (Euclidean simplex)
Having p + 1 affinely independent points v1,v2, . . . ,vp+1 ∈ En, anEuclidean simplex σ = 〈v1, . . . ,vp+1〉 is a set of points given by aformula:
v = α1v1 + . . .+ αp+1vp+1,
where αi ≥ 0, ∑i αi = 1 (σ is the convex hull of v1, . . . ,vp+1).
• σ is a closed set in En.• p is the dimension of σ (equivalently σ is an Euclidean
p-simplex).
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Simplex
Definition (Euclidean simplex)
Having p + 1 affinely independent points v1,v2, . . . ,vp+1 ∈ En, anEuclidean simplex σ = 〈v1, . . . ,vp+1〉 is a set of points given by aformula:
v = α1v1 + . . .+ αp+1vp+1,
where αi ≥ 0, ∑i αi = 1 (σ is the convex hull of v1, . . . ,vp+1).
• σ is a closed set in En.
• p is the dimension of σ (equivalently σ is an Euclideanp-simplex).
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Simplex
Definition (Euclidean simplex)
Having p + 1 affinely independent points v1,v2, . . . ,vp+1 ∈ En, anEuclidean simplex σ = 〈v1, . . . ,vp+1〉 is a set of points given by aformula:
v = α1v1 + . . .+ αp+1vp+1,
where αi ≥ 0, ∑i αi = 1 (σ is the convex hull of v1, . . . ,vp+1).
• σ is a closed set in En.• p is the dimension of σ (equivalently σ is an Euclidean
p-simplex).
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Simplices
We call a 0-simplex a vertex, a 1-simplex an edge, a 2-simplex a faceand a 3-simplex a tetrahedron.
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Faces
Definition (vertex, q-face)
We call each point vi a vertex of σ , and each simplex 〈vi1 , . . . ,viq+1〉(0≤ q ≤ p, 1≤ ik ≤ p + 1) a q-face of σ (or simply a face of σ , if noambiguity arises).
• We also call all the (p−1)-faces of a p-simplex σp its boundaryfaces.
• The faces of σ that are not equal to σ itself are called its properfaces.
• The union of all the boundary faces of a simplex σ is called theboundary of σ .
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Faces
Definition (vertex, q-face)
We call each point vi a vertex of σ , and each simplex 〈vi1 , . . . ,viq+1〉(0≤ q ≤ p, 1≤ ik ≤ p + 1) a q-face of σ (or simply a face of σ , if noambiguity arises).
• We also call all the (p−1)-faces of a p-simplex σp its boundaryfaces.
• The faces of σ that are not equal to σ itself are called its properfaces.
• The union of all the boundary faces of a simplex σ is called theboundary of σ .
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Faces
Definition (vertex, q-face)
We call each point vi a vertex of σ , and each simplex 〈vi1 , . . . ,viq+1〉(0≤ q ≤ p, 1≤ ik ≤ p + 1) a q-face of σ (or simply a face of σ , if noambiguity arises).
• We also call all the (p−1)-faces of a p-simplex σp its boundaryfaces.
• The faces of σ that are not equal to σ itself are called its properfaces.
• The union of all the boundary faces of a simplex σ is called theboundary of σ .
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Faces
Definition (vertex, q-face)
We call each point vi a vertex of σ , and each simplex 〈vi1 , . . . ,viq+1〉(0≤ q ≤ p, 1≤ ik ≤ p + 1) a q-face of σ (or simply a face of σ , if noambiguity arises).
• We also call all the (p−1)-faces of a p-simplex σp its boundaryfaces.
• The faces of σ that are not equal to σ itself are called its properfaces.
• The union of all the boundary faces of a simplex σ is called theboundary of σ .
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Simplex sets
• For arbitrary, finite set of simplices Σ we define its dimension, asthe maximum dimension of the simplices in Σ:
dim(Σ) = max{dim(σ) : σ ∈ Σ}.
• We also define a k -subset of Σ as a set of all k -simplices in Σ:
filterk (Σ) = {σi ∈ Σ : dim(σi ) = k}.
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Simplex sets
• For arbitrary, finite set of simplices Σ we define its dimension, asthe maximum dimension of the simplices in Σ:
dim(Σ) = max{dim(σ) : σ ∈ Σ}.
• We also define a k -subset of Σ as a set of all k -simplices in Σ:
filterk (Σ) = {σi ∈ Σ : dim(σi ) = k}.
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Euclidean simplicial complex
Definition (Euclidean simplicial complex)
A finite set Σ of Euclidean simplices forms a (finite) Euclideansimplicial complex if the following two conditions hold:
1. Σ is closed: for each simplex σ ∈ Σ, all faces of σ are also in Σ.2. The intersection σi ∩σj of any two simplices σi ,σj ∈ Σ is either
empty or is a face of both σi and σj .
• Any subset K′ ⊂ K that is itself a simplicial complex is called asubcomplex of K.
• In particular, for any nonnegative integer k , the subset K(k) ⊂ Kconsisting of all simplices of dimension less than or equal to k isa subcomplex, called the k -skeleton of K.
• The 0-skeleton of K is called a vertex set of K and denoted V (K).
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Euclidean simplicial complex
Definition (Euclidean simplicial complex)
A finite set Σ of Euclidean simplices forms a (finite) Euclideansimplicial complex if the following two conditions hold:
1. Σ is closed: for each simplex σ ∈ Σ, all faces of σ are also in Σ.2. The intersection σi ∩σj of any two simplices σi ,σj ∈ Σ is either
empty or is a face of both σi and σj .
• Any subset K′ ⊂ K that is itself a simplicial complex is called asubcomplex of K.
• In particular, for any nonnegative integer k , the subset K(k) ⊂ Kconsisting of all simplices of dimension less than or equal to k isa subcomplex, called the k -skeleton of K.
• The 0-skeleton of K is called a vertex set of K and denoted V (K).
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Euclidean simplicial complex
Definition (Euclidean simplicial complex)
A finite set Σ of Euclidean simplices forms a (finite) Euclideansimplicial complex if the following two conditions hold:
1. Σ is closed: for each simplex σ ∈ Σ, all faces of σ are also in Σ.2. The intersection σi ∩σj of any two simplices σi ,σj ∈ Σ is either
empty or is a face of both σi and σj .
• Any subset K′ ⊂ K that is itself a simplicial complex is called asubcomplex of K.
• In particular, for any nonnegative integer k , the subset K(k) ⊂ Kconsisting of all simplices of dimension less than or equal to k isa subcomplex, called the k -skeleton of K.
• The 0-skeleton of K is called a vertex set of K and denoted V (K).
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Euclidean simplicial complex
Definition (Euclidean simplicial complex)
A finite set Σ of Euclidean simplices forms a (finite) Euclideansimplicial complex if the following two conditions hold:
1. Σ is closed: for each simplex σ ∈ Σ, all faces of σ are also in Σ.2. The intersection σi ∩σj of any two simplices σi ,σj ∈ Σ is either
empty or is a face of both σi and σj .
• Any subset K′ ⊂ K that is itself a simplicial complex is called asubcomplex of K.
• In particular, for any nonnegative integer k , the subset K(k) ⊂ Kconsisting of all simplices of dimension less than or equal to k isa subcomplex, called the k -skeleton of K.
• The 0-skeleton of K is called a vertex set of K and denoted V (K).
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Euclidean simplicial complex
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Topological relationsFor a p-simplex σp in a simplicial complex K we define the followingtopological relations:
• for p > q, the boundary relation Bp,q(σp) is the set of all q-facesof σp:
Bp,q(σp) = filterq{σ ∈ K : vert(σ)⊂ vert(σ
p)},
• for p < q, the coboundary relation Cp,q(σp) is the set of allq-simplices that have σp as a face:
Cp,q(σp) = filterq{σ ∈ K : vert(σ
p)⊂ vert(σ)},
• for p > 0, the adjacency relation Ap(σp) is the set of allp-simplices, which are (p−1)-adjacent to σp (which meansthose simplices, that share a (p−1)-face with σp):
Ap(σp) = filterp{σ ∈ K : |vert(σ
p)∩vert(σ)|= p},
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Topological relationsFor a p-simplex σp in a simplicial complex K we define the followingtopological relations:
• for p > q, the boundary relation Bp,q(σp) is the set of all q-facesof σp:
Bp,q(σp) = filterq{σ ∈ K : vert(σ)⊂ vert(σ
p)},
• for p < q, the coboundary relation Cp,q(σp) is the set of allq-simplices that have σp as a face:
Cp,q(σp) = filterq{σ ∈ K : vert(σ
p)⊂ vert(σ)},
• for p > 0, the adjacency relation Ap(σp) is the set of allp-simplices, which are (p−1)-adjacent to σp (which meansthose simplices, that share a (p−1)-face with σp):
Ap(σp) = filterp{σ ∈ K : |vert(σ
p)∩vert(σ)|= p},
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Topological relationsFor a p-simplex σp in a simplicial complex K we define the followingtopological relations:
• for p > q, the boundary relation Bp,q(σp) is the set of all q-facesof σp:
Bp,q(σp) = filterq{σ ∈ K : vert(σ)⊂ vert(σ
p)},
• for p < q, the coboundary relation Cp,q(σp) is the set of allq-simplices that have σp as a face:
Cp,q(σp) = filterq{σ ∈ K : vert(σ
p)⊂ vert(σ)},
• for p > 0, the adjacency relation Ap(σp) is the set of allp-simplices, which are (p−1)-adjacent to σp (which meansthose simplices, that share a (p−1)-face with σp):
Ap(σp) = filterp{σ ∈ K : |vert(σ
p)∩vert(σ)|= p},
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Topological relationsFor a p-simplex σp in a simplicial complex K we define the followingtopological relations:
• for p > q, the boundary relation Bp,q(σp) is the set of all q-facesof σp:
Bp,q(σp) = filterq{σ ∈ K : vert(σ)⊂ vert(σ
p)},
• for p < q, the coboundary relation Cp,q(σp) is the set of allq-simplices that have σp as a face:
Cp,q(σp) = filterq{σ ∈ K : vert(σ
p)⊂ vert(σ)},
• for p > 0, the adjacency relation Ap(σp) is the set of allp-simplices, which are (p−1)-adjacent to σp (which meansthose simplices, that share a (p−1)-face with σp):
Ap(σp) = filterp{σ ∈ K : |vert(σ
p)∩vert(σ)|= p},
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Star
Definition (star)
We define the star of a simplex σ as a set of all the simplices in K,which have σ as a face:
st(σp) = {σ ∈ K : vert(σ
p)⊂ vert(σ)}=n⋃
q=p+1
Cp,q(σp).
For the sake of convenience, we also define a star of an arbitrarysubset Σ of K, as the union of the stars of all simplices in Σ:
st(Σ) =⋃
σi∈Σ
st(σi ).
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Star
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Closure
Definition (closure)
We define the closure of a simplex σp ∈ K as a set
cl(σp) =
p⋃q=0
Bp,q(σp).
The closure of a simplex set Σ⊂ K is expressed as a set
cl(Σ) =⋃
σi∈Σ
cl(σi ).
Equivalently, we can define the closure of a simplex σ ∈ K (simplexset Σ⊂ K) as the smallest subcomplex of K containing σ
(including Σ).
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Closure
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Link
Definition (link)
The link of a simplex σ is defined as the set of all the the simplices inthe closure of the star of σ , which do not share a face with σ :
lk(σ) = cl(st(σ))− st(cl(σ)).
It can be proven that for every simplex σ ∈ K, lk(σ) is a subcomplex.
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Link
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Carrier
DefinitionThe carrier ‖K‖ of a simplicial complex K (also called the polyhedron‖K‖) is a subset of En defined by the union, as point sets, of all thesimplices in K.
DefinitionFor each point v ∈ ‖K‖ there exists exactly one simplex σ ∈ Kcontaining v in its relative interior. This simplex is denoted by supp(v)and called the support of the point v .
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Manifoldness
Definition (notion of manifoldness)
We say that a point v ∈ A⊂ En is p-manifold if there exists aneighbourhood U of v such that A∩U is homeomorphic to Rp orR(p−1)× (0,+∞). Otherwise we call v non-manifold.
We say that a simplex σ ∈ K is p-manifold, if every point of the relativeinterior of this simplex is p-manifold with regard to the carrier of K.E.g. obviously each n-simplex is n-manifold, each (n−1)-simplex isn-manifold if it is a face of at least one n-simplex, etc.
We also say that an n-dimensional simplicial complex K in En ismanifold, if each of its simplices is n-manifold.
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Manifoldness
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Orientations
We introduce the following equivalence relation in the set Pσ of allorderings (vi1 , . . . ,vip+1) of the vertices of σ = 〈v1, . . . ,vp+1〉:
(vi1 , . . . ,vip+1)∼ (vπ(i1), . . . ,vπ(ip+1))
iff π : {1,2, . . . ,p + 1} −→ {1,2, . . . ,p + 1} is an even permutationoperator.
We call each element of the quotient set Oσ = Pσ/∼ an orientation ofa simplex σ .
If p > 0, |Oσ |= 2, meaning that there are two possible orientations forany simplex defined on a set of p + 1 points from En.
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Oriented volume
We define an oriented volume of σ = [v1, . . . ,vp+1]
V (σ) = V (v1, . . . ,vp+1)
=1p!
det(v1−v2,v2−v3, . . . ,vp−vp+1,vp+1−v1).
It can be proven, that for an even permutation operator π
V (vπ(1), . . . ,vπ(p+1)) = V (v1, . . . ,vp+1),
and for an odd permutation operator π ′
V (vπ ′(1), . . . ,vπ ′(p+1)) =−V (v1, . . . ,vp+1),
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Natural and induced orientation
Definition (natural orientation)
The orientation of σ , for which V (σ) > 0 is called the naturalorientation.
Definition (induced orientation)
The p-simplex σp = [v1,v2, . . . ,vp+1] determines an orientation ofeach of its (p−1)-faces, called the induced orientation, by thefollowing rule: the induced orientation on the faceσ
p−1i = 〈v1, . . . ,vi−1,vi+1, . . . ,vp+1〉 is defined to be
(−1)i+1[v1, . . . ,vi−1,vi+1, . . . ,vp+1].
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Theory Meshes Implementation
Consistency
• Let K be an n-dimensional simplicial complex in which every(n−1)-simplex is a face of no more than two n-simplices.
• If σni ,σ
nj ∈ K are two n-simplices that share an (n−1)-face σn−1,
we say that orientations of σni and σn
j are consistent if theyinduce opposite orientations on σn−1.
• An orientation of K is a choice of orientation of each n-simplex insuch a way that any two simplices that interesect in an(n−1)-face are consistently oriented.
• If a complex K admits an orientation, it is said to be orientable.
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Theory Meshes Implementation
Consistency
• Let K be an n-dimensional simplicial complex in which every(n−1)-simplex is a face of no more than two n-simplices.
• If σni ,σ
nj ∈ K are two n-simplices that share an (n−1)-face σn−1,
we say that orientations of σni and σn
j are consistent if theyinduce opposite orientations on σn−1.
• An orientation of K is a choice of orientation of each n-simplex insuch a way that any two simplices that interesect in an(n−1)-face are consistently oriented.
• If a complex K admits an orientation, it is said to be orientable.
M. K. Misztal Simplicial Complexes: Theory and Implementation
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Theory Meshes Implementation
Consistency
• Let K be an n-dimensional simplicial complex in which every(n−1)-simplex is a face of no more than two n-simplices.
• If σni ,σ
nj ∈ K are two n-simplices that share an (n−1)-face σn−1,
we say that orientations of σni and σn
j are consistent if theyinduce opposite orientations on σn−1.
• An orientation of K is a choice of orientation of each n-simplex insuch a way that any two simplices that interesect in an(n−1)-face are consistently oriented.
• If a complex K admits an orientation, it is said to be orientable.
M. K. Misztal Simplicial Complexes: Theory and Implementation
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Theory Meshes Implementation
Consistency
• Let K be an n-dimensional simplicial complex in which every(n−1)-simplex is a face of no more than two n-simplices.
• If σni ,σ
nj ∈ K are two n-simplices that share an (n−1)-face σn−1,
we say that orientations of σni and σn
j are consistent if theyinduce opposite orientations on σn−1.
• An orientation of K is a choice of orientation of each n-simplex insuch a way that any two simplices that interesect in an(n−1)-face are consistently oriented.
• If a complex K admits an orientation, it is said to be orientable.
M. K. Misztal Simplicial Complexes: Theory and Implementation
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Theory Meshes Implementation
Triangle meshes
Definition (triangle mesh)
A dimension 2 simplicial complex K⊂ En (where n ≥ 2), such thatevery 0 or 1-simplex σ ∈ K is a face of a 2-simplex σ2 ∈ K is called atriangle mesh.
Triangle meshes inherit the notions of manifoldness and orientabilityfrom simplicial complexes.
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Theory Meshes Implementation
Triangle mesh operations
Triangle mesh operations include:
• smoothing: displacing vertices without changing connectivity,performend in order to improve mesh quality;
• edge flips: mesh reconnection without changing vertexplacement;
• edge splits: introducing a new vertex on an edge;• face splits: introducing a new vertex on a face;• edge collapse: removing an edge and its adjacent triangles;
M. K. Misztal Simplicial Complexes: Theory and Implementation
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Theory Meshes Implementation
Triangle mesh operations
Triangle mesh operations include:• smoothing: displacing vertices without changing connectivity,
performend in order to improve mesh quality;
• edge flips: mesh reconnection without changing vertexplacement;
• edge splits: introducing a new vertex on an edge;• face splits: introducing a new vertex on a face;• edge collapse: removing an edge and its adjacent triangles;
M. K. Misztal Simplicial Complexes: Theory and Implementation
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Theory Meshes Implementation
Triangle mesh operations
Triangle mesh operations include:• smoothing: displacing vertices without changing connectivity,
performend in order to improve mesh quality;• edge flips: mesh reconnection without changing vertex
placement;
• edge splits: introducing a new vertex on an edge;• face splits: introducing a new vertex on a face;• edge collapse: removing an edge and its adjacent triangles;
M. K. Misztal Simplicial Complexes: Theory and Implementation
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Theory Meshes Implementation
Triangle mesh operations
Triangle mesh operations include:• smoothing: displacing vertices without changing connectivity,
performend in order to improve mesh quality;• edge flips: mesh reconnection without changing vertex
placement;• edge splits: introducing a new vertex on an edge;
• face splits: introducing a new vertex on a face;• edge collapse: removing an edge and its adjacent triangles;
M. K. Misztal Simplicial Complexes: Theory and Implementation
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Theory Meshes Implementation
Triangle mesh operations
Triangle mesh operations include:• smoothing: displacing vertices without changing connectivity,
performend in order to improve mesh quality;• edge flips: mesh reconnection without changing vertex
placement;• edge splits: introducing a new vertex on an edge;• face splits: introducing a new vertex on a face;
• edge collapse: removing an edge and its adjacent triangles;
M. K. Misztal Simplicial Complexes: Theory and Implementation
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Theory Meshes Implementation
Triangle mesh operations
Triangle mesh operations include:• smoothing: displacing vertices without changing connectivity,
performend in order to improve mesh quality;• edge flips: mesh reconnection without changing vertex
placement;• edge splits: introducing a new vertex on an edge;• face splits: introducing a new vertex on a face;• edge collapse: removing an edge and its adjacent triangles;
M. K. Misztal Simplicial Complexes: Theory and Implementation
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Theory Meshes Implementation
Edge flip
M. K. Misztal Simplicial Complexes: Theory and Implementation
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Theory Meshes Implementation
Edge split
M. K. Misztal Simplicial Complexes: Theory and Implementation
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Theory Meshes Implementation
Face split
M. K. Misztal Simplicial Complexes: Theory and Implementation
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Theory Meshes Implementation
Edge collapse
M. K. Misztal Simplicial Complexes: Theory and Implementation
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Theory Meshes Implementation
Tetrahedral meshes
Definition (tetrahedral mesh)
A dimension 3 simplicial complex K⊂ En (where n ≥ 3), such thatevery 0, 1 or 2-simplex σ ∈ K is a face of a 3-simplex σ3 ∈ K is calleda tetrahedral mesh.
Tetrahedral meshes inherit the notions of manifoldness andorientability from simplicial complexes.
Triangle mesh operations generalize (although not always easily) totetrahedral meshes.
M. K. Misztal Simplicial Complexes: Theory and Implementation
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Theory Meshes Implementation
Tetrahedral mesh
M. K. Misztal Simplicial Complexes: Theory and Implementation
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Theory Meshes Implementation
Data structures
• The main purpose of data structures representing a simplicialcomplex K is to store data associated with simplices in K.
• Depending on the purpose, not all of the simplex types might berepresented in the data structure (for example: indexed face set,for simplicial complexes of dimension 2, with no dangling edges).
• If want to ensure efficient traversal, incidence information has tobe stored together with the simplices.
• Examples include: quad-edge, half-edge (for 2-manifoldtriangular meshes).
M. K. Misztal Simplicial Complexes: Theory and Implementation
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Theory Meshes Implementation
Data structures
• The main purpose of data structures representing a simplicialcomplex K is to store data associated with simplices in K.
• Depending on the purpose, not all of the simplex types might berepresented in the data structure (for example: indexed face set,for simplicial complexes of dimension 2, with no dangling edges).
• If want to ensure efficient traversal, incidence information has tobe stored together with the simplices.
• Examples include: quad-edge, half-edge (for 2-manifoldtriangular meshes).
M. K. Misztal Simplicial Complexes: Theory and Implementation
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Theory Meshes Implementation
Data structures
• The main purpose of data structures representing a simplicialcomplex K is to store data associated with simplices in K.
• Depending on the purpose, not all of the simplex types might berepresented in the data structure (for example: indexed face set,for simplicial complexes of dimension 2, with no dangling edges).
• If want to ensure efficient traversal, incidence information has tobe stored together with the simplices.
• Examples include: quad-edge, half-edge (for 2-manifoldtriangular meshes).
M. K. Misztal Simplicial Complexes: Theory and Implementation
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Theory Meshes Implementation
Data structures
• The main purpose of data structures representing a simplicialcomplex K is to store data associated with simplices in K.
• Depending on the purpose, not all of the simplex types might berepresented in the data structure (for example: indexed face set,for simplicial complexes of dimension 2, with no dangling edges).
• If want to ensure efficient traversal, incidence information has tobe stored together with the simplices.
• Examples include: quad-edge, half-edge (for 2-manifoldtriangular meshes).
M. K. Misztal Simplicial Complexes: Theory and Implementation
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Theory Meshes Implementation
Incidence simplicial data structure
• The incidence simplicial (IS) data structure is adimension-independent, compact data structure designed forrepresenting arbitrary simplicial complexes K.
• Each simplex in K has its representation in IS data structure.• We store with each p-simplex σp ∈ K (for p > 1) the unordered
set of handles to its p + 1 (p−1)-dimensional facesσ
p−11 , . . . ,σp−1
p+1 (the boundary relation Bp,p−1(σp)).
• In order to make the traversal efficient, partial coboundaryrelation C∗p,p+1(σp) is also stored with every p-simplex σp ∈ K, forp < n.
• Partial coboundary relation C∗p,p+1(σp) consists of(p + 1)-simplices from st(σp) connecting σp with its link, one pereach connected component in lk(σp).
M. K. Misztal Simplicial Complexes: Theory and Implementation
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Theory Meshes Implementation
Incidence simplicial data structure
• The incidence simplicial (IS) data structure is adimension-independent, compact data structure designed forrepresenting arbitrary simplicial complexes K.
• Each simplex in K has its representation in IS data structure.
• We store with each p-simplex σp ∈ K (for p > 1) the unorderedset of handles to its p + 1 (p−1)-dimensional facesσ
p−11 , . . . ,σp−1
p+1 (the boundary relation Bp,p−1(σp)).
• In order to make the traversal efficient, partial coboundaryrelation C∗p,p+1(σp) is also stored with every p-simplex σp ∈ K, forp < n.
• Partial coboundary relation C∗p,p+1(σp) consists of(p + 1)-simplices from st(σp) connecting σp with its link, one pereach connected component in lk(σp).
M. K. Misztal Simplicial Complexes: Theory and Implementation
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Theory Meshes Implementation
Incidence simplicial data structure
• The incidence simplicial (IS) data structure is adimension-independent, compact data structure designed forrepresenting arbitrary simplicial complexes K.
• Each simplex in K has its representation in IS data structure.• We store with each p-simplex σp ∈ K (for p > 1) the unordered
set of handles to its p + 1 (p−1)-dimensional facesσ
p−11 , . . . ,σp−1
p+1 (the boundary relation Bp,p−1(σp)).
• In order to make the traversal efficient, partial coboundaryrelation C∗p,p+1(σp) is also stored with every p-simplex σp ∈ K, forp < n.
• Partial coboundary relation C∗p,p+1(σp) consists of(p + 1)-simplices from st(σp) connecting σp with its link, one pereach connected component in lk(σp).
M. K. Misztal Simplicial Complexes: Theory and Implementation
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Theory Meshes Implementation
Incidence simplicial data structure
• The incidence simplicial (IS) data structure is adimension-independent, compact data structure designed forrepresenting arbitrary simplicial complexes K.
• Each simplex in K has its representation in IS data structure.• We store with each p-simplex σp ∈ K (for p > 1) the unordered
set of handles to its p + 1 (p−1)-dimensional facesσ
p−11 , . . . ,σp−1
p+1 (the boundary relation Bp,p−1(σp)).
• In order to make the traversal efficient, partial coboundaryrelation C∗p,p+1(σp) is also stored with every p-simplex σp ∈ K, forp < n.
• Partial coboundary relation C∗p,p+1(σp) consists of(p + 1)-simplices from st(σp) connecting σp with its link, one pereach connected component in lk(σp).
M. K. Misztal Simplicial Complexes: Theory and Implementation
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Theory Meshes Implementation
Incidence simplicial data structure
• The incidence simplicial (IS) data structure is adimension-independent, compact data structure designed forrepresenting arbitrary simplicial complexes K.
• Each simplex in K has its representation in IS data structure.• We store with each p-simplex σp ∈ K (for p > 1) the unordered
set of handles to its p + 1 (p−1)-dimensional facesσ
p−11 , . . . ,σp−1
p+1 (the boundary relation Bp,p−1(σp)).
• In order to make the traversal efficient, partial coboundaryrelation C∗p,p+1(σp) is also stored with every p-simplex σp ∈ K, forp < n.
• Partial coboundary relation C∗p,p+1(σp) consists of(p + 1)-simplices from st(σp) connecting σp with its link, one pereach connected component in lk(σp).
M. K. Misztal Simplicial Complexes: Theory and Implementation
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Theory Meshes Implementation
Our implementation
• Our implementation of the IS data structure is restricted tosimplicial complexes of dimension three or less.
• Our implementation is orientation-aware: we identify an orientedsimplex σp with an ordered tuple of its (p−1)-faces:[
σp−11 , . . . ,σp−1
p+1
],
which implies:
σp =
[vert(σ
p)/vert(σ
p−11
), . . . ,vert
(σ
p)/vert(σ
p−1p+1
)],
where:
vert(σ
d)=d+1⋃i=1
vert(σ
d−1i
),
M. K. Misztal Simplicial Complexes: Theory and Implementation
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Theory Meshes Implementation
Our implementation
• Our implementation of the IS data structure is restricted tosimplicial complexes of dimension three or less.
• Our implementation is orientation-aware: we identify an orientedsimplex σp with an ordered tuple of its (p−1)-faces:[
σp−11 , . . . ,σp−1
p+1
],
which implies:
σp =
[vert(σ
p)/vert(σ
p−11
), . . . ,vert
(σ
p)/vert(σ
p−1p+1
)],
where:
vert(σ
d)=d+1⋃i=1
vert(σ
d−1i
),
M. K. Misztal Simplicial Complexes: Theory and Implementation
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Theory Meshes Implementation
Our implemetation
It can be seen that:• C∗2,3(σ2) = C2,3(σ2),
• if σp (p < 2) is 3-manifold, then |C∗p,p+1(σp)|= 1.
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Theory Meshes Implementation
Operations
The operations for traversal and manipulation of the simplicialcomplex include:
• star – evaluation of the star of a simplex;• closure – evaluation of the closure of a simplex or a set of
simplices;• link – evaluation of the link of a simplex;• boundary – evaluation of the boundary of the simplex;• orient faces consistently/oppositely – enforcing a
consistent/opposite orientation on all (p−1)-faces of a p-simplexσp;
• orient co-faces consistently/oppositely – enforcing aconsistent/opposite orientation on all (p + 1)-simplices having agiven p-simplex σp as a face;
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Theory Meshes Implementation
Operations
The operations for traversal and manipulation of the simplicialcomplex include:
• star – evaluation of the star of a simplex;
• closure – evaluation of the closure of a simplex or a set ofsimplices;
• link – evaluation of the link of a simplex;• boundary – evaluation of the boundary of the simplex;• orient faces consistently/oppositely – enforcing a
consistent/opposite orientation on all (p−1)-faces of a p-simplexσp;
• orient co-faces consistently/oppositely – enforcing aconsistent/opposite orientation on all (p + 1)-simplices having agiven p-simplex σp as a face;
M. K. Misztal Simplicial Complexes: Theory and Implementation
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Theory Meshes Implementation
Operations
The operations for traversal and manipulation of the simplicialcomplex include:
• star – evaluation of the star of a simplex;• closure – evaluation of the closure of a simplex or a set of
simplices;
• link – evaluation of the link of a simplex;• boundary – evaluation of the boundary of the simplex;• orient faces consistently/oppositely – enforcing a
consistent/opposite orientation on all (p−1)-faces of a p-simplexσp;
• orient co-faces consistently/oppositely – enforcing aconsistent/opposite orientation on all (p + 1)-simplices having agiven p-simplex σp as a face;
M. K. Misztal Simplicial Complexes: Theory and Implementation
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Theory Meshes Implementation
Operations
The operations for traversal and manipulation of the simplicialcomplex include:
• star – evaluation of the star of a simplex;• closure – evaluation of the closure of a simplex or a set of
simplices;• link – evaluation of the link of a simplex;
• boundary – evaluation of the boundary of the simplex;• orient faces consistently/oppositely – enforcing a
consistent/opposite orientation on all (p−1)-faces of a p-simplexσp;
• orient co-faces consistently/oppositely – enforcing aconsistent/opposite orientation on all (p + 1)-simplices having agiven p-simplex σp as a face;
M. K. Misztal Simplicial Complexes: Theory and Implementation
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Theory Meshes Implementation
Operations
The operations for traversal and manipulation of the simplicialcomplex include:
• star – evaluation of the star of a simplex;• closure – evaluation of the closure of a simplex or a set of
simplices;• link – evaluation of the link of a simplex;• boundary – evaluation of the boundary of the simplex;
• orient faces consistently/oppositely – enforcing aconsistent/opposite orientation on all (p−1)-faces of a p-simplexσp;
• orient co-faces consistently/oppositely – enforcing aconsistent/opposite orientation on all (p + 1)-simplices having agiven p-simplex σp as a face;
M. K. Misztal Simplicial Complexes: Theory and Implementation
![Page 68: Simplicial Complexes: Theory and Implementation · Simplicial Complexes: Theory and Implementation Marek K. Misztal Informatics and Mathematical Modelling, Technical University of](https://reader034.fdocuments.in/reader034/viewer/2022050608/5faeef81ff41942b2e4cb491/html5/thumbnails/68.jpg)
Theory Meshes Implementation
Operations
The operations for traversal and manipulation of the simplicialcomplex include:
• star – evaluation of the star of a simplex;• closure – evaluation of the closure of a simplex or a set of
simplices;• link – evaluation of the link of a simplex;• boundary – evaluation of the boundary of the simplex;• orient faces consistently/oppositely – enforcing a
consistent/opposite orientation on all (p−1)-faces of a p-simplexσp;
• orient co-faces consistently/oppositely – enforcing aconsistent/opposite orientation on all (p + 1)-simplices having agiven p-simplex σp as a face;
M. K. Misztal Simplicial Complexes: Theory and Implementation
![Page 69: Simplicial Complexes: Theory and Implementation · Simplicial Complexes: Theory and Implementation Marek K. Misztal Informatics and Mathematical Modelling, Technical University of](https://reader034.fdocuments.in/reader034/viewer/2022050608/5faeef81ff41942b2e4cb491/html5/thumbnails/69.jpg)
Theory Meshes Implementation
Operations
The operations for traversal and manipulation of the simplicialcomplex include:
• star – evaluation of the star of a simplex;• closure – evaluation of the closure of a simplex or a set of
simplices;• link – evaluation of the link of a simplex;• boundary – evaluation of the boundary of the simplex;• orient faces consistently/oppositely – enforcing a
consistent/opposite orientation on all (p−1)-faces of a p-simplexσp;
• orient co-faces consistently/oppositely – enforcing aconsistent/opposite orientation on all (p + 1)-simplices having agiven p-simplex σp as a face;
M. K. Misztal Simplicial Complexes: Theory and Implementation
![Page 70: Simplicial Complexes: Theory and Implementation · Simplicial Complexes: Theory and Implementation Marek K. Misztal Informatics and Mathematical Modelling, Technical University of](https://reader034.fdocuments.in/reader034/viewer/2022050608/5faeef81ff41942b2e4cb491/html5/thumbnails/70.jpg)
Theory Meshes Implementation
References
• J. M. Lee. Introduction to topological manifolds. 2000.
• L. de Floriani, A. Hui, D. Panozzo and D. Canino. Adimension-independent data structure for simplicial complexes.2010.
• M. K. Misztal. Deformable simplicial complexes. PhD thesis,2010.
M. K. Misztal Simplicial Complexes: Theory and Implementation
![Page 71: Simplicial Complexes: Theory and Implementation · Simplicial Complexes: Theory and Implementation Marek K. Misztal Informatics and Mathematical Modelling, Technical University of](https://reader034.fdocuments.in/reader034/viewer/2022050608/5faeef81ff41942b2e4cb491/html5/thumbnails/71.jpg)
Theory Meshes Implementation
References
• J. M. Lee. Introduction to topological manifolds. 2000.• L. de Floriani, A. Hui, D. Panozzo and D. Canino. A
dimension-independent data structure for simplicial complexes.2010.
• M. K. Misztal. Deformable simplicial complexes. PhD thesis,2010.
M. K. Misztal Simplicial Complexes: Theory and Implementation
![Page 72: Simplicial Complexes: Theory and Implementation · Simplicial Complexes: Theory and Implementation Marek K. Misztal Informatics and Mathematical Modelling, Technical University of](https://reader034.fdocuments.in/reader034/viewer/2022050608/5faeef81ff41942b2e4cb491/html5/thumbnails/72.jpg)
Theory Meshes Implementation
References
• J. M. Lee. Introduction to topological manifolds. 2000.• L. de Floriani, A. Hui, D. Panozzo and D. Canino. A
dimension-independent data structure for simplicial complexes.2010.
• M. K. Misztal. Deformable simplicial complexes. PhD thesis,2010.
M. K. Misztal Simplicial Complexes: Theory and Implementation