# Catmull–Clark subdivision surface

Catmull–Clark subdivision surface
You are encouraged to solve this task according to the task description, using any language you may know.

Implement the Catmull-Clark surface subdivision (description on Wikipedia), which is an algorithm that maps from a surface (described as a set of points and a set of polygons with vertices at those points) to another more refined surface. The resulting surface will always consist of a mesh of quadrilaterals.

The process for computing the new locations of the points works as follows when the surface is free of holes:

Starting cubic mesh; the meshes below are derived from this.
After one round of the Catmull-Clark algorithm applied to a cubic mesh.
After two rounds of the Catmull-Clark algorithm. As can be seen, this is converging to a surface that looks nearly spherical.
1. for each face, a face point is created which is the average of all the points of the face.
2. for each edge, an edge point is created which is the average between the center of the edge and the center of the segment made with the face points of the two adjacent faces.
3. for each vertex point, its coordinates are updated from (new_coords):
1. the old coordinates (old_coords),
2. the average of the face points of the faces the point belongs to (avg_face_points),
3. the average of the centers of edges the point belongs to (avg_mid_edges),
4. how many faces a point belongs to (n), then use this formula:
```     m1 = (n - 3) / n
m2 = 1 / n
m3 = 2 / n
new_coords = (m1 * old_coords)
+ (m2 * avg_face_points)
+ (m3 * avg_mid_edges)
```

Then each face is replaced by new faces made with the new points,

• for a triangle face (a,b,c):
```   (a, edge_pointab, face_pointabc, edge_pointca)
(b, edge_pointbc, face_pointabc, edge_pointab)
(c, edge_pointca, face_pointabc, edge_pointbc)
```
• for a quad face (a,b,c,d):
```   (a, edge_pointab, face_pointabcd, edge_pointda)
(b, edge_pointbc, face_pointabcd, edge_pointab)
(c, edge_pointcd, face_pointabcd, edge_pointbc)
(d, edge_pointda, face_pointabcd, edge_pointcd)
```

When there is a hole, we can detect it as follows:

• an edge is the border of a hole if it belongs to only one face,
• a point is on the border of a hole if nfaces != nedges with nfaces the number of faces the point belongs to, and nedges the number of edges a point belongs to.

On the border of a hole the subdivision occurs as follows:

1. for the edges that are on the border of a hole, the edge point is just the middle of the edge.
2. for the vertex points that are on the border of a hole, the new coordinates are calculated as follows:
1. in all the edges the point belongs to, only take in account the middles of the edges that are on the border of the hole
2. calculate the average between these points (on the hole boundary) and the old coordinates (also on the hole boundary).

For edges and vertices not next to a hole, the standard algorithm from above is used.

## C

Only the subdivision part. The full source is way too long to be shown here. Lots of macros, you'll have to see the full code to know what's what.

`vertex face_point(face f){	int i;	vertex v; 	if (!f->avg) {		f->avg = vertex_new();		foreach(i, v, f->v)			if (!i) f->avg->pos = v->pos;			else    vadd(f->avg->pos, v->pos); 		vdiv(f->avg->pos, len(f->v));	}	return f->avg;} #define hole_edge(e) (len(e->f)==1)vertex edge_point(edge e){	int i;	face f; 	if (!e->e_pt) {		e->e_pt = vertex_new();		e->avg = e->v[0]->pos;		vadd(e->avg, e->v[1]->pos);		e->e_pt->pos = e->avg; 		if (!hole_edge(e)) {			foreach (i, f, e->f)				vadd(e->e_pt->pos, face_point(f)->pos);			vdiv(e->e_pt->pos, 4);		} else			vdiv(e->e_pt->pos, 2); 		vdiv(e->avg, 2);	} 	return e->e_pt;} #define hole_vertex(v) (len((v)->f) != len((v)->e))vertex updated_point(vertex v){	int i, n = 0;	edge e;	face f;	coord_t sum = {0, 0, 0}; 	if (v->v_new) return v->v_new; 	v->v_new = vertex_new();	if (hole_vertex(v)) {		v->v_new->pos = v->pos;		foreach(i, e, v->e) {			if (!hole_edge(e)) continue;			vadd(v->v_new->pos, edge_point(e)->pos);			n++;		}		vdiv(v->v_new->pos, n + 1);	} else {		n = len(v->f);		foreach(i, f, v->f)			vadd(sum, face_point(f)->pos);		foreach(i, e, v->e)			vmadd(sum, edge_point(e)->pos, 2, sum);		vdiv(sum, n);		vmadd(sum, v->pos, n - 3, sum);		vdiv(sum, n);		v->v_new->pos = sum;	} 	return v->v_new;} model catmull(model m){	int i, j, a, b, c, d;	face f;	vertex v, x; 	model nm = model_new();	foreach (i, f, m->f) {		foreach(j, v, f->v) {			_get_idx(a, updated_point(v));			_get_idx(b, edge_point(elem(f->e, (j + 1) % len(f->e))));			_get_idx(c, face_point(f));			_get_idx(d, edge_point(elem(f->e, j)));			model_add_face(nm, 4, a, b, c, d);		}	}	return nm;}`

`{-# LANGUAGE GeneralizedNewtypeDeriving #-}{-# LANGUAGE ScopedTypeVariables #-} import Data.Arrayimport Data.Foldable (length, concat, sum)import Data.List (genericLength)import Data.Maybe (mapMaybe)import Prelude hiding (length, concat, sum)import qualified Data.Map.Strict as Map {-A SimpleMesh consists of only vertices and faces that refer to them.A Mesh extends the SimpleMesh to contain edges as well as references toadjoining mesh components for each other component, such as a vertexalso contains what faces it belongs to.An isolated edge can be represented as a degenerate face with 2 vertices.Faces with 0 or 1 vertices can be thrown out, as they do not contribute tothe result (they can also propagate NaNs).-} newtype VertexId = VertexId { getVertexId :: Int } deriving (Ix, Ord, Eq, Show)newtype EdgeId = EdgeId { getEdgeId :: Int } deriving (Ix, Ord, Eq, Show)newtype FaceId = FaceId { getFaceId :: Int } deriving (Ix, Ord, Eq, Show) data Vertex a = Vertex  { vertexPoint :: a  , vertexEdges :: [EdgeId]  , vertexFaces :: [FaceId]  } deriving Show data Edge = Edge  { edgeVertexA :: VertexId  , edgeVertexB :: VertexId  , edgeFaces :: [FaceId]  } deriving Show data Face = Face  { faceVertices :: [VertexId]  , faceEdges :: [EdgeId]  } deriving Show type VertexArray a = Array VertexId (Vertex a)type EdgeArray = Array EdgeId Edgetype FaceArray = Array FaceId Face data Mesh a = Mesh  { meshVertices :: VertexArray a  , meshEdges :: EdgeArray  , meshFaces :: FaceArray  } deriving Show data SimpleVertex a = SimpleVertex { sVertexPoint :: a } deriving Showdata SimpleFace = SimpleFace { sFaceVertices :: [VertexId] } deriving Show type SimpleVertexArray a = Array VertexId (SimpleVertex a)type SimpleFaceArray = Array FaceId SimpleFace data SimpleMesh a = SimpleMesh  { sMeshVertices :: SimpleVertexArray a  , sMeshFaces :: SimpleFaceArray  } deriving Show -- Generic helpers.fmap1 :: Functor f => (t -> a -> b) -> (t -> f a) -> t -> f bfmap1 g h x = fmap (g x) (h x) aZipWith :: Ix i1 => (a -> b -> e) -> Array i1 a -> Array i b -> Array i1 eaZipWith f a b = listArray (bounds a) \$ zipWith f (elems a) (elems b) average :: (Foldable f, Fractional a) => f a -> aaverage xs = (sum xs) / (fromIntegral \$ length xs) -- Intermediary point types for ultimately converting into a point `a`.newtype FacePoint a = FacePoint { getFacePoint :: a } deriving Shownewtype EdgeCenterPoint a = EdgeCenterPoint { getEdgeCenterPoint :: a } deriving Shownewtype EdgePoint a = EdgePoint { getEdgePoint :: a } deriving Shownewtype VertexPoint a = VertexPoint { getVertexPoint :: a } deriving Show type FacePointArray a = Array FaceId (FacePoint a)type EdgePointArray a = Array EdgeId (EdgePoint a)type EdgeCenterPointArray a = Array EdgeId (EdgeCenterPoint a)type IsEdgeHoleArray = Array EdgeId Booltype VertexPointArray a = Array VertexId (VertexPoint a) -- Subdivision helpers.facePoint :: Fractional a => Mesh a -> Face -> FacePoint afacePoint mesh = FacePoint . average . (fmap \$ vertexPointById mesh) . faceVertices allFacePoints :: Fractional a => Mesh a -> FacePointArray aallFacePoints = fmap1 facePoint meshFaces vertexPointById :: Mesh a -> VertexId -> avertexPointById mesh = vertexPoint . (meshVertices mesh !) edgeCenterPoint :: Fractional a => Mesh a -> Edge -> EdgeCenterPoint aedgeCenterPoint mesh (Edge ea eb _)  = EdgeCenterPoint . average \$ fmap (vertexPointById mesh) [ea, eb] allEdgeCenterPoints :: Fractional a => Mesh a -> EdgeCenterPointArray aallEdgeCenterPoints = fmap1 edgeCenterPoint meshEdges allIsEdgeHoles :: Mesh a -> IsEdgeHoleArrayallIsEdgeHoles = fmap ((< 2) . length . edgeFaces) . meshEdges edgePoint :: Fractional a => Edge -> FacePointArray a -> EdgeCenterPoint a -> EdgePoint aedgePoint (Edge _ _ [_]) _ (EdgeCenterPoint ecp) = EdgePoint ecpedgePoint (Edge _ _ faceIds) facePoints (EdgeCenterPoint ecp)  = EdgePoint \$ average [ecp, average \$ fmap (getFacePoint . (facePoints !)) faceIds] allEdgePoints :: Fractional a => Mesh a -> FacePointArray a -> EdgeCenterPointArray a -> EdgePointArray aallEdgePoints mesh fps ecps = aZipWith (\e ecp -> edgePoint e fps ecp) (meshEdges mesh) ecps vertexPoint' :: Fractional a => Vertex a -> FacePointArray a -> EdgeCenterPointArray a -> IsEdgeHoleArray -> VertexPoint avertexPoint' vertex facePoints ecps iehs  | length faceIds == length edgeIds = VertexPoint newCoords  | otherwise = VertexPoint avgHoleEcps  where    newCoords = (oldCoords * m1) + (avgFacePoints * m2) + (avgMidEdges * m3)    oldCoords = vertexPoint vertex    avgFacePoints = average \$ fmap (getFacePoint . (facePoints !)) faceIds    avgMidEdges = average \$ fmap (getEdgeCenterPoint . (ecps !)) edgeIds    m1 = (n - 3) / n    m2 = 1 / n    m3 = 2 / n    n = genericLength faceIds    faceIds = vertexFaces vertex    edgeIds = vertexEdges vertex    avgHoleEcps = average . (oldCoords:) . fmap (getEdgeCenterPoint . (ecps !)) \$ filter (iehs !) edgeIds allVertexPoints :: Fractional a => Mesh a -> FacePointArray a -> EdgeCenterPointArray a -> IsEdgeHoleArray -> VertexPointArray aallVertexPoints mesh fps ecps iehs = fmap (\v -> vertexPoint' v fps ecps iehs) (meshVertices mesh) -- For each vertex in a face, generate a set of new faces from it with its vertex point,-- neighbor edge points, and face point. The new faces will refer to vertices in the-- combined vertex array.newFaces :: Face -> FaceId -> Int -> Int -> [SimpleFace]newFaces (Face vertexIds edgeIds) faceId epOffset vpOffset  = take (genericLength vertexIds)  \$ zipWith3 newFace (cycle vertexIds) (cycle edgeIds) (drop 1 (cycle edgeIds))  where    f = VertexId . (+ epOffset) . getEdgeId    newFace vid epA epB = SimpleFace      [ VertexId . (+ vpOffset) \$ getVertexId vid      , f epA      , VertexId \$ getFaceId faceId      , f epB] subdivide :: Fractional a => SimpleMesh a -> SimpleMesh asubdivide simpleMesh  = SimpleMesh combinedVertices (listArray (FaceId 0, FaceId (genericLength faces - 1)) faces)  where    mesh = makeComplexMesh simpleMesh    fps = allFacePoints mesh    ecps = allEdgeCenterPoints mesh    eps = allEdgePoints mesh fps ecps    iehs = allIsEdgeHoles mesh    vps = allVertexPoints mesh fps ecps iehs    edgePointOffset = length fps    vertexPointOffset = edgePointOffset + length eps    combinedVertices      = listArray (VertexId 0, VertexId (vertexPointOffset + length vps - 1))      . fmap SimpleVertex      \$ concat [ fmap getFacePoint \$ elems fps               , fmap getEdgePoint \$ elems eps               , fmap getVertexPoint \$ elems vps]    faces      = concat \$ zipWith (\face fid -> newFaces face fid edgePointOffset vertexPointOffset)      (elems \$ meshFaces mesh) (fmap FaceId [0..]) -- Transform to a Mesh by filling in the missing references and generating edges.-- Faces can be updated with their edges, but must be ordered.-- Edge and face order does not matter for vertices.-- TODO: Discard degenerate faces (ones with 0 to 2 vertices/edges),-- or we could transform these into single edges or vertices.makeComplexMesh :: forall a. SimpleMesh a -> Mesh amakeComplexMesh (SimpleMesh sVertices sFaces) = Mesh vertices edges faces  where    makeEdgesFromFace :: SimpleFace -> FaceId -> [Edge]    makeEdgesFromFace (SimpleFace vertexIds) fid      = take (genericLength vertexIds)      \$ zipWith (\a b -> Edge a b [fid]) verts (drop 1 verts)      where        verts = cycle vertexIds     edgeKey :: VertexId -> VertexId -> (VertexId, VertexId)    edgeKey a b = (min a b, max a b)     sFacesList :: [SimpleFace]    sFacesList = elems sFaces     fids :: [FaceId]    fids = fmap FaceId [0..]     eids :: [EdgeId]    eids = fmap EdgeId [0..]     faceEdges :: [[Edge]]    faceEdges = zipWith makeEdgesFromFace sFacesList fids     edgeMap :: Map.Map (VertexId, VertexId) Edge    edgeMap      = Map.fromListWith (\(Edge a b fidsA) (Edge _ _ fidsB) -> Edge a b (fidsA ++ fidsB))      . fmap (\edg[email protected](Edge a b _) -> (edgeKey a b, edge))      \$ concat faceEdges     edges :: EdgeArray    edges = listArray (EdgeId 0, EdgeId \$ (Map.size edgeMap) - 1) \$ Map.elems edgeMap     edgeIdMap :: Map.Map (VertexId, VertexId) EdgeId    edgeIdMap = Map.fromList \$ zipWith (\(Edge a b _) eid -> ((edgeKey a b), eid)) (elems edges) eids     faceEdgeIds :: [[EdgeId]]    faceEdgeIds = fmap (mapMaybe (\(Edge a b _) -> Map.lookup (edgeKey a b) edgeIdMap)) faceEdges     faces :: FaceArray    faces      = listArray (FaceId 0, FaceId \$ (length sFaces) - 1)      \$ zipWith (\(SimpleFace verts) edgeIds -> Face verts edgeIds) sFacesList faceEdgeIds     vidsToFids :: Map.Map VertexId [FaceId]    vidsToFids      = Map.fromListWith (++)      . concat      \$ zipWith (\(SimpleFace vertexIds) fid -> fmap (\vid -> (vid, [fid])) vertexIds) sFacesList fids     vidsToEids :: Map.Map VertexId [EdgeId]    vidsToEids      = Map.fromListWith (++)      . concat      \$ zipWith (\(Edge a b _) eid -> [(a, [eid]), (b, [eid])]) (elems edges) eids     simpleToComplexVert :: SimpleVertex a -> VertexId -> Vertex a    simpleToComplexVert (SimpleVertex point) vid      = Vertex point      (Map.findWithDefault [] vid vidsToEids)      (Map.findWithDefault [] vid vidsToFids)     vertices :: VertexArray a    vertices      = listArray (bounds sVertices)      \$ zipWith simpleToComplexVert (elems sVertices) (fmap VertexId [0..]) pShowSimpleMesh :: Show a => SimpleMesh a -> StringpShowSimpleMesh (SimpleMesh vertices faces)  = "Vertices:\n" ++ (arrShow vertices sVertexPoint)  ++ "Faces:\n" ++ (arrShow faces (fmap getVertexId . sFaceVertices))  where    arrShow a f = concatMap ((++ "\n") . show . (\(i, e) -> (i, f e))) . zip [0 :: Int ..] \$ elems a -- Testing types.data Point a = Point a a a deriving (Show) instance Functor Point where  fmap f (Point x y z) = Point (f x) (f y) (f z) zipPoint :: (a -> b -> c) -> Point a -> Point b -> Point czipPoint f (Point x y z) (Point x' y' z') = Point (f x x') (f y y') (f z z') instance Num a => Num (Point a) where  (+) = zipPoint (+)  (-) = zipPoint (-)  (*) = zipPoint (*)  negate = fmap negate  abs = fmap abs  signum = fmap signum  fromInteger i = let i' = fromInteger i in Point i' i' i' instance Fractional a => Fractional (Point a) where  recip = fmap recip  fromRational r = let r' = fromRational r in Point r' r' r' testCube :: SimpleMesh (Point Double)testCube = SimpleMesh vertices faces  where    vertices = listArray (VertexId 0, VertexId 7)      \$ fmap SimpleVertex      [ Point (-1) (-1) (-1)      , Point (-1) (-1) 1      , Point (-1) 1 (-1)      , Point (-1) 1 1      , Point 1 (-1) (-1)      , Point 1 (-1) 1      , Point 1 1 (-1)      , Point 1 1 1]    faces = listArray (FaceId 0, FaceId 5)      \$ fmap (SimpleFace . (fmap VertexId))      [ [0, 4, 5, 1]      , [4, 6, 7, 5]      , [6, 2, 3, 7]      , [2, 0, 1, 3]      , [1, 5, 7, 3]      , [0, 2, 6, 4]] testCubeWithHole :: SimpleMesh (Point Double)testCubeWithHole  = SimpleMesh (sMeshVertices testCube) (ixmap (FaceId 0, FaceId 4) id (sMeshFaces testCube)) testTriangle :: SimpleMesh (Point Double)testTriangle = SimpleMesh vertices faces  where    vertices = listArray (VertexId 0, VertexId 2)      \$ fmap SimpleVertex      [ Point 0 0 0      , Point 0 0 1      , Point 0 1 0]    faces = listArray (FaceId 0, FaceId 0)      \$ fmap (SimpleFace . (fmap VertexId))      [ [0, 1, 2]] main :: IO ()main = putStr . pShowSimpleMesh \$ subdivide testCube`
Output:
```Vertices:
(0,Point 0.0 (-1.0) 0.0)
(1,Point 1.0 0.0 0.0)
(2,Point 0.0 1.0 0.0)
(3,Point (-1.0) 0.0 0.0)
(4,Point 0.0 0.0 1.0)
(5,Point 0.0 0.0 (-1.0))
(6,Point (-0.75) (-0.75) 0.0)
(7,Point (-0.75) 0.0 (-0.75))
(8,Point 0.0 (-0.75) (-0.75))
(9,Point (-0.75) 0.0 0.75)
(10,Point 0.0 (-0.75) 0.75)
(11,Point (-0.75) 0.75 0.0)
(12,Point 0.0 0.75 (-0.75))
(13,Point 0.0 0.75 0.75)
(14,Point 0.75 (-0.75) 0.0)
(15,Point 0.75 0.0 (-0.75))
(16,Point 0.75 0.0 0.75)
(17,Point 0.75 0.75 0.0)
(18,Point (-0.5555555555555556) (-0.5555555555555556) (-0.5555555555555556))
(19,Point (-0.5555555555555556) (-0.5555555555555556) 0.5555555555555556)
(20,Point (-0.5555555555555556) 0.5555555555555556 (-0.5555555555555556))
(21,Point (-0.5555555555555556) 0.5555555555555556 0.5555555555555556)
(22,Point 0.5555555555555556 (-0.5555555555555556) (-0.5555555555555556))
(23,Point 0.5555555555555556 (-0.5555555555555556) 0.5555555555555556)
(24,Point 0.5555555555555556 0.5555555555555556 (-0.5555555555555556))
(25,Point 0.5555555555555556 0.5555555555555556 0.5555555555555556)
Faces:
(0,[18,8,0,14])
(1,[22,14,0,10])
(2,[23,10,0,6])
(3,[19,6,0,8])
(4,[22,15,1,17])
(5,[24,17,1,16])
(6,[25,16,1,14])
(7,[23,14,1,15])
(8,[24,12,2,11])
(9,[20,11,2,13])
(10,[21,13,2,17])
(11,[25,17,2,12])
(12,[20,7,3,6])
(13,[18,6,3,9])
(14,[19,9,3,11])
(15,[21,11,3,7])
(16,[19,10,4,16])
(17,[23,16,4,13])
(18,[25,13,4,9])
(19,[21,9,4,10])
(20,[18,7,5,12])
(21,[20,12,5,15])
(22,[24,15,5,8])
(23,[22,8,5,7])```

## J

`avg=: +/ % # havePoints=: e."1/~ [email protected]# catmullclark=:3 :0  'mesh points'=. y  face_point=. avg"2 mesh{points  point_face=. |: mesh havePoints points  avg_face_points=. point_face [email protected]#"1 2 face_point  edges=. ~.,/ meshEdges=. mesh /:[email protected],"+1|."1 mesh  edge_face=. *./"2 edges e."0 1/ mesh  edge_center=. avg"2 edges{points  edge_point=. (0.5*edge_center) + 0.25 * edge_face +/ .* face_point    point_edge=. |: edges havePoints points  avg_mid_edges=.  point_edge [email protected]#"1 2 edge_center  n=. +/"1 point_edge  'm3 m2 m1'=. (2,1,:n-3)%"1 n  new_coords=. (m1 * points) + (m2 * avg_face_points) + (m3 * avg_mid_edges)  pts=. face_point,edge_point,new_coords  c0=. (#edge_point)+ e0=. #face_point  msh=. (,c0+mesh),.(,e0+edges i. meshEdges),.((#i.)~/\$mesh),.,e0+_1|."1 edges i. meshEdges  msh;pts)`

Example use:

`NB.cubepoints=: _1+2*#:i.8mesh=: 1 A."1 I.(,1-|.)8&[email protected]#&0 1">4 2 1    catmullclark mesh;points┌──────────┬─────────────────────────────┐│22  6 0  9│        1         0         0││23  7 0  6│        0         1         0││25  8 0  7│        0         0         1││24  9 0  8│        0         0        _1││20 10 1 12│        0        _1         0││21 11 1 10│       _1         0         0││25  8 1 11│     0.75     _0.75         0││24 12 1  8│     0.75         0      0.75││19 13 2 14│     0.75      0.75         0││21 11 2 13│     0.75         0     _0.75││25  7 2 11│    _0.75      0.75         0││23 14 2  7│        0      0.75      0.75││18 15 3 16│        0      0.75     _0.75││20 12 3 15│    _0.75         0      0.75││24  9 3 12│        0     _0.75      0.75││22 16 3  9│    _0.75         0     _0.75││18 17 4 16│        0     _0.75     _0.75││19 14 4 17│    _0.75     _0.75         0││23  6 4 14│_0.555556 _0.555556 _0.555556││22 16 4  6│_0.555556 _0.555556  0.555556││18 17 5 15│_0.555556  0.555556 _0.555556││19 13 5 17│_0.555556  0.555556  0.555556││21 10 5 13│ 0.555556 _0.555556 _0.555556││20 15 5 10│ 0.555556 _0.555556  0.555556││          │ 0.555556  0.555556 _0.555556││          │ 0.555556  0.555556  0.555556│└──────────┴─────────────────────────────┘`

## Mathematica

This implementation supports tris, quads, and higher polys, as well as surfaces with holes. The function relies on three externally defined general functionality functions:

`subSetQ[large_,small_] := MemberQ[large,small]subSetQ[large_,small_List] := [email protected]@(MemberQ[large,#]&/@small) containing[groupList_,item_]:= Flatten[Position[groupList,group_/;subSetQ[group,item]]] ReplaceFace[face_]:=Transpose[Prepend[Transpose[{#[[1]],face,#[[2]]}&/@Transpose[Partition[face,2,1,1]//{#,RotateRight[#]}&]],face]]`

subSetQ[small,large] is a boolean test for whether small is a subset of large. Note this is not a general purpose implimentation and only serves this purpose under the constrictions of the following program.

containing[{obj1,obj2,...},item] Will return a list of indices of the objects containing item, where objects are faces or edges and item is edges or vertexes. faces containing a given vertex, faces containing a given edge, edges containing a given point. It is used for each such purpose in the code called via infix notation, the specific usage is easily distinguised by variable names. For example faces~containing~edge would be a list of the indices for the faces containing the given edge.

ReplaceFace[face] replaces the face with a list of descriptions for the new faces. It will return a list containing mixed objects, vertexes, edges and faces where edges and faces referes to the new vertexes to be generated by the code. When the new vertexes have been appended to the updated old vertexes, these mixed objects will be recalcluated into correct indices into the new vertex list by the later defined function newIndex[].

`CatMullClark[{Points_,faces_}]:=Block[{avgFacePoints,avgEdgePoints,updatedPoints,newEdgePoints,newPoints,edges,newFaces,weights,pointUpdate,edgeUpdate,newIndex},edges = DeleteDuplicates[Flatten[Partition[#,2,1,-1]&/@faces,1],Sort[#1]==Sort[#2]&];avgFacePoints=Mean[Points[[#]]] &/@ faces;avgEdgePoints=Mean[Points[[#]]] &/@ edges; weights[vertex_]:= Count[faces,vertex,2]//{(#-3),1,2}/#&;pointUpdate[vertex_]:= 	If[Length[faces~containing~vertex]!=Length[edges~containing~vertex],		Mean[avgEdgePoints[[Select[edges~containing~vertex,holeQ[edges[[#]],faces]&]]]],		Total[weights[vertex]{ Points[[vertex]], Mean[avgFacePoints[[faces~containing~vertex]]], Mean[avgEdgePoints[[edges~containing~vertex]]]}]	]; edgeUpdate[edge_]:= 	If[Length[faces~containing~edge]==1,		Mean[Points[[edge]]],		Mean[Points[[Flatten[{edge, faces[[faces~containing~edge]]}]]]]	]; updatedPoints = pointUpdate/@Range[1,Length[Points]];newEdgePoints = edgeUpdate/@edges;newPoints = Join[updatedPoints,avgFacePoints,newEdgePoints]; newIndex[edge_/;Length[edge]==2]  := Length[Points]+Length[faces]+Position[Sort/@edges,[email protected]][[1,1]]newIndex[face_] := Length[Points]+Position[faces,face][[1,1]] newFaces  = Flatten[Map[newIndex[#,{Points,edges,faces}]&,ReplaceFace/@faces,{-2}],1];{newPoints,newFaces}]`

The implimentation can be tested with polygons with and without holes by using the polydata

`{points,faces}=PolyhedronData["Cube",{"VertexCoordinates","FaceIndices"}]; Function[iteration,Graphics3D[(Polygon[iteration[[1]][[#]]]&/@iteration[[2]])]]/@NestList[CatMullClark,{points,faces},3]//GraphicsRow`

For a surface with holes the resulting iterative subdivision will be:

`faces = Delete[faces, 6];Function[iteration, Graphics3D[    (Polygon[iteration[[1]][[#]]] & /@ iteration[[2]])    ]] /@ NestList[CatMullClark, {points, faces}, 3] // GraphicsRow`

This code was written in Mathematica 8.

## OCaml

 This example is incorrect. Please fix the code and remove this message.Details: wrong output data

The implementation below only supports quad faces, but it does handle surfaces with holes.

This code uses a module called Dynar (for dynamic array) because it needs a structure similar to arrays but with which we can push a new element at the end. (The source of this module is given in the sub-page.)

In the sub-page there is also a program in OCaml+OpenGL which displays a cube subdivided 2 times with this algorithm.

`open Dynar let add3 (x1, y1, z1) (x2, y2, z2) (x3, y3, z3) =  ( (x1 +. x2 +. x3),    (y1 +. y2 +. y3),    (z1 +. z2 +. z3) ) let mul m (x,y,z) = (m *. x, m *. y, m *. z) let avg pts =  let n, (x,y,z) =    List.fold_left      (fun (n, (xt,yt,zt)) (xi,yi,zi) ->         succ n, (xt +. xi, yt +. yi, zt +. zi))      (1, List.hd pts) (List.tl pts)  in  let n = float_of_int n in  (x /. n, y /. n, z /. n)  let catmull ~points ~faces =  let da_points = Dynar.of_array points in  let new_faces = Dynar.of_array [| |] in  let push_face face = Dynar.push new_faces face in  let h1 = Hashtbl.create 43 in  let h2 = Hashtbl.create 43 in  let h3 = Hashtbl.create 43 in  let h4 = Hashtbl.create 43 in  let blg = Array.make (Array.length points) 0 in (* how many faces a point belongs to *)  let f_incr p = blg.(p) <- succ blg.(p) in  let eblg = Array.make (Array.length points) 0 in (* how many edges a point belongs to *)  let e_incr p = eblg.(p) <- succ eblg.(p) in  let edge a b = (min a b, max a b) in  (* suitable for hash-table keys *)  let mid_edge p1 p2 =    let x1, y1, z1 = points.(p1)    and x2, y2, z2 = points.(p2) in    ( (x1 +. x2) /. 2.0,      (y1 +. y2) /. 2.0,      (z1 +. z2) /. 2.0 )  in  let mid_face p1 p2 p3 p4 =    let x1, y1, z1 = points.(p1)    and x2, y2, z2 = points.(p2)    and x3, y3, z3 = points.(p3)    and x4, y4, z4 = points.(p4) in    ( (x1 +. x2 +. x3 +. x4) /. 4.0,      (y1 +. y2 +. y3 +. y4) /. 4.0,      (z1 +. z2 +. z3 +. z4) /. 4.0 )  in  Array.iteri (fun i (a,b,c,d) ->    f_incr a; f_incr b; f_incr c; f_incr d;     let face_point = mid_face a b c d in    let face_pi = pushi da_points face_point in    Hashtbl.add h3 a face_point;    Hashtbl.add h3 b face_point;    Hashtbl.add h3 c face_point;    Hashtbl.add h3 d face_point;     let process_edge a b =      let ab = edge a b in      if not(Hashtbl.mem h1 ab)      then begin        let mid_ab = mid_edge a b in        let index = pushi da_points mid_ab in        Hashtbl.add h1 ab (index, mid_ab, [face_point]);        Hashtbl.add h2 a mid_ab;        Hashtbl.add h2 b mid_ab;        Hashtbl.add h4 mid_ab 1;        (index)      end      else begin        let index, mid_ab, fpl = Hashtbl.find h1 ab in        Hashtbl.replace h1 ab (index, mid_ab, face_point::fpl);        Hashtbl.add h4 mid_ab (succ(Hashtbl.find h4 mid_ab));        (index)      end    in    let mid_ab = process_edge a b    and mid_bc = process_edge b c    and mid_cd = process_edge c d    and mid_da = process_edge d a in     push_face (a, mid_ab, face_pi, mid_da);    push_face (b, mid_bc, face_pi, mid_ab);    push_face (c, mid_cd, face_pi, mid_bc);    push_face (d, mid_da, face_pi, mid_cd);  ) faces;   Hashtbl.iter (fun (a,b) (index, mid_ab, fpl) ->    e_incr a; e_incr b;    if List.length fpl = 2 then      da_points.ar.(index) <- avg (mid_ab::fpl)  ) h1;   Array.iteri (fun i old_vertex ->    let n = blg.(i)    and e_n = eblg.(i) in    (* if the vertex doesn't belongs to as many faces than edges       this means that this is a hole *)    if n = e_n then    begin      let avg_face_points =        let face_point_list = Hashtbl.find_all h3 i in        (avg face_point_list)      in      let avg_mid_edges =         let mid_edge_list = Hashtbl.find_all h2 i in        (avg mid_edge_list)      in      let n = float_of_int n in      let m1 = (n -. 3.0) /. n      and m2 = 1.0 /. n      and m3 = 2.0 /. n in      da_points.ar.(i) <-          add3 (mul m1 old_vertex)               (mul m2 avg_face_points)               (mul m3 avg_mid_edges)    end    else begin      let mid_edge_list = Hashtbl.find_all h2 i in      let mid_edge_list =        (* only average mid-edges near the hole *)        List.fold_left (fun acc mid_edge ->          match Hashtbl.find h4 mid_edge with          | 1 -> mid_edge::acc          | _ -> acc        ) [] mid_edge_list      in      da_points.ar.(i) <- avg (old_vertex :: mid_edge_list)    end  ) points;   (Dynar.to_array da_points,   Dynar.to_array new_faces);;`

### Another implementation

Another implementation which should work with holes, but has only been tested on a cube

Works with: OCaml version 4.02+
`type point = { x: float; y : float; z : float }let zero = { x = 0.0; y = 0.0; z = 0.0 }let add a b = { x = a.x+.b.x; y = a.y+.b.y; z = a.z+.b.z }let mul a k = { x = a.x*.k; y = a.y*.k; z= a.z*.k }let div p k = mul p (1.0/.k) type face = Face of point listtype edge = Edge of point*point let make_edge a b = Edge (min a b, max a b)let make_face a b c d = Face [a;b;c;d] let centroid plist = div (List.fold_left add zero plist) (float (List.length plist))let mid_edge (Edge (p1,p2)) = div (add p1 p2) 2.0let face_point (Face pl) = centroid pllet point_in_face p (Face pl) = List.mem p pllet point_in_edge p (Edge (p1,p2)) = p = p1 || p = p2let edge_in_face (Edge (p1,p2)) f = point_in_face p1 f && point_in_face p2 f let border_edge faces e =   List.length (List.filter (edge_in_face e) faces) < 2 let edge_point faces e =   if border_edge faces e then mid_edge e else   let adjacent = List.filter (edge_in_face e) faces in   let fps = List.map face_point adjacent in   centroid [mid_edge e; centroid fps] let mod_vertex faces edges p =   let v_edges = List.filter (point_in_edge p) edges in   let v_faces = List.filter (point_in_face p) faces in   let n = List.length v_faces in   let is_border = n <> (List.length v_edges) in   if is_border then      let border_mids = List.map mid_edge (List.filter (border_edge faces) v_edges) in      (* description ambiguity: average (border+p) or average(average(border),p) ?? *)      centroid (p :: border_mids)   else      let avg_face = centroid (List.map face_point v_faces) in      let avg_mid = centroid (List.map mid_edge v_edges) in      div (add (add (mul p (float(n-3))) avg_face) (mul avg_mid 2.0)) (float n) let edges_of_face (Face pl) =   let rec next acc = function      | [] -> invalid_arg "empty face"      | a :: [] -> List.rev (make_edge a (List.hd pl) :: acc)      | a :: (b :: _ as xs) -> next (make_edge a b :: acc) xs in   next [] pl let catmull_clark faces =   let module EdgeSet = Set.Make(struct type t = edge let compare = compare end) in   let edges = EdgeSet.elements (EdgeSet.of_list (List.concat (List.map edges_of_face faces))) in   let mod_face ((Face pl) as face) =      let fp = face_point face in      let ep = List.map (edge_point faces) (edges_of_face face) in      let e_tl = List.hd (List.rev ep) in      let vl = List.map (mod_vertex faces edges) pl in      let add_facet (e', acc) v e = e, (make_face e' v e fp :: acc) in      let _, new_faces = List.fold_left2 add_facet (e_tl, []) vl ep in      List.rev new_faces in   List.concat (List.map mod_face faces) let show_faces fl =   let pr_point p = Printf.printf " (%.4f, %.4f, %.4f)" p.x p.y p.z in   let pr_face (Face pl) = print_string "Face:"; List.iter pr_point pl; print_string "\n" in   (print_string "surface {\n"; List.iter pr_face fl; print_string "}\n") let c p q r = let s i = if i = 0 then -1.0 else 1.0 in { x = s p; y = s q; z = s r } ;;let cube = [   Face [c 0 0 0; c 0 0 1; c 0 1 1; c 0 1 0]; Face [c 1 0 0; c 1 0 1; c 1 1 1; c 1 1 0];   Face [c 0 0 0; c 1 0 0; c 1 0 1; c 0 0 1]; Face [c 0 1 0; c 1 1 0; c 1 1 1; c 0 1 1];   Face [c 0 0 0; c 0 1 0; c 1 1 0; c 1 0 0]; Face [c 0 0 1; c 0 1 1; c 1 1 1; c 1 0 1] ] inshow_faces cube;show_faces (catmull_clark cube)`

with output:

```surface {
Face: (-1.0000, -1.0000, -1.0000) (-1.0000, -1.0000, 1.0000) (-1.0000, 1.0000, 1.0000) (-1.0000, 1.0000, -1.0000)
Face: (1.0000, -1.0000, -1.0000) (1.0000, -1.0000, 1.0000) (1.0000, 1.0000, 1.0000) (1.0000, 1.0000, -1.0000)
Face: (-1.0000, -1.0000, -1.0000) (1.0000, -1.0000, -1.0000) (1.0000, -1.0000, 1.0000) (-1.0000, -1.0000, 1.0000)
Face: (-1.0000, 1.0000, -1.0000) (1.0000, 1.0000, -1.0000) (1.0000, 1.0000, 1.0000) (-1.0000, 1.0000, 1.0000)
Face: (-1.0000, -1.0000, -1.0000) (-1.0000, 1.0000, -1.0000) (1.0000, 1.0000, -1.0000) (1.0000, -1.0000, -1.0000)
Face: (-1.0000, -1.0000, 1.0000) (-1.0000, 1.0000, 1.0000) (1.0000, 1.0000, 1.0000) (1.0000, -1.0000, 1.0000)
}
surface {
Face: (-0.7500, 0.0000, -0.7500) (-0.5556, -0.5556, -0.5556) (-0.7500, -0.7500, 0.0000) (-1.0000, 0.0000, 0.0000)
Face: (-0.7500, -0.7500, 0.0000) (-0.5556, -0.5556, 0.5556) (-0.7500, 0.0000, 0.7500) (-1.0000, 0.0000, 0.0000)
Face: (-0.7500, 0.0000, 0.7500) (-0.5556, 0.5556, 0.5556) (-0.7500, 0.7500, 0.0000) (-1.0000, 0.0000, 0.0000)
Face: (-0.7500, 0.7500, 0.0000) (-0.5556, 0.5556, -0.5556) (-0.7500, 0.0000, -0.7500) (-1.0000, 0.0000, 0.0000)
Face: (0.7500, 0.0000, -0.7500) (0.5556, -0.5556, -0.5556) (0.7500, -0.7500, 0.0000) (1.0000, 0.0000, 0.0000)
Face: (0.7500, -0.7500, 0.0000) (0.5556, -0.5556, 0.5556) (0.7500, 0.0000, 0.7500) (1.0000, 0.0000, 0.0000)
Face: (0.7500, 0.0000, 0.7500) (0.5556, 0.5556, 0.5556) (0.7500, 0.7500, 0.0000) (1.0000, 0.0000, 0.0000)
Face: (0.7500, 0.7500, 0.0000) (0.5556, 0.5556, -0.5556) (0.7500, 0.0000, -0.7500) (1.0000, 0.0000, 0.0000)
Face: (-0.7500, -0.7500, 0.0000) (-0.5556, -0.5556, -0.5556) (0.0000, -0.7500, -0.7500) (0.0000, -1.0000, 0.0000)
Face: (0.0000, -0.7500, -0.7500) (0.5556, -0.5556, -0.5556) (0.7500, -0.7500, 0.0000) (0.0000, -1.0000, 0.0000)
Face: (0.7500, -0.7500, 0.0000) (0.5556, -0.5556, 0.5556) (0.0000, -0.7500, 0.7500) (0.0000, -1.0000, 0.0000)
Face: (0.0000, -0.7500, 0.7500) (-0.5556, -0.5556, 0.5556) (-0.7500, -0.7500, 0.0000) (0.0000, -1.0000, 0.0000)
Face: (-0.7500, 0.7500, 0.0000) (-0.5556, 0.5556, -0.5556) (0.0000, 0.7500, -0.7500) (0.0000, 1.0000, 0.0000)
Face: (0.0000, 0.7500, -0.7500) (0.5556, 0.5556, -0.5556) (0.7500, 0.7500, 0.0000) (0.0000, 1.0000, 0.0000)
Face: (0.7500, 0.7500, 0.0000) (0.5556, 0.5556, 0.5556) (0.0000, 0.7500, 0.7500) (0.0000, 1.0000, 0.0000)
Face: (0.0000, 0.7500, 0.7500) (-0.5556, 0.5556, 0.5556) (-0.7500, 0.7500, 0.0000) (0.0000, 1.0000, 0.0000)
Face: (0.0000, -0.7500, -0.7500) (-0.5556, -0.5556, -0.5556) (-0.7500, 0.0000, -0.7500) (0.0000, 0.0000, -1.0000)
Face: (-0.7500, 0.0000, -0.7500) (-0.5556, 0.5556, -0.5556) (0.0000, 0.7500, -0.7500) (0.0000, 0.0000, -1.0000)
Face: (0.0000, 0.7500, -0.7500) (0.5556, 0.5556, -0.5556) (0.7500, 0.0000, -0.7500) (0.0000, 0.0000, -1.0000)
Face: (0.7500, 0.0000, -0.7500) (0.5556, -0.5556, -0.5556) (0.0000, -0.7500, -0.7500) (0.0000, 0.0000, -1.0000)
Face: (0.0000, -0.7500, 0.7500) (-0.5556, -0.5556, 0.5556) (-0.7500, 0.0000, 0.7500) (0.0000, 0.0000, 1.0000)
Face: (-0.7500, 0.0000, 0.7500) (-0.5556, 0.5556, 0.5556) (0.0000, 0.7500, 0.7500) (0.0000, 0.0000, 1.0000)
Face: (0.0000, 0.7500, 0.7500) (0.5556, 0.5556, 0.5556) (0.7500, 0.0000, 0.7500) (0.0000, 0.0000, 1.0000)
Face: (0.7500, 0.0000, 0.7500) (0.5556, -0.5556, 0.5556) (0.0000, -0.7500, 0.7500) (0.0000, 0.0000, 1.0000)
}```

## Python

` """ Input and output are assumed to be in this form based on the talkpage for the task: input_points = [  [-1.0,  1.0,  1.0],  [-1.0, -1.0,  1.0],  [ 1.0, -1.0,  1.0],  [ 1.0,  1.0,  1.0],  [ 1.0, -1.0, -1.0],  [ 1.0,  1.0, -1.0],  [-1.0, -1.0, -1.0],  [-1.0,  1.0, -1.0]] input_faces = [  [0, 1, 2, 3],  [3, 2, 4, 5],  [5, 4, 6, 7],  [7, 0, 3, 5],  [7, 6, 1, 0],  [6, 1, 2, 4],] So, the graph is a list of points and a list of faces.Each face is a list of indexes into the points list. """ from mpl_toolkits.mplot3d import axes3dimport matplotlib.pyplot as pltimport numpy as npimport sys def center_point(p1, p2):    """     returns a point in the center of the     segment ended by points p1 and p2    """    cp = []    for i in range(3):        cp.append((p1[i]+p2[i])/2)     return cp def sum_point(p1, p2):    """     adds points p1 and p2    """    sp = []    for i in range(3):        sp.append(p1[i]+p2[i])     return sp def div_point(p, d):    """     divide point p by d    """    sp = []    for i in range(3):        sp.append(p[i]/d)     return sp def mul_point(p, m):    """     multiply point p by m    """    sp = []    for i in range(3):        sp.append(p[i]*m)     return sp def get_face_points(input_points, input_faces):    """    From http://rosettacode.org/wiki/Catmull%E2%80%93Clark_subdivision_surface     1. for each face, a face point is created which is the average of all the points of the face.    """     # 3 dimensional space     NUM_DIMENSIONS = 3     # face_points will have one point for each face     face_points = []     for curr_face in input_faces:        face_point = [0.0, 0.0, 0.0]        for curr_point_index in curr_face:            curr_point = input_points[curr_point_index]            # add curr_point to face_point            # will divide later            for i in range(NUM_DIMENSIONS):                face_point[i] += curr_point[i]        # divide by number of points for average        num_points = len(curr_face)        for i in range(NUM_DIMENSIONS):            face_point[i] /= num_points        face_points.append(face_point)     return face_points def get_edges_faces(input_points, input_faces):    """     Get list of edges and the one or two adjacent faces in a list.    also get center point of edge     Each edge would be [pointnum_1, pointnum_2, facenum_1, facenum_2, center]     """     # will have [pointnum_1, pointnum_2, facenum]     edges = []     # get edges from each face     for facenum in range(len(input_faces)):        face = input_faces[facenum]        num_points = len(face)        # loop over index into face        for pointindex in range(num_points):            # if not last point then edge is curr point and next point            if pointindex < num_points - 1:                pointnum_1 = face[pointindex]                pointnum_2 = face[pointindex+1]            else:                # for last point edge is curr point and first point                pointnum_1 = face[pointindex]                pointnum_2 = face[0]            # order points in edge by lowest point number            if pointnum_1 > pointnum_2:                temp = pointnum_1                pointnum_1 = pointnum_2                pointnum_2 = temp            edges.append([pointnum_1, pointnum_2, facenum])     # sort edges by pointnum_1, pointnum_2, facenum     edges = sorted(edges)     # merge edges with 2 adjacent faces    # [pointnum_1, pointnum_2, facenum_1, facenum_2] or    # [pointnum_1, pointnum_2, facenum_1, None]     num_edges = len(edges)    eindex = 0    merged_edges = []     while eindex < num_edges:        e1 = edges[eindex]        # check if not last edge        if eindex < num_edges - 1:            e2 = edges[eindex+1]            if e1[0] == e2[0] and e1[1] == e2[1]:                merged_edges.append([e1[0],e1[1],e1[2],e2[2]])                eindex += 2            else:                merged_edges.append([e1[0],e1[1],e1[2],None])                eindex += 1        else:            merged_edges.append([e1[0],e1[1],e1[2],None])            eindex += 1     # add edge centers     edges_centers = []     for me in merged_edges:        p1 = input_points[me[0]]        p2 = input_points[me[1]]        cp = center_point(p1, p2)        edges_centers.append(me+[cp])     return edges_centers def get_edge_points(input_points, edges_faces, face_points):    """    for each edge, an edge point is created which is the average     between the center of the edge and the center of the segment made    with the face points of the two adjacent faces.    """     edge_points = []     for edge in edges_faces:        # get center of edge        cp = edge[4]        # get center of two facepoints        fp1 = face_points[edge[2]]        # if not two faces just use one facepoint        # should not happen for solid like a cube        if edge[3] == None:            fp2 = fp1        else:            fp2 = face_points[edge[3]]        cfp = center_point(fp1, fp2)        # get average between center of edge and        # center of facepoints        edge_point = center_point(cp, cfp)        edge_points.append(edge_point)           return edge_points def get_avg_face_points(input_points, input_faces, face_points):    """     for each point calculate     the average of the face points of the faces the point belongs to (avg_face_points)     create a list of lists of two numbers [facepoint_sum, num_points] by going through the    points in all the faces.     then create the avg_face_points list of point by dividing point_sum (x, y, z) by num_points     """     # initialize list with [[0.0, 0.0, 0.0], 0]     num_points = len(input_points)     temp_points = []     for pointnum in range(num_points):        temp_points.append([[0.0, 0.0, 0.0], 0])     # loop through faces updating temp_points     for facenum in range(len(input_faces)):        fp = face_points[facenum]        for pointnum in input_faces[facenum]:            tp = temp_points[pointnum][0]            temp_points[pointnum][0] = sum_point(tp,fp)            temp_points[pointnum][1] += 1     # divide to create avg_face_points     avg_face_points = []     for tp in temp_points:       afp = div_point(tp[0], tp[1])       avg_face_points.append(afp)     return avg_face_points def get_avg_mid_edges(input_points, edges_faces):    """     the average of the centers of edges the point belongs to (avg_mid_edges)     create list with entry for each point     each entry has two elements. one is a point that is the sum of the centers of the edges    and the other is the number of edges. after going through all edges divide by    number of edges.     """     # initialize list with [[0.0, 0.0, 0.0], 0]     num_points = len(input_points)     temp_points = []     for pointnum in range(num_points):        temp_points.append([[0.0, 0.0, 0.0], 0])     # go through edges_faces using center updating each point     for edge in edges_faces:        cp = edge[4]        for pointnum in [edge[0], edge[1]]:            tp = temp_points[pointnum][0]            temp_points[pointnum][0] = sum_point(tp,cp)            temp_points[pointnum][1] += 1     # divide out number of points to get average     avg_mid_edges = []     for tp in temp_points:       ame = div_point(tp[0], tp[1])       avg_mid_edges.append(ame)     return avg_mid_edges def get_points_faces(input_points, input_faces):    # initialize list with 0     num_points = len(input_points)     points_faces = []     for pointnum in range(num_points):        points_faces.append(0)     # loop through faces updating points_faces     for facenum in range(len(input_faces)):        for pointnum in input_faces[facenum]:            points_faces[pointnum] += 1     return points_faces def get_new_points(input_points, points_faces, avg_face_points, avg_mid_edges):    """     m1 = (n - 3) / n    m2 = 1 / n    m3 = 2 / n    new_coords = (m1 * old_coords)               + (m2 * avg_face_points)               + (m3 * avg_mid_edges)     """     new_points =[]     for pointnum in range(len(input_points)):        n = points_faces[pointnum]        m1 = (n - 3) / n        m2 = 1 / n        m3 = 2 / n        old_coords = input_points[pointnum]        p1 = mul_point(old_coords, m1)        afp = avg_face_points[pointnum]        p2 = mul_point(afp, m2)        ame = avg_mid_edges[pointnum]        p3 = mul_point(ame, m3)        p4 = sum_point(p1, p2)        new_coords = sum_point(p4, p3)         new_points.append(new_coords)     return new_points def switch_nums(point_nums):    """    Returns tuple of point numbers    sorted least to most    """    if point_nums[0] < point_nums[1]:        return point_nums    else:        return (point_nums[1], point_nums[0])     def cmc_subdiv(input_points, input_faces):    # 1. for each face, a face point is created which is the average of all the points of the face.    # each entry in the returned list is a point (x, y, z).     face_points = get_face_points(input_points, input_faces)     # get list of edges with 1 or 2 adjacent faces    # [pointnum_1, pointnum_2, facenum_1, facenum_2, center] or    # [pointnum_1, pointnum_2, facenum_1, None, center]     edges_faces = get_edges_faces(input_points, input_faces)     # get edge points, a list of points     edge_points = get_edge_points(input_points, edges_faces, face_points)     # the average of the face points of the faces the point belongs to (avg_face_points)                     avg_face_points = get_avg_face_points(input_points, input_faces, face_points)     # the average of the centers of edges the point belongs to (avg_mid_edges)     avg_mid_edges = get_avg_mid_edges(input_points, edges_faces)      # how many faces a point belongs to     points_faces = get_points_faces(input_points, input_faces)     """     m1 = (n - 3) / n    m2 = 1 / n    m3 = 2 / n    new_coords = (m1 * old_coords)               + (m2 * avg_face_points)               + (m3 * avg_mid_edges)     """     new_points = get_new_points(input_points, points_faces, avg_face_points, avg_mid_edges)     """     Then each face is replaced by new faces made with the new points,     for a triangle face (a,b,c):       (a, edge_point ab, face_point abc, edge_point ca)       (b, edge_point bc, face_point abc, edge_point ab)       (c, edge_point ca, face_point abc, edge_point bc)     for a quad face (a,b,c,d):       (a, edge_point ab, face_point abcd, edge_point da)       (b, edge_point bc, face_point abcd, edge_point ab)       (c, edge_point cd, face_point abcd, edge_point bc)       (d, edge_point da, face_point abcd, edge_point cd)     face_points is a list indexed by face number so that is    easy to get.     edge_points is a list indexed by the edge number    which is an index into edges_faces.     need to add face_points and edge points to     new_points and get index into each.     then create two new structures     face_point_nums - list indexes by facenum    whose value is the index into new_points     edge_point num - dictionary with key (pointnum_1, pointnum_2)    and value is index into new_points     """     # add face points to new_points     face_point_nums = []     # point num after next append to new_points    next_pointnum = len(new_points)     for face_point in face_points:        new_points.append(face_point)        face_point_nums.append(next_pointnum)        next_pointnum += 1     # add edge points to new_points     edge_point_nums = dict()     for edgenum in range(len(edges_faces)):        pointnum_1 = edges_faces[edgenum][0]        pointnum_2 = edges_faces[edgenum][1]        edge_point = edge_points[edgenum]        new_points.append(edge_point)        edge_point_nums[(pointnum_1, pointnum_2)] = next_pointnum        next_pointnum += 1     # new_points now has the points to output. Need new    # faces     """     just doing this case for now:     for a quad face (a,b,c,d):       (a, edge_point ab, face_point abcd, edge_point da)       (b, edge_point bc, face_point abcd, edge_point ab)       (c, edge_point cd, face_point abcd, edge_point bc)       (d, edge_point da, face_point abcd, edge_point cd)     new_faces will be a list of lists where the elements are like this:     [pointnum_1, pointnum_2, pointnum_3, pointnum_4]     """     new_faces =[]     for oldfacenum in range(len(input_faces)):        oldface = input_faces[oldfacenum]        # 4 point face        if len(oldface) == 4:            a = oldface[0]            b = oldface[1]            c = oldface[2]            d = oldface[3]            face_point_abcd = face_point_nums[oldfacenum]            edge_point_ab = edge_point_nums[switch_nums((a, b))]            edge_point_da = edge_point_nums[switch_nums((d, a))]            edge_point_bc = edge_point_nums[switch_nums((b, c))]            edge_point_cd = edge_point_nums[switch_nums((c, d))]            new_faces.append((a, edge_point_ab, face_point_abcd, edge_point_da))            new_faces.append((b, edge_point_bc, face_point_abcd, edge_point_ab))            new_faces.append((c, edge_point_cd, face_point_abcd, edge_point_bc))            new_faces.append((d, edge_point_da, face_point_abcd, edge_point_cd))         return new_points, new_faces  def graph_output(output_points, output_faces):     fig = plt.figure()    ax = fig.add_subplot(111, projection='3d')     """     Plot each face     """     for facenum in range(len(output_faces)):        curr_face = output_faces[facenum]        xcurr = []        ycurr = []        zcurr = []        for pointnum in range(len(curr_face)):            xcurr.append(output_points[curr_face[pointnum]][0])            ycurr.append(output_points[curr_face[pointnum]][1])            zcurr.append(output_points[curr_face[pointnum]][2])        xcurr.append(output_points[curr_face[0]][0])        ycurr.append(output_points[curr_face[0]][1])        zcurr.append(output_points[curr_face[0]][2])         ax.plot(xcurr,ycurr,zcurr,color='b')     plt.show()  # cube input_points = [  [-1.0,  1.0,  1.0],  [-1.0, -1.0,  1.0],  [ 1.0, -1.0,  1.0],  [ 1.0,  1.0,  1.0],  [ 1.0, -1.0, -1.0],  [ 1.0,  1.0, -1.0],  [-1.0, -1.0, -1.0],  [-1.0,  1.0, -1.0]] input_faces = [  [0, 1, 2, 3],  [3, 2, 4, 5],  [5, 4, 6, 7],  [7, 0, 3, 5],  [7, 6, 1, 0],  [6, 1, 2, 4],] if len(sys.argv) != 2:    print("Should have one argument integer number of iterations")    sys.exit()else:    iterations = int(sys.argv[1])     output_points, output_faces = input_points, input_faces     for i in range(iterations):        output_points, output_faces = cmc_subdiv(output_points, output_faces) graph_output(output_points, output_faces) `

## Tcl

This code handles both holes and arbitrary polygons in the input data.

`package require Tcl 8.5 # Use math functions and operators as commands (Lisp-like).namespace path {tcl::mathfunc tcl::mathop} # Add 3 points.proc add3 {A B C} {    lassign \$A Ax Ay Az    lassign \$B Bx By Bz    lassign \$C Cx Cy Cz    list [+ \$Ax \$Bx \$Cx] [+ \$Ay \$By \$Cy] [+ \$Az \$Bz \$Cz]} # Multiply a point by a constant.proc mulC {m A} {    lassign \$A x y z    list [* \$m \$x] [* \$m \$y] [* \$m \$z]} # Take the centroid of a set of points.# Note that each of the arguments is a *list* of coordinate triples# This makes things easier later.proc centroid args {    set x [set y [set z 0.0]]    foreach plist \$args {	incr n [llength \$plist]	foreach p \$plist {	    lassign \$p px py pz	    set x [+ \$x \$px]	    set y [+ \$y \$py]	    set z [+ \$z \$pz]	}    }    set n [double \$n]    list [/ \$x \$n] [/ \$y \$n] [/ \$z \$n]} # Select from the list the value from each of the indices in the *lists*# in the trailing arguments.proc selectFrom {list args} {    foreach is \$args {foreach i \$is {lappend r [lindex \$list \$i]}}    return \$r} # Rotate a list.proc lrot {list {n 1}} {    set n [% \$n [llength \$list]]    list {*}[lrange \$list \$n end] {*}[lrange \$list 0 [incr n -1]]} # Generate an edge by putting the smaller coordinate index first.proc edge {a b} {    list [min \$a \$b] [max \$a \$b]} # Perform one step of Catmull-Clark subdivision of a surface.proc CatmullClark {points faces} {    # Generate the new face-points and list of edges, plus some lookup tables.    set edges {}    foreach f \$faces {	set ps [selectFrom \$points \$f]	set fp [centroid \$ps]	lappend facepoints \$fp	foreach p \$ps {	    lappend fp4p(\$p) \$fp	}	foreach p1 \$f p2 [lrot \$f] {	    set e [edge \$p1 \$p2]	    if {\$e ni \$edges} {		lappend edges \$e	    }	    lappend fp4e(\$e) \$fp	}    }     # Generate the new edge-points and mid-points of edges, and a few more    # lookup tables.    set i [+ [llength \$points] [llength \$faces]]    foreach e \$edges {	set ep [selectFrom \$points \$e]	if {[llength \$fp4e(\$e)] > 1} {	    set mid [centroid \$ep \$fp4e(\$e)]	} else {	    set mid [centroid \$ep]	    foreach p \$ep {		lappend ep_heavy(\$p) \$mid	    }	}	lappend edgepoints \$mid	set en4e(\$e) \$i	foreach p \$ep {	    lappend ep4p(\$p) \$mid	}	incr i    }     # Generate the new vertex points with our lookup tables.    foreach p \$points {	if {[llength \$fp4p(\$p)] >= 4} {	    set n [llength \$fp4p(\$p)]	    lappend newPoints [add3 [mulC [/ [- \$n 3.0] \$n] \$p] \		    [mulC [/ 1.0 \$n] [centroid \$fp4p(\$p)]] \		    [mulC [/ 2.0 \$n] [centroid \$ep4p(\$p)]]]	} else {	    # Update a point on the edge of a hole. This formula is not	    # described on the WP page, but produces a nice result.	    lappend newPoints [centroid \$ep_heavy(\$p) [list \$p \$p]]	}    }     # Now compute the new set of quadrilateral faces.    set i [llength \$points]    foreach f \$faces {	foreach a \$f b [lrot \$f] c [lrot \$f -1] {	    lappend newFaces [list \		    \$a \$en4e([edge \$a \$b]) \$i \$en4e([edge \$c \$a])]	}	incr i    }     list [concat \$newPoints \$facepoints \$edgepoints] \$newFaces}`

The test code for this solution is available as well. The example there produces the following partial toroid output image: