Catmull–Clark subdivision surface

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.

for each face, a face point is created which is the average of all the points of the face.
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.
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_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)

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 n_faces != n_edges with n_faces the number of faces the point belongs to, and n_edges the number of edges a point belongs to.

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

for the edges that are on the border of a hole, the edge point is just the middle of the edge.
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. <lang c>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; }</lang>

J

<lang j>havePoints=: e."1/~ i.@# catmullclark=:3 :0

 'mesh points'=.y
 face_point=: avg"2 mesh{points
 point_face=: |: mesh havePoints points
 avg_face_points=: point_face avg@#"1 2 face_point
 edges=: ~.,/meshEdges=:mesh /:~@,"+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 avg@#"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+edge i. meshEdges),.((#i.)~/$mesh),.,e0+_1|."1 edge i. meshEdges
 msh;pts

)</lang>

Example use:

<lang j>NB.cube points=: _1+2*#:i.8 mesh=:1 A."1 I.(,1-|.)8&$@#&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│ └──────────┴─────────────────────────────┘</lang>

Mathematica

This example is in need of improvement:

Reformat the code to be more readable; to make its structure more visible.

This implementation supports tris, quads, and higher polys, as well as surfaces with holes.

<lang Mathematica>CatmullClark[{v_, i_}] := Block[{e, vc, fp, ep, vp},

 e = Function[a, {a, Select[Transpose[{i, Range@Length@i}], 
       Length@Intersection[#1, a] == 2 &]All, 2}] /@ 
   Union[Sort /@ Flatten[Partition[#, 2, 1, 1] & /@ i, 1]]; 
 vc = Table[{n, Select[Transpose[{i, Range@Length@i}], MemberQ[#1, n] &]All, 2, 
    Select[Transpose[{eAll, 1, Range@Length@eAll, 1}], MemberQ[#1, n] &]All, 2}, {n, Length@v}]; 
 fp = Mean[v#] & /@ i;
 ep = If[Length[#2] == 1, Mean[v[[#1]]], Mean@Join[v[[#1]], fp[[#2]]]] & /@ e; 
 vp = If[Length[#2] != Length[#3], Mean@Join[{v[[#1]]}, ep[[Select[#3, 
          Length[e#, 2] != 2 &]]]], ((Length@#2 - 3) v[[#1]] + Mean@fp[[#2]] +
           2 Mean@ep[[#3]])/Length@#2] & /@ vc;
 {Join[vp, ep, fp], Flatten[Function[a, Function[
       b, {a1, #1 + Length[vc], b + Length[vc] + Length[e], #2 + Length[vc]} &@
        Sort[Select[Transpose[{e, Range@Length@e}], MemberQ[#1, 1, a1] && MemberQ[#1, 2, b] &], 
          With[{f = i[[Intersection[#1, 2, #21, 2]1]], 
             n = Intersection[#1, 1, #21, 1]1}, 
            Xor[Abs[#] == 1, # < 0] &@(Position[f, Complement[#1, 1, {n}]1] - 
                Position[f, n])1, 1] &]All, 2] /@ a2] /@ vc, 1]}]

v = PolyhedronData["Cube", "VertexCoordinates"] // N i = PolyhedronData["Cube", "FaceIndices"] NestList[CatmullClark, {v, i}, 4]; Graphics3D[{FaceForm[{Opacity[0.3]}, {Opacity[0.1]}], GraphicsComplex[#1, Polygon[#2]]}] & /@ % Graphics3D[{EdgeForm[], FaceForm[White, Black],

   GraphicsComplex[#1, Polygon[#2], VertexNormals -> #1]}, Boxed -> False] & /@ %%

</lang>

The last few lines, after the function definition, do a test by using the built-in polyhedron data to generate the vertices and face indices. Then it repeatedly applies the method and graphs the results. Note that this was written in Mathematica 7, although it should be easy enough to port to maybe v5.2.

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.

<lang ocaml>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)

</lang>

Tcl

This code handles both holes and arbitrary polygons in the input data. <lang tcl>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] set mid [centroid $ep] if {[llength $fp4e($e)] > 1} { lappend edgepoints [centroid $ep $fp4e($e)] } else { 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 {

set n [llength $fp4p($p)] if {$n == [llength $ep4p($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 { lappend newPoints [centroid [list $p] $ep4p($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

}</lang>

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