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genmarchingcubes.go
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genmarchingcubes.go
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// Package genmarchingcubes implements the marching cubes algorithm.
//
// Algorithm:
// * https://en.wikipedia.org/wiki/Marching_cubes
//
// This implementation is based on:
// * https://github.com/soypat/sdf/blob/main/render/marchingcubes.go
// * https://github.com/fogleman/mc/blob/master/mc.go
package genmarchingcubes
import (
"bufio"
"fmt"
"math"
"os"
"github.com/Flokey82/go_gens/vectors"
)
const eps = 1e-12
type Triangle struct {
V1, V2, V3 vectors.Vec3
}
func (t *Triangle) toList() []vectors.Vec3 {
return []vectors.Vec3{t.V1, t.V2, t.V3}
}
// MarchingCubesGrid traverses the data grid with the given width, height, and depth,
// and evaluates each cube and returns the relevant triangles to represent the grid.
func MarchingCubesGrid(w, h, d int, data []float64, value float64) []Triangle {
var triangles []Triangle
for z0 := 0; z0 < d-1; z0++ {
z1 := z0 + 1
for y0 := 0; y0 < h-1; y0++ {
y1 := y0 + 1
for x0 := 0; x0 < w-1; x0++ {
x1 := x0 + 1
// Collect the values represented by the eight corners of the cube.
v := [8]float64{
data[x0+y0*w+z0*w*h],
data[x1+y0*w+z0*w*h],
data[x1+y1*w+z0*w*h],
data[x0+y1*w+z0*w*h],
data[x0+y0*w+z1*w*h],
data[x1+y0*w+z1*w*h],
data[x1+y1*w+z1*w*h],
data[x0+y1*w+z1*w*h],
}
// Encode the corners of the current cube as a set of
// eight vectors.
px0, py0, pz0 := float64(x0), float64(y0), float64(z0)
px1, py1, pz1 := float64(x1), float64(y1), float64(z1)
p := [8]vectors.Vec3{
{X: px0, Y: py0, Z: pz0},
{X: px1, Y: py0, Z: pz0},
{X: px1, Y: py1, Z: pz0},
{X: px0, Y: py1, Z: pz0},
{X: px0, Y: py0, Z: pz1},
{X: px1, Y: py0, Z: pz1},
{X: px1, Y: py1, Z: pz1},
{X: px0, Y: py1, Z: pz1},
}
// Determine the triangles that represent the current cube.
triangles = append(triangles, Polygonize(p, v, value)...)
}
}
}
return triangles
}
func ExportToOBJ(filename string, tri []Triangle) error {
// Open/create the destination file.
f, err := os.Create(filename)
if err != nil {
return err
}
defer f.Close()
// Initialize a new bufio writer.
wr := bufio.NewWriter(f)
// Make sure to flush the buffer on exit.
defer wr.Flush()
var vertices []vectors.Vec3
var faces [][]int
for _, t := range tri {
var faceIndices []int
for _, v := range t.toList() {
vertices = append(vertices, v)
faceIndices = append(faceIndices, len(vertices))
}
// Append the face vertex indices to the list of faces.
faces = append(faces, faceIndices)
}
// Write all the vertices to the file.
for _, v := range vertices {
// NOTE: I switched Y and Z since importing into Blender would have the Y axis as the up axis.
if _, err := wr.WriteString(fmt.Sprintf("v %f %f %f \n", v.X, v.Z, v.Y)); err != nil {
return err
}
}
// Write all the faces to the file.
for _, f := range faces {
if _, err := wr.WriteString(fmt.Sprintf("f %d %d %d \n", f[0], f[1], f[2])); err != nil {
return err
}
}
return nil
}
// Polygonize returns the triangles that represent the given cube represented by
// the given vectors and values.
func Polygonize(p [8]vectors.Vec3, v [8]float64, x float64) []Triangle {
// Calculate the index by checking which corners of the cube
// have a value that exceeds the threshold 'x'.
// So we figure out which of the 0..255 patterns we have.
var index int
for i := 0; i < 8; i++ {
if v[i] < x {
index |= 1 << uint(i)
}
}
// If the edge table entry for the index is set to zero, there's
// nothing left to do.
if edgeTable[index] == 0 {
return nil
}
// We build up a point index which contains the midpoints of all edges
// of the cube. These will be the vertices of the triangles.
var points [12]vectors.Vec3
// Iterate over the edges of the cube and work out the
// interpolated points on the edges
for i := 0; i < 12; i++ {
// If the edge (bit) is set for the current edge in the table (indicating
// that the edge is in use), calculate the midpoint of the edge and add it
// to the point index.
bit := 1 << uint(i)
if edgeTable[index]&bit != 0 {
// Look up which vertices build this edge.
a := pairTable[i][0]
b := pairTable[i][1]
// Interpolate the midpoint based on the two vertices that form the edge.
points[i] = Interpolate(p[a], p[b], v[a], v[b], x)
}
}
// Use the triangle table to look up what edge midpoints we use for the
// current index, then construct the triangles by looking the actual
// points in the points index we build above.
table := triangleTable[index]
count := len(table) / 3
result := make([]Triangle, count)
for i := 0; i < count; i++ {
v1 := points[table[i*3+0]]
v2 := points[table[i*3+1]]
v3 := points[table[i*3+2]]
result[i] = Triangle{v1, v2, v3}
}
return result
}
// Interpolate returns the point that is close enough to the threshold value or interpolates
// between the two points if the threshold is between the two points.
// Inspired by: https://github.com/soypat/sdf/blob/main/render/marchingcubes.go
func Interpolate(p1, p2 vectors.Vec3, v1, v2, x float64) vectors.Vec3 {
closeToV1 := math.Abs(x-v1) < eps
closeToV2 := math.Abs(x-v2) < eps
if closeToV1 && !closeToV2 {
return p1
}
if !closeToV1 && closeToV2 {
return p2
}
var t float64
if closeToV1 && closeToV2 {
t = 0.5 // Pick the half way point
} else {
t = (x - v1) / (v2 - v1) // linear interpolation
}
return vectors.Vec3{
X: p1.X + t*(p2.X-p1.X),
Y: p1.Y + t*(p2.Y-p1.Y),
Z: p1.Z + t*(p2.Z-p1.Z),
}
}
// These are the vertex pairs for the edges
var pairTable = [12][2]int{
{0, 1},
{1, 2},
{2, 3},
{3, 0},
{4, 5},
{5, 6},
{6, 7},
{7, 4},
{0, 4},
{1, 5},
{2, 6},
{3, 7},
}
// 8 vertices -> 256 possible inside/outside combinations
// A 1 bit in the value indicates an edge with a line end point.
// 12 edges -> 12 bit values, note the fwd/rev symmetry
var edgeTable = [256]int{
0x0000, 0x0109, 0x0203, 0x030a, 0x0406, 0x050f, 0x0605, 0x070c,
0x080c, 0x0905, 0x0a0f, 0x0b06, 0x0c0a, 0x0d03, 0x0e09, 0x0f00,
0x0190, 0x0099, 0x0393, 0x029a, 0x0596, 0x049f, 0x0795, 0x069c,
0x099c, 0x0895, 0x0b9f, 0x0a96, 0x0d9a, 0x0c93, 0x0f99, 0x0e90,
0x0230, 0x0339, 0x0033, 0x013a, 0x0636, 0x073f, 0x0435, 0x053c,
0x0a3c, 0x0b35, 0x083f, 0x0936, 0x0e3a, 0x0f33, 0x0c39, 0x0d30,
0x03a0, 0x02a9, 0x01a3, 0x00aa, 0x07a6, 0x06af, 0x05a5, 0x04ac,
0x0bac, 0x0aa5, 0x09af, 0x08a6, 0x0faa, 0x0ea3, 0x0da9, 0x0ca0,
0x0460, 0x0569, 0x0663, 0x076a, 0x0066, 0x016f, 0x0265, 0x036c,
0x0c6c, 0x0d65, 0x0e6f, 0x0f66, 0x086a, 0x0963, 0x0a69, 0x0b60,
0x05f0, 0x04f9, 0x07f3, 0x06fa, 0x01f6, 0x00ff, 0x03f5, 0x02fc,
0x0dfc, 0x0cf5, 0x0fff, 0x0ef6, 0x09fa, 0x08f3, 0x0bf9, 0x0af0,
0x0650, 0x0759, 0x0453, 0x055a, 0x0256, 0x035f, 0x0055, 0x015c,
0x0e5c, 0x0f55, 0x0c5f, 0x0d56, 0x0a5a, 0x0b53, 0x0859, 0x0950,
0x07c0, 0x06c9, 0x05c3, 0x04ca, 0x03c6, 0x02cf, 0x01c5, 0x00cc,
0x0fcc, 0x0ec5, 0x0dcf, 0x0cc6, 0x0bca, 0x0ac3, 0x09c9, 0x08c0,
0x08c0, 0x09c9, 0x0ac3, 0x0bca, 0x0cc6, 0x0dcf, 0x0ec5, 0x0fcc,
0x00cc, 0x01c5, 0x02cf, 0x03c6, 0x04ca, 0x05c3, 0x06c9, 0x07c0,
0x0950, 0x0859, 0x0b53, 0x0a5a, 0x0d56, 0x0c5f, 0x0f55, 0x0e5c,
0x015c, 0x0055, 0x035f, 0x0256, 0x055a, 0x0453, 0x0759, 0x0650,
0x0af0, 0x0bf9, 0x08f3, 0x09fa, 0x0ef6, 0x0fff, 0x0cf5, 0x0dfc,
0x02fc, 0x03f5, 0x00ff, 0x01f6, 0x06fa, 0x07f3, 0x04f9, 0x05f0,
0x0b60, 0x0a69, 0x0963, 0x086a, 0x0f66, 0x0e6f, 0x0d65, 0x0c6c,
0x036c, 0x0265, 0x016f, 0x0066, 0x076a, 0x0663, 0x0569, 0x0460,
0x0ca0, 0x0da9, 0x0ea3, 0x0faa, 0x08a6, 0x09af, 0x0aa5, 0x0bac,
0x04ac, 0x05a5, 0x06af, 0x07a6, 0x00aa, 0x01a3, 0x02a9, 0x03a0,
0x0d30, 0x0c39, 0x0f33, 0x0e3a, 0x0936, 0x083f, 0x0b35, 0x0a3c,
0x053c, 0x0435, 0x073f, 0x0636, 0x013a, 0x0033, 0x0339, 0x0230,
0x0e90, 0x0f99, 0x0c93, 0x0d9a, 0x0a96, 0x0b9f, 0x0895, 0x099c,
0x069c, 0x0795, 0x049f, 0x0596, 0x029a, 0x0393, 0x0099, 0x0190,
0x0f00, 0x0e09, 0x0d03, 0x0c0a, 0x0b06, 0x0a0f, 0x0905, 0x080c,
0x070c, 0x0605, 0x050f, 0x0406, 0x030a, 0x0203, 0x0109, 0x0000,
}
// specify the edges used to create the triangle(s)
// Max number of edges is 15.
var triangleTable = [256][]int{
{},
{0, 8, 3},
{0, 1, 9},
{1, 8, 3, 9, 8, 1},
{1, 2, 10},
{0, 8, 3, 1, 2, 10},
{9, 2, 10, 0, 2, 9},
{2, 8, 3, 2, 10, 8, 10, 9, 8},
{3, 11, 2},
{0, 11, 2, 8, 11, 0},
{1, 9, 0, 2, 3, 11},
{1, 11, 2, 1, 9, 11, 9, 8, 11},
{3, 10, 1, 11, 10, 3},
{0, 10, 1, 0, 8, 10, 8, 11, 10},
{3, 9, 0, 3, 11, 9, 11, 10, 9},
{9, 8, 10, 10, 8, 11},
{4, 7, 8},
{4, 3, 0, 7, 3, 4},
{0, 1, 9, 8, 4, 7},
{4, 1, 9, 4, 7, 1, 7, 3, 1},
{1, 2, 10, 8, 4, 7},
{3, 4, 7, 3, 0, 4, 1, 2, 10},
{9, 2, 10, 9, 0, 2, 8, 4, 7},
{2, 10, 9, 2, 9, 7, 2, 7, 3, 7, 9, 4},
{8, 4, 7, 3, 11, 2},
{11, 4, 7, 11, 2, 4, 2, 0, 4},
{9, 0, 1, 8, 4, 7, 2, 3, 11},
{4, 7, 11, 9, 4, 11, 9, 11, 2, 9, 2, 1},
{3, 10, 1, 3, 11, 10, 7, 8, 4},
{1, 11, 10, 1, 4, 11, 1, 0, 4, 7, 11, 4},
{4, 7, 8, 9, 0, 11, 9, 11, 10, 11, 0, 3},
{4, 7, 11, 4, 11, 9, 9, 11, 10},
{9, 5, 4},
{9, 5, 4, 0, 8, 3},
{0, 5, 4, 1, 5, 0},
{8, 5, 4, 8, 3, 5, 3, 1, 5},
{1, 2, 10, 9, 5, 4},
{3, 0, 8, 1, 2, 10, 4, 9, 5},
{5, 2, 10, 5, 4, 2, 4, 0, 2},
{2, 10, 5, 3, 2, 5, 3, 5, 4, 3, 4, 8},
{9, 5, 4, 2, 3, 11},
{0, 11, 2, 0, 8, 11, 4, 9, 5},
{0, 5, 4, 0, 1, 5, 2, 3, 11},
{2, 1, 5, 2, 5, 8, 2, 8, 11, 4, 8, 5},
{10, 3, 11, 10, 1, 3, 9, 5, 4},
{4, 9, 5, 0, 8, 1, 8, 10, 1, 8, 11, 10},
{5, 4, 0, 5, 0, 11, 5, 11, 10, 11, 0, 3},
{5, 4, 8, 5, 8, 10, 10, 8, 11},
{9, 7, 8, 5, 7, 9},
{9, 3, 0, 9, 5, 3, 5, 7, 3},
{0, 7, 8, 0, 1, 7, 1, 5, 7},
{1, 5, 3, 3, 5, 7},
{9, 7, 8, 9, 5, 7, 10, 1, 2},
{10, 1, 2, 9, 5, 0, 5, 3, 0, 5, 7, 3},
{8, 0, 2, 8, 2, 5, 8, 5, 7, 10, 5, 2},
{2, 10, 5, 2, 5, 3, 3, 5, 7},
{7, 9, 5, 7, 8, 9, 3, 11, 2},
{9, 5, 7, 9, 7, 2, 9, 2, 0, 2, 7, 11},
{2, 3, 11, 0, 1, 8, 1, 7, 8, 1, 5, 7},
{11, 2, 1, 11, 1, 7, 7, 1, 5},
{9, 5, 8, 8, 5, 7, 10, 1, 3, 10, 3, 11},
{5, 7, 0, 5, 0, 9, 7, 11, 0, 1, 0, 10, 11, 10, 0},
{11, 10, 0, 11, 0, 3, 10, 5, 0, 8, 0, 7, 5, 7, 0},
{11, 10, 5, 7, 11, 5},
{10, 6, 5},
{0, 8, 3, 5, 10, 6},
{9, 0, 1, 5, 10, 6},
{1, 8, 3, 1, 9, 8, 5, 10, 6},
{1, 6, 5, 2, 6, 1},
{1, 6, 5, 1, 2, 6, 3, 0, 8},
{9, 6, 5, 9, 0, 6, 0, 2, 6},
{5, 9, 8, 5, 8, 2, 5, 2, 6, 3, 2, 8},
{2, 3, 11, 10, 6, 5},
{11, 0, 8, 11, 2, 0, 10, 6, 5},
{0, 1, 9, 2, 3, 11, 5, 10, 6},
{5, 10, 6, 1, 9, 2, 9, 11, 2, 9, 8, 11},
{6, 3, 11, 6, 5, 3, 5, 1, 3},
{0, 8, 11, 0, 11, 5, 0, 5, 1, 5, 11, 6},
{3, 11, 6, 0, 3, 6, 0, 6, 5, 0, 5, 9},
{6, 5, 9, 6, 9, 11, 11, 9, 8},
{5, 10, 6, 4, 7, 8},
{4, 3, 0, 4, 7, 3, 6, 5, 10},
{1, 9, 0, 5, 10, 6, 8, 4, 7},
{10, 6, 5, 1, 9, 7, 1, 7, 3, 7, 9, 4},
{6, 1, 2, 6, 5, 1, 4, 7, 8},
{1, 2, 5, 5, 2, 6, 3, 0, 4, 3, 4, 7},
{8, 4, 7, 9, 0, 5, 0, 6, 5, 0, 2, 6},
{7, 3, 9, 7, 9, 4, 3, 2, 9, 5, 9, 6, 2, 6, 9},
{3, 11, 2, 7, 8, 4, 10, 6, 5},
{5, 10, 6, 4, 7, 2, 4, 2, 0, 2, 7, 11},
{0, 1, 9, 4, 7, 8, 2, 3, 11, 5, 10, 6},
{9, 2, 1, 9, 11, 2, 9, 4, 11, 7, 11, 4, 5, 10, 6},
{8, 4, 7, 3, 11, 5, 3, 5, 1, 5, 11, 6},
{5, 1, 11, 5, 11, 6, 1, 0, 11, 7, 11, 4, 0, 4, 11},
{0, 5, 9, 0, 6, 5, 0, 3, 6, 11, 6, 3, 8, 4, 7},
{6, 5, 9, 6, 9, 11, 4, 7, 9, 7, 11, 9},
{10, 4, 9, 6, 4, 10},
{4, 10, 6, 4, 9, 10, 0, 8, 3},
{10, 0, 1, 10, 6, 0, 6, 4, 0},
{8, 3, 1, 8, 1, 6, 8, 6, 4, 6, 1, 10},
{1, 4, 9, 1, 2, 4, 2, 6, 4},
{3, 0, 8, 1, 2, 9, 2, 4, 9, 2, 6, 4},
{0, 2, 4, 4, 2, 6},
{8, 3, 2, 8, 2, 4, 4, 2, 6},
{10, 4, 9, 10, 6, 4, 11, 2, 3},
{0, 8, 2, 2, 8, 11, 4, 9, 10, 4, 10, 6},
{3, 11, 2, 0, 1, 6, 0, 6, 4, 6, 1, 10},
{6, 4, 1, 6, 1, 10, 4, 8, 1, 2, 1, 11, 8, 11, 1},
{9, 6, 4, 9, 3, 6, 9, 1, 3, 11, 6, 3},
{8, 11, 1, 8, 1, 0, 11, 6, 1, 9, 1, 4, 6, 4, 1},
{3, 11, 6, 3, 6, 0, 0, 6, 4},
{6, 4, 8, 11, 6, 8},
{7, 10, 6, 7, 8, 10, 8, 9, 10},
{0, 7, 3, 0, 10, 7, 0, 9, 10, 6, 7, 10},
{10, 6, 7, 1, 10, 7, 1, 7, 8, 1, 8, 0},
{10, 6, 7, 10, 7, 1, 1, 7, 3},
{1, 2, 6, 1, 6, 8, 1, 8, 9, 8, 6, 7},
{2, 6, 9, 2, 9, 1, 6, 7, 9, 0, 9, 3, 7, 3, 9},
{7, 8, 0, 7, 0, 6, 6, 0, 2},
{7, 3, 2, 6, 7, 2},
{2, 3, 11, 10, 6, 8, 10, 8, 9, 8, 6, 7},
{2, 0, 7, 2, 7, 11, 0, 9, 7, 6, 7, 10, 9, 10, 7},
{1, 8, 0, 1, 7, 8, 1, 10, 7, 6, 7, 10, 2, 3, 11},
{11, 2, 1, 11, 1, 7, 10, 6, 1, 6, 7, 1},
{8, 9, 6, 8, 6, 7, 9, 1, 6, 11, 6, 3, 1, 3, 6},
{0, 9, 1, 11, 6, 7},
{7, 8, 0, 7, 0, 6, 3, 11, 0, 11, 6, 0},
{7, 11, 6},
{7, 6, 11},
{3, 0, 8, 11, 7, 6},
{0, 1, 9, 11, 7, 6},
{8, 1, 9, 8, 3, 1, 11, 7, 6},
{10, 1, 2, 6, 11, 7},
{1, 2, 10, 3, 0, 8, 6, 11, 7},
{2, 9, 0, 2, 10, 9, 6, 11, 7},
{6, 11, 7, 2, 10, 3, 10, 8, 3, 10, 9, 8},
{7, 2, 3, 6, 2, 7},
{7, 0, 8, 7, 6, 0, 6, 2, 0},
{2, 7, 6, 2, 3, 7, 0, 1, 9},
{1, 6, 2, 1, 8, 6, 1, 9, 8, 8, 7, 6},
{10, 7, 6, 10, 1, 7, 1, 3, 7},
{10, 7, 6, 1, 7, 10, 1, 8, 7, 1, 0, 8},
{0, 3, 7, 0, 7, 10, 0, 10, 9, 6, 10, 7},
{7, 6, 10, 7, 10, 8, 8, 10, 9},
{6, 8, 4, 11, 8, 6},
{3, 6, 11, 3, 0, 6, 0, 4, 6},
{8, 6, 11, 8, 4, 6, 9, 0, 1},
{9, 4, 6, 9, 6, 3, 9, 3, 1, 11, 3, 6},
{6, 8, 4, 6, 11, 8, 2, 10, 1},
{1, 2, 10, 3, 0, 11, 0, 6, 11, 0, 4, 6},
{4, 11, 8, 4, 6, 11, 0, 2, 9, 2, 10, 9},
{10, 9, 3, 10, 3, 2, 9, 4, 3, 11, 3, 6, 4, 6, 3},
{8, 2, 3, 8, 4, 2, 4, 6, 2},
{0, 4, 2, 4, 6, 2},
{1, 9, 0, 2, 3, 4, 2, 4, 6, 4, 3, 8},
{1, 9, 4, 1, 4, 2, 2, 4, 6},
{8, 1, 3, 8, 6, 1, 8, 4, 6, 6, 10, 1},
{10, 1, 0, 10, 0, 6, 6, 0, 4},
{4, 6, 3, 4, 3, 8, 6, 10, 3, 0, 3, 9, 10, 9, 3},
{10, 9, 4, 6, 10, 4},
{4, 9, 5, 7, 6, 11},
{0, 8, 3, 4, 9, 5, 11, 7, 6},
{5, 0, 1, 5, 4, 0, 7, 6, 11},
{11, 7, 6, 8, 3, 4, 3, 5, 4, 3, 1, 5},
{9, 5, 4, 10, 1, 2, 7, 6, 11},
{6, 11, 7, 1, 2, 10, 0, 8, 3, 4, 9, 5},
{7, 6, 11, 5, 4, 10, 4, 2, 10, 4, 0, 2},
{3, 4, 8, 3, 5, 4, 3, 2, 5, 10, 5, 2, 11, 7, 6},
{7, 2, 3, 7, 6, 2, 5, 4, 9},
{9, 5, 4, 0, 8, 6, 0, 6, 2, 6, 8, 7},
{3, 6, 2, 3, 7, 6, 1, 5, 0, 5, 4, 0},
{6, 2, 8, 6, 8, 7, 2, 1, 8, 4, 8, 5, 1, 5, 8},
{9, 5, 4, 10, 1, 6, 1, 7, 6, 1, 3, 7},
{1, 6, 10, 1, 7, 6, 1, 0, 7, 8, 7, 0, 9, 5, 4},
{4, 0, 10, 4, 10, 5, 0, 3, 10, 6, 10, 7, 3, 7, 10},
{7, 6, 10, 7, 10, 8, 5, 4, 10, 4, 8, 10},
{6, 9, 5, 6, 11, 9, 11, 8, 9},
{3, 6, 11, 0, 6, 3, 0, 5, 6, 0, 9, 5},
{0, 11, 8, 0, 5, 11, 0, 1, 5, 5, 6, 11},
{6, 11, 3, 6, 3, 5, 5, 3, 1},
{1, 2, 10, 9, 5, 11, 9, 11, 8, 11, 5, 6},
{0, 11, 3, 0, 6, 11, 0, 9, 6, 5, 6, 9, 1, 2, 10},
{11, 8, 5, 11, 5, 6, 8, 0, 5, 10, 5, 2, 0, 2, 5},
{6, 11, 3, 6, 3, 5, 2, 10, 3, 10, 5, 3},
{5, 8, 9, 5, 2, 8, 5, 6, 2, 3, 8, 2},
{9, 5, 6, 9, 6, 0, 0, 6, 2},
{1, 5, 8, 1, 8, 0, 5, 6, 8, 3, 8, 2, 6, 2, 8},
{1, 5, 6, 2, 1, 6},
{1, 3, 6, 1, 6, 10, 3, 8, 6, 5, 6, 9, 8, 9, 6},
{10, 1, 0, 10, 0, 6, 9, 5, 0, 5, 6, 0},
{0, 3, 8, 5, 6, 10},
{10, 5, 6},
{11, 5, 10, 7, 5, 11},
{11, 5, 10, 11, 7, 5, 8, 3, 0},
{5, 11, 7, 5, 10, 11, 1, 9, 0},
{10, 7, 5, 10, 11, 7, 9, 8, 1, 8, 3, 1},
{11, 1, 2, 11, 7, 1, 7, 5, 1},
{0, 8, 3, 1, 2, 7, 1, 7, 5, 7, 2, 11},
{9, 7, 5, 9, 2, 7, 9, 0, 2, 2, 11, 7},
{7, 5, 2, 7, 2, 11, 5, 9, 2, 3, 2, 8, 9, 8, 2},
{2, 5, 10, 2, 3, 5, 3, 7, 5},
{8, 2, 0, 8, 5, 2, 8, 7, 5, 10, 2, 5},
{9, 0, 1, 5, 10, 3, 5, 3, 7, 3, 10, 2},
{9, 8, 2, 9, 2, 1, 8, 7, 2, 10, 2, 5, 7, 5, 2},
{1, 3, 5, 3, 7, 5},
{0, 8, 7, 0, 7, 1, 1, 7, 5},
{9, 0, 3, 9, 3, 5, 5, 3, 7},
{9, 8, 7, 5, 9, 7},
{5, 8, 4, 5, 10, 8, 10, 11, 8},
{5, 0, 4, 5, 11, 0, 5, 10, 11, 11, 3, 0},
{0, 1, 9, 8, 4, 10, 8, 10, 11, 10, 4, 5},
{10, 11, 4, 10, 4, 5, 11, 3, 4, 9, 4, 1, 3, 1, 4},
{2, 5, 1, 2, 8, 5, 2, 11, 8, 4, 5, 8},
{0, 4, 11, 0, 11, 3, 4, 5, 11, 2, 11, 1, 5, 1, 11},
{0, 2, 5, 0, 5, 9, 2, 11, 5, 4, 5, 8, 11, 8, 5},
{9, 4, 5, 2, 11, 3},
{2, 5, 10, 3, 5, 2, 3, 4, 5, 3, 8, 4},
{5, 10, 2, 5, 2, 4, 4, 2, 0},
{3, 10, 2, 3, 5, 10, 3, 8, 5, 4, 5, 8, 0, 1, 9},
{5, 10, 2, 5, 2, 4, 1, 9, 2, 9, 4, 2},
{8, 4, 5, 8, 5, 3, 3, 5, 1},
{0, 4, 5, 1, 0, 5},
{8, 4, 5, 8, 5, 3, 9, 0, 5, 0, 3, 5},
{9, 4, 5},
{4, 11, 7, 4, 9, 11, 9, 10, 11},
{0, 8, 3, 4, 9, 7, 9, 11, 7, 9, 10, 11},
{1, 10, 11, 1, 11, 4, 1, 4, 0, 7, 4, 11},
{3, 1, 4, 3, 4, 8, 1, 10, 4, 7, 4, 11, 10, 11, 4},
{4, 11, 7, 9, 11, 4, 9, 2, 11, 9, 1, 2},
{9, 7, 4, 9, 11, 7, 9, 1, 11, 2, 11, 1, 0, 8, 3},
{11, 7, 4, 11, 4, 2, 2, 4, 0},
{11, 7, 4, 11, 4, 2, 8, 3, 4, 3, 2, 4},
{2, 9, 10, 2, 7, 9, 2, 3, 7, 7, 4, 9},
{9, 10, 7, 9, 7, 4, 10, 2, 7, 8, 7, 0, 2, 0, 7},
{3, 7, 10, 3, 10, 2, 7, 4, 10, 1, 10, 0, 4, 0, 10},
{1, 10, 2, 8, 7, 4},
{4, 9, 1, 4, 1, 7, 7, 1, 3},
{4, 9, 1, 4, 1, 7, 0, 8, 1, 8, 7, 1},
{4, 0, 3, 7, 4, 3},
{4, 8, 7},
{9, 10, 8, 10, 11, 8},
{3, 0, 9, 3, 9, 11, 11, 9, 10},
{0, 1, 10, 0, 10, 8, 8, 10, 11},
{3, 1, 10, 11, 3, 10},
{1, 2, 11, 1, 11, 9, 9, 11, 8},
{3, 0, 9, 3, 9, 11, 1, 2, 9, 2, 11, 9},
{0, 2, 11, 8, 0, 11},
{3, 2, 11},
{2, 3, 8, 2, 8, 10, 10, 8, 9},
{9, 10, 2, 0, 9, 2},
{2, 3, 8, 2, 8, 10, 0, 1, 8, 1, 10, 8},
{1, 10, 2},
{1, 3, 8, 9, 1, 8},
{0, 9, 1},
{0, 3, 8},
{},
}