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Basic Mutation

This tutorial explores the surface mesh data structure in geometry-central, via a simple example which splits and flips edges to perform 4:1 subidivision of a triangle mesh.

View full, runnable source code in the tutorial repository.

Basic setup

To begin we include headers, bring in namespaces, See the first tutorial for more info on these steps.

#include "geometrycentral/surface/manifold_surface_mesh.h"
#include "geometrycentral/surface/meshio.h"
#include "geometrycentral/surface/surface_mesh.h"

#include "polyscope/polyscope.h"
#include "polyscope/surface_mesh.h"

using namespace geometrycentral;
using namespace geometrycentral::surface;

Again as before, load an input mesh from file, and visualize it using Polyscope. Here, we’ll require that the mesh be manifold.

std::unique_ptr<ManifoldSurfaceMesh> mesh;
std::unique_ptr<VertexPositionGeometry> geometry;
std::tie(mesh, geometry) = readManifoldSurfaceMesh(args::get(inputFilename));

polyscope::registerSurfaceMesh("input mesh", geometry->vertexPositions,

// call polyscope::show(); to inspect the mesh at this point

Now, we’ll evaluate a 4:1 sudivision via the following algorithm:

  • split each edge and connect to opposite vertices
  • flip any new edge connecting a new to old vertex

First, we’ll create two MeshData<> containers. They associate data with mesh elements. We initialize them both with true values, because initially all vertices (resp. edges) are original. We’ll also track a list of edges to flip in the second phase.

VertexData<bool> isOrigVert(*mesh, true);
EdgeData<bool> isOrigEdge(*mesh, true);
std::vector<Edge> toFlip;

Now our main algorithm loop (commented inline).

for (Edge e : mesh->edges()) { // loop over all edges
  if (!isOrigEdge[e]) continue; // don't keep processing new edges

  // gather both vertices incident on the edge, and their positions
  Vertex oldA = e.halfedge().tipVertex();
  Vertex oldB = e.halfedge().tailVertex();
  Vector3 oldAPos = geometry->vertexPositions[oldA];
  Vector3 oldBPos = geometry->vertexPositions[oldB];

  // split the edge
  Vertex newV = mesh->splitEdgeTriangular(e).vertex();
  isOrigVert[newV] = false;

  // position the new vertex
  Vector3 newPos = 0.5 * (oldAPos + oldBPos);
  geometry->vertexPositions[newV] = newPos;

  // iterate through the edges incident on the new vertex
  for (Edge e : newV.adjacentEdges()) {
    isOrigEdge[e] = false;                  // mark the new edges
    Vertex otherV = e.otherVertex(newV);    // other side of edge

    // if this is a new edge between an old an new vertex, save for flipping
    if (isOrigVert[otherV] && otherV != oldA && otherV != oldB) {

and finally, we can actually flip the edges to complete the second phase

for (Edge e : toFlip) {

we now register this new mesh with geometry-central

auto* psMesh =
    polyscope::registerSurfaceMesh("subdiv mesh", 


and as expected get the subdivided mesh.

Before: before subdivison

After: after subdivison