## Archive for February, 2017

### Quad meshing in matlab using gptoolbox

Friday, February 17th, 2017

This morning I played around with a very simple way of quad meshing a triangle mesh. This is more or less the “ideal pipeline” for quad meshing via parameterization. It will likely only work for very simple meshes. I’m pleasantly surprised that it works at all.

% Load a mesh with boundary and no topological (donut) holes
V = V*axisangle2matrix([1 0 0],-pi/2);
% Construct the LSCM matrix
[~,Q] = lscm(V,F,[],[]);
M = massmatrix(V,F);
% Use Fiedler vector as parameterization
[EV,ED] = eigs(Q,repdiag(M,2),5,'sm');
U = reshape(EV(:,end-3),[],2);
% try to find "canonical" rotation
[sU,sS,sV] = svd(U'*U);
U = U*sU;
% Lay down a grid in 2D
h = ((max(U(:,1))-min(U(:,1)))/(160-2));
[X,Y] = meshgrid(min(U(:,1))-h:h:max(U(:,1))+h,min(U(:,2))-h:h:max(U(:,2))+h);
% Find all quads that are strictly inside the parameterized mesh
[QQ,QU] = surf2patch(X,Y,0*Y);QU = QU(:,1:2);
[sqrD,I,C] = signed_distance([QU QU(:,1)*0],[U U(:,1)*0],F);
QQ = QQ(all(abs(sqrD(QQ))<1e-10,2),:);
[QU,I] = remove_unreferenced(QU,QQ);
QQ = fliplr(I(QQ));
% Snap boundary vertices of quad mesh to boundary of mesh, and
% solve little Dirichlet problem to diffuse the displacements smoothly
QF = [QQ(:,[1 2 3]);QQ(:,[1 3 4])];
L = cotmatrix(QU,QF);
b = unique(outline(QQ));
[~,~,QU(b,:)] = point_mesh_squared_distance(QU(b,:),U,outline(F));
% Locate each quad mesh vertex in the parameterized mesh
[sqrD,I,C] = signed_distance([QU QU(:,1)*0],[U U(:,1)*0],F);
B = barycentric_coordinates(C(:,1:2),U(F(I,1),:),U(F(I,2),:),U(F(I,3),:));
% Interpolate 3D positions at quad mesh vertex locations
QV = V(F(I,1),:).*B(:,1) + V(F(I,2),:).*B(:,2) + V(F(I,3),:).*B(:,3);


To create the image above use:

clf;
hold on;
tq = tsurf(QQ,QV,'FaceVertexCData',repmat(blue,size(QV,1),1),'EdgeColor','none',fsoft,fphong);
tt = tsurf(F,V-[0.8 0 0],'FaceVertexCData',repmat(1-0.8*(1-blue),size(V,1),1),'EdgeAlpha',0.5,fsoft,fphong);
axis equal;
camproj('persp');
view(-38,17);
drawn;
camlight;
plot_edges(QV,[QQ(:,2:3);QQ(:,[4 1])],'Color',orange);
plot_edges(QV,[QQ(:,1:2);QQ(:,3:4)],'w');hold off;
l = light('Position',[-10 0 10],'Style','infinite');