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This example shows how to create, edit, and query Delaunay triangulations using the delaunayTriangulation class. The Delaunay triangulation is the most widely used triangulation in scientific computing. The properties associated with the triangulation provide a basis for solving a variety of geometric problems. Construction of constrained Delaunay triangulations is also shown, together with an applications covering medial axis computation and mesh morphing.

This example shows you how to compute a 2D Delaunay triangulation and how to plot the triangulation together with the vertex and triangle labels.

x = rand(10,1); y = rand(10,1); dt = delaunayTriangulation(x,y)

dt = delaunayTriangulation with properties: Points: [10x2 double] ConnectivityList: [11x3 double] Constraints: []

triplot(dt); % % Display the Vertex and Triangle labels on the plot hold on vxlabels = arrayfun(@(n) {sprintf('P%d', n)}, (1:10)'); Hpl = text(x, y, vxlabels, 'FontWeight', 'bold', 'HorizontalAlignment',... 'center', 'BackgroundColor', 'none'); ic = incenter(dt); numtri = size(dt,1); trilabels = arrayfun(@(x) {sprintf('T%d', x)}, (1:numtri)'); Htl = text(ic(:,1), ic(:,2), trilabels, 'FontWeight', 'bold', ... 'HorizontalAlignment', 'center', 'Color', 'blue'); hold off

This example shows you how to compute a 3D Delaunay triangulation and how to plot the triangulation.

X = rand(10,3)

```
X =
0.6557 0.7060 0.4387
0.0357 0.0318 0.3816
0.8491 0.2769 0.7655
0.9340 0.0462 0.7952
0.6787 0.0971 0.1869
0.7577 0.8235 0.4898
0.7431 0.6948 0.4456
0.3922 0.3171 0.6463
0.6555 0.9502 0.7094
0.1712 0.0344 0.7547
```

dt = delaunayTriangulation(X)

dt = delaunayTriangulation with properties: Points: [10x3 double] ConnectivityList: [22x4 double] Constraints: []

tetramesh(dt, 'FaceColor', 'cyan');

% To display large tetrahedral meshes use the convexHull method to % compute the boundary triangulation and plot it using trisurf. % For example; % triboundary = convexHull(dt) % trisurf(triboundary, X(:,1), X(:,2), X(:,3), 'FaceColor', 'cyan')

There are two ways to access the triangulation data structure. One way is via the Triangulation property, the other way is using indexing.

Create a 2D Delaunay triangulation from 10 random points.

X = rand(10,2)

```
X =
0.2760 0.7513
0.6797 0.2551
0.6551 0.5060
0.1626 0.6991
0.1190 0.8909
0.4984 0.9593
0.9597 0.5472
0.3404 0.1386
0.5853 0.1493
0.2238 0.2575
```

dt = delaunayTriangulation(X)

dt = delaunayTriangulation with properties: Points: [10x2 double] ConnectivityList: [12x3 double] Constraints: []

```
% The triangulation datastructure is;
dt.ConnectivityList
```

```
ans =
4 10 1
3 8 9
10 4 5
4 1 5
1 6 5
3 10 8
1 3 6
2 9 7
6 3 7
1 10 3
```

% Indexing is a shorthand way to query the triangulation. The format is % dt(i, j) where j is the j'th vertex of the i'th triangle, standard % indexing rules apply. % The triangulation datastructure is dt(:,:)

```
ans =
4 10 1
3 8 9
10 4 5
4 1 5
1 6 5
3 10 8
1 3 6
2 9 7
6 3 7
1 10 3
```

The second triangle is;

dt(2,:)

```
ans =
3 8 9
```

The third vertex of the second triangle is;

dt(2,3)

ans = 9

The first three triangles;

dt(1:3,:)

```
ans =
4 10 1
3 8 9
10 4 5
```

This example shows you how to use index-based subscripting to insert or remove points. It is more efficient to edit a delaunayTriangulation to make minor modifications as opposed to recreating a new delaunayTriangulation from scratch, this is especially true if the dataset is large.

% Construct a Delaunay triangulation from % 10 random points within a unit square x = rand(10,1); y = rand(10,1); dt = delaunayTriangulation(x,y)

dt = delaunayTriangulation with properties: Points: [10x2 double] ConnectivityList: [12x3 double] Constraints: []

```
% Insert 5 additional random points
dt.Points(end+(1:5),:) = rand(5,2)
```

dt = delaunayTriangulation with properties: Points: [15x2 double] ConnectivityList: [21x3 double] Constraints: []

Replace the fifth point

dt.Points(5,:) = [0, 0]

dt = delaunayTriangulation with properties: Points: [15x2 double] ConnectivityList: [21x3 double] Constraints: []

Remove the fourth point

dt.Points(4,:) = []

dt = delaunayTriangulation with properties: Points: [14x2 double] ConnectivityList: [19x3 double] Constraints: []

This example shows you how to create a simple constrained Delaunay triangulation and illustrates the effect of the constraints.

X = [0 0; 16 0; 16 2; 2 2; 2 3; 8 3; 8 5; 0 5]; C = [1 2; 2 3; 3 4; 4 5; 5 6; 6 7; 7 8; 8 1]; dt = delaunayTriangulation(X, C); subplot(2,1,1); triplot(dt); axis([-1 17 -1 6]); xlabel('Constrained Delaunay triangulation', 'fontweight','b'); % Plot the constrained edges in red hold on; plot(X(C'),X(C'+size(X,1)),'-r', 'LineWidth', 2); hold off; % Now delete the constraints and plot the unconstrained Delaunay dt.Constraints = []; subplot(2,1,2); triplot(dt); axis([-1 17 -1 6]); xlabel('Unconstrained Delaunay triangulation', 'fontweight','b');

Load a map of the perimeter of the conterminous United States. Construct a constrained Delaunay triangulation representing the polygon. This triangulation spans a domain that is bounded by the convex hull of the set of points. Filter out the triangles that are within the domain of the polygon and plot them. Note: The dataset contains duplicate datapoints; that is two or more datapoints have the same location. The duplicate points are rejected and the delaunayTriangulation reformats the constraints accordingly.

clf load usapolygon % Define an edge constraint between two successive % points that make up the polygonal boundary. nump = numel(uslon); C = [(1:(nump-1))' (2:nump)'; nump 1]; dt = delaunayTriangulation(uslon, uslat, C);

Warning: Duplicate data points have been detected and removed. The Triangulation indices and constraints are defined with respect to the unique set of points in delaunayTriangulation.

Warning: Intersecting edge constraints have been split, this may have added new points into the triangulation.

io = dt.isInterior(); patch('faces',dt(io,:), 'vertices', dt.Points, 'FaceColor','r'); axis equal; axis([-130 -60 20 55]); xlabel('Constrained Delaunay Triangulation of usapolygon', 'fontweight','b');

This example highlights the use of a Delaunay triangulation to reconstruct a polygonal boundary from a cloud of points. The reconstruction is based on the elegant Crust algorithm.

Reference: N. Amenta, M. Bern, and D. Eppstein. The crust and the beta-skeleton: combinatorial curve reconstruction. Graphical Models and Image Processing, 60:125-135, 1998.

```
% Create a set of points representing the point cloud
numpts = 192;
t = linspace( -pi, pi, numpts+1 )';
t(end) = [];
r = 0.1 + 5*sqrt( cos( 6*t ).^2 + (0.7).^2 );
x = r.*cos(t);
y = r.*sin(t);
ri = randperm(numpts);
x = x(ri);
y = y(ri);
```

```
% Construct a Delaunay Triangulation of the point set.
dt = delaunayTriangulation(x,y);
tri = dt(:,:);
```

% Insert the location of the Voronoi vertices into the existing % triangulation V = dt.voronoiDiagram(); % Remove the infinite vertex V(1,:) = []; numv = size(V,1); dt.Points(end+(1:numv),:) = V;

Warning: Duplicate data points have been detected and removed. The Triangulation indices are defined with respect to the unique set of points in delaunayTriangulation.

% The Delaunay edges that connect pairs of sample points represent the % boundary. delEdges = dt.edges(); validx = delEdges(:,1) <= numpts; validy = delEdges(:,2) <= numpts; boundaryEdges = delEdges((validx & validy), :)'; xb = x(boundaryEdges); yb = y(boundaryEdges); clf; triplot(tri,x,y); axis equal; hold on; plot(x,y,'*r'); plot(xb,yb, '-r'); xlabel('Curve reconstruction from a point cloud', 'fontweight','b'); hold off;

This example shows how to create an approximate Medial Axis of a polygonal domain using a constrained Delaunay triangulation. The Medial Axis of a polygon is defined by the locus of the center of a maximal disk within the polygon interior.

% Construct a constrained Delaunay triangulation of a sample of points % on the domain boundary. load trimesh2d dt = delaunayTriangulation(x,y,Constraints); inside = dt.isInterior();

% Construct a triangulation to represent the domain triangles. tr = triangulation(dt(inside, :), dt.Points); % Construct a set of edges that join the circumcenters of neighboring % triangles; the additional logic constructs a unique set of such edges. numt = size(tr,1); T = (1:numt)'; neigh = tr.neighbors(); cc = tr.circumcenter(); xcc = cc(:,1); ycc = cc(:,2); idx1 = T < neigh(:,1); idx2 = T < neigh(:,2); idx3 = T < neigh(:,3); neigh = [T(idx1) neigh(idx1,1); T(idx2) neigh(idx2,2); T(idx3) neigh(idx3,3)]';

% Plot the domain triangles in green, the domain boundary in blue and the % medial axis in red. clf; triplot(tr, 'g'); hold on; plot(xcc(neigh), ycc(neigh), '-r', 'LineWidth', 1.5); axis([-10 310 -10 310]); axis equal; plot(x(Constraints'),y(Constraints'), '-b', 'LineWidth', 1.5); xlabel('Medial Axis of a Polygonal Domain', 'fontweight','b'); hold off;

This example shows how to morph a mesh of a 2D domain to accommodate a modification to the domain boundary.

**Step 1:** Load the data. The mesh to be morphed is defined by trife, xfe, yfe, which is a triangulation in face-vertex format.

load trimesh2d clf; triplot(trife,xfe,yfe); axis equal; axis([-10 310 -10 310]); axis equal; xlabel('Initial Mesh', 'fontweight','b');

**Step 2:** Construct a background triangulation - a Constrained Delaunay triangulation of the set of points representing the mesh boundary. For each vertex of the mesh, compute a descriptor that defines it's location with respect to the background triangulation. The descriptor is the enclosing triangle together with the barycentric coordinates with respect to that triangle.

dt = delaunayTriangulation(x,y,Constraints); clf; triplot(dt); axis equal; axis([-10 310 -10 310]); axis equal; xlabel('Background Triangulation', 'fontweight','b');

descriptors.tri = pointLocation(dt,xfe, yfe); descriptors.baryCoords = cartesianToBarycentric(dt,descriptors.tri, [xfe yfe]);

**Step 3:** Edit the background triangulation to incorporate the desired modification to the domain boundary.

cc1 = [210 90]; circ1 = (143:180)'; x(circ1) = (x(circ1)-cc1(1))*0.6 + cc1(1); y(circ1) = (y(circ1)-cc1(2))*0.6 + cc1(2); tr = triangulation(dt(:,:),x,y); clf; triplot(tr); axis([-10 310 -10 310]); axis equal; xlabel('Edited Background Triangulation - Hole Size Reduced', 'fontweight','b');

**Step 4:** Convert the descriptors back to Cartesian coordinates using the deformed background triangulation as a basis for evaluation.

Xnew = barycentricToCartesian(tr,descriptors.tri, descriptors.baryCoords); tr = triangulation(trife, Xnew); clf; triplot(tr); axis([-10 310 -10 310]); axis equal; xlabel('Morphed Mesh', 'fontweight','b');

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