3D Delaunay triangulation set constraint on free boundary

I am trying to create and visualise the envelope of an unstructured point cloud that has been measured using a structured approach:

There measurements are made in spherical coordinates and map a half sphere. From that I am able to find the connectivity of the free boundary. I would like to apply that constraint on the measurement (the half sphere is deformed by the measured distance)

I have scratched the script below but I cannot figure out how to apply the constraint.

I am also trying other methods such as alphaShape, but without success.

clear all;close all;clc

load myData.mat
% sphere 2 cart 
[x,y,z] = createPointCloud(alphaAngle, betaAngle,Distance);  
% sphere 2 cart
[xU,yU,zU] = createPointCloud(alphaAngle, betaAngle,ones(length(alphaAngle),1));

figure
ax1(1) = subplot(2,2,1); 
scatter3(ax1(1),xU,yU,zU,10,'blue',MarkerFaceColor='flat');
set(gca,'DataAspectRatio',[1 1 1]);xlabel('X (m)');ylabel('Y (m)');zlabel('Z (m)');view(3);

ax1(2) = subplot(2,2,2); 
scatter3(ax1(2),x,y,z,10,'blue',MarkerFaceColor='flat');
set(gca,'DataAspectRatio',[1 1 1]);xlabel('X (m)');ylabel('Y (m)');zlabel('Z (m)');view(3);


% triangulation of the half sphere
DTU = delaunayTriangulation(xU,yU,zU);

ax1(3) = subplot(2,2,3);
scatter3(ax1(3),xU,yU,zU,10,'blue',MarkerFaceColor='flat'); hold on
trisurf(DTU.freeBoundary,DTU.Points(:,1),DTU.Points(:,2),DTU.Points(:,3),0.05,'FaceAlpha',0.05,'FaceColor','blue');

% triangulation of the measurement point
DT = delaunayTriangulation(x,y,z);
ax1(4) = subplot(2,2,4);
scatter3(ax1(4),x,y,z,10,'blue',MarkerFaceColor='flat'); hold on
trisurf(DT.freeBoundary,DT.Points(:,1),DT.Points(:,2),DT.Points(:,3),0.05,'FaceAlpha',0.05,'FaceColor','blue');
linkprop(ax1, {'View'})

% get constraint to apply on the 3D triangulation
FU = DTU.freeBoundary;

% how to apply the constraint to the measurement data?


function [x,y,z] = createPointCloud(alphaAngle, betaAngle,Distance)    
    rho = Distance;
    phi = pi/2-alphaAngle;
    theta = betaAngle;
   
    x = rho .* sin(phi) .* cos(theta);
    y = rho .* sin(phi) .* sin(theta);
    z = rho .* cos(phi);
    
    x = [x',x']';
    y = [y',-y']';
    z = [z',z']';
    input= [x y z] ;
    output = unique(input,'rows');
    x = output(:,1);
    y = output(:,2);
    z = output(:,3);
end

Answer 1

[Prathamesh]

Hi Sylvain, I understand that you are trying to visualise the envelope of an unstructured point cloud. The measurements are made in spherical coordinates and map a half sphere. The connectivity of the free boundary has been successfully found. So now you want to apply constraint on the measurement.

Solution:

1. Convert spherical to Cartesian coordinates for both datasets for 3D visualization.
2. Use Delaunay triangulation on undeformed data to find its free boundary.
3. Visualize the undeformed half-sphere and its boundary as a reference.
4. Use ‘alphaShape’ to create a smooth outline around deformed data; adjust ‘alphaVal’ for detail.

While direct application of boundary constraints to the deformed data is complex, the alpha shape provides a smooth approximation of the envelope, respecting the original structure's general shape.

Below is the code with comments provided where necessary.

clear all; close all; clc;
load myData.mat

% Convert spherical coordinates to Cartesian coordinates
[x, y, z] = createPointCloud(alphaAngle, betaAngle, Distance);  
[xU, yU, zU] = createPointCloud(alphaAngle, betaAngle, ones(size(Distance)));

% Triangulate the undeformed point cloud
DTU = delaunayTriangulation(xU, yU, zU);
FU = DTU.freeBoundary;  % Extract free boundary

% Visualize the undeformed point cloud and its boundary
figure;
subplot(1, 2, 1);
trisurf(FU, xU, yU, zU, 'FaceColor', 'cyan', 'EdgeColor', 'none', 'FaceAlpha', 0.5);
hold on;
scatter3(xU, yU, zU, 10, 'blue', 'filled');
title('Undeformed Half-Sphere');
xlabel('X'); ylabel('Y'); zlabel('Z');
axis equal; view(3);

% Visualize the deformed point cloud
subplot(1, 2, 2);
scatter3(x, y, z, 10, 'red', 'filled');
title('Deformed Measurement Points');
xlabel('X'); ylabel('Y'); zlabel('Z');
axis equal; view(3);

% Create an alpha shape to visualize the envelope of the deformed data
alphaVal = 1.5;  % Adjust this value as needed
shp = alphaShape(x, y, z, alphaVal);

% Plot the alpha shape
figure;
plot(shp, 'FaceColor', 'green', 'EdgeColor', 'none', 'FaceAlpha', 0.5);
hold on;
scatter3(x, y, z, 10, 'red', 'filled');
title('Alpha Shape of Deformed Data');
xlabel('X'); ylabel('Y'); zlabel('Z');
axis equal; view(3);

function [x, y, z] = createPointCloud(alphaAngle, betaAngle, Distance)
    % Convert spherical coordinates (alpha, beta, Distance) to Cartesian coordinates (x, y, z)
    rho = Distance;
    phi = pi/2 - alphaAngle;
    theta = betaAngle;

    x = rho .* sin(phi) .* cos(theta);
    y = rho .* sin(phi) .* sin(theta);
    z = rho .* cos(phi);

    % Ensure unique points
    input = [x, y, z];
    output = unique(input, 'rows');
    x = output(:, 1);
    y = output(:, 2);
    z = output(:, 3);
end

screenshot of my output

[Sylvain]

Thank you @Prathamesh, Unfortunatelly, this does not work as expected. The alphaShape has no constratraint on the connectivity compared to the delaunay method. Hence trying to find the connectivity of the half sphere is pointless, unless you can use this latest data.

I have created an app with sliders to change dynamically the alpha radius, hole threshold and region threshold for the alpha shape but without succeess. I end up having the same issue: the surface can have unfilled both holes and dual walls.

I think that sticking to the delaunay triangulation would be wiser, as long as I can transfer the connectivity from the half sphere to the deformed half sphere.