# Tutorial for SurfaceIntersection Function

By Jarek Tuszynski (jaroslaw.w.tuszynski@leidos.com)

Triangle/triangle intersection using the algorithm proposed by Tomas Möller (1997), implemented as highly vectorized MATLAB code. The algorithm was expanded to include calculation of the intersection surface.

[intMatrix, intSurface] = SurfaceIntersection(surface1, surface2) calculates the intersection of surfaces 1 and 2. Code can either return just the matrix indicating which face of surface1 intersected with face of surface2, which is calculated using Tomas Moller algorithm, or can also return the actual line or surface of intersection. In case when parts of the surface 1 and 2 lay on the same plane the intersection is a 2D area instead of 1D edge. In such a case the intersection area will be triangulated and intSurface.edges will hold the edges of the triangulation surface and intSurface.faces will hold the faces.

## References

Based on Triangle/triangle intersection test routine by Tomas Möller, 1997. See article "A Fast Triangle-Triangle Intersection Test", Journal of Graphics Tools, 2(2), 1997 http://web.stanford.edu/class/cs277/resources/papers/Moller1997b.pdf http://fileadmin.cs.lth.se/cs/Personal/Tomas_Akenine-Moller/code/opttritri.txt

## Change History

• 2014-11-17 - original version

## Licence

```format compact; % viewing preference
clear variables;
```
```Copyright (c) 2014, Jaroslaw Tuszynski

Redistribution and use in source and binary forms, with or without
modification, are permitted provided that the following conditions are
met:

* Redistributions of source code must retain the above copyright
notice, this list of conditions and the following disclaimer.
* Redistributions in binary form must reproduce the above copyright
notice, this list of conditions and the following disclaimer in
the documentation and/or other materials provided with the distribution

THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE
LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
POSSIBILITY OF SUCH DAMAGE.
```

## Define three surfaces and plot them

```% Create Surface #1
[x,y] = pol2cart((0:2)'*2*pi/3,40);
Surface1.vertices = [0 0 110; x y [0; 0; 0]];
Surface1.faces    = [1 2 3; 1 3 4; 1 4 2; 2 3 4];
% Create Surface #2
TR = triangulation(tet,X);
[Surface2.faces, Surface2.vertices] = freeBoundary(TR);
% Create Surface #3
Surface3.vertices = [x y [30; 30; 30]; y x [60; 60; 60]];
Surface3.faces    = [1:3;4:6];
% Plot them
clf; hold on
S=Surface1; trisurf(S.faces, S.vertices(:,1),S.vertices(:,2),S.vertices(:,3),'FaceAlpha', 0.5, 'FaceColor', 'r');
S=Surface2; trisurf(S.faces, S.vertices(:,1),S.vertices(:,2),S.vertices(:,3),'FaceAlpha', 0.5, 'FaceColor', 'g');
S=Surface3; trisurf(S.faces, S.vertices(:,1),S.vertices(:,2),S.vertices(:,3),'FaceAlpha', 0.5, 'FaceColor', 'b');
view([3 1 1])
axis equal
title ('Three test surfaces')
legend({'#1', '#2', '#3'});
```

## Run SurfaceIntersection and plot the results

Parts of Surface #1 and #2 are on the same plane and the intersection is a 2D area instead of collection of 1D edges

```[intersect12, Surf12] = SurfaceIntersection(Surface1, Surface2);
[intersect13, Surf13] = SurfaceIntersection(Surface1, Surface3);
[intersect23, Surf23] = SurfaceIntersection(Surface3, Surface2);

clf; hold on
S=Surf12; trisurf(S.faces, S.vertices(:,1),S.vertices(:,2),S.vertices(:,3),'EdgeColor', 'r', 'FaceColor', 'r');
S=Surf13; trisurf(S.faces, S.vertices(:,1),S.vertices(:,2),S.vertices(:,3),'EdgeColor', 'g', 'FaceColor', 'g');
S=Surf23; trisurf(S.faces, S.vertices(:,1),S.vertices(:,2),S.vertices(:,3),'EdgeColor', 'b', 'FaceColor', 'b');
title ('Surface/Surface intersections')
legend({'#1/#2', '#1/#3', '#2/#3'});
view([3 1 1])
axis equal
```

## Test SurfaceIntersection on surfaces laying on the same plane (coplanar)

```clf; hold on
% Define surface #1
SurfaceA.vertices = rand([10,2]);
Sa = delaunayTriangulation(SurfaceA.vertices);
SurfaceA.faces = Sa.ConnectivityList;
SurfaceA.vertices(:,3)=0;
% Define surface #2
SurfaceB.vertices = rand([10,2]);
Sb = delaunayTriangulation(SurfaceB.vertices);
SurfaceB.faces = Sb.ConnectivityList;
SurfaceB.vertices(:,3)=0;
% Calculate intersection
[~, SurfAB] = SurfaceIntersection(SurfaceA, SurfaceB);
% Plot all
S=SurfaceA; trisurf(S.faces, S.vertices(:,1),S.vertices(:,2),S.vertices(:,3),'EdgeColor', 'k', 'FaceAlpha', 0.5, 'FaceColor', 'r');
S=SurfaceB; trisurf(S.faces, S.vertices(:,1),S.vertices(:,2),S.vertices(:,3),'EdgeColor', 'k', 'FaceAlpha', 0.5, 'FaceColor', 'b');
S=SurfAB;   trisurf(S.faces, S.vertices(:,1),S.vertices(:,2),S.vertices(:,3),'EdgeColor', 'k', 'FaceAlpha', 0.5, 'FaceColor', 'm');
legend({'Surface #1', 'Surface #2', 'Intersection'});
axis equal
title('Coplanar surfaces with 10 points each')
```

## Test SurfaceIntersection on surfaces laying on the same plane (coplanar)

```clf; hold on
% Define surface #1
SurfaceA.vertices = rand([100,2]);
Sa = delaunayTriangulation(SurfaceA.vertices);
SurfaceA.faces = Sa.ConnectivityList;
SurfaceA.vertices(:,3)=0;
% Define surface #2
SurfaceB.vertices = rand([100,2]);
Sb = delaunayTriangulation(SurfaceB.vertices);
SurfaceB.faces = Sb.ConnectivityList;
SurfaceB.vertices(:,3)=0;
% Calculate intersection
[~, SurfAB] = SurfaceIntersection(SurfaceA, SurfaceB);
% Plot all
S=SurfaceA; trisurf(S.faces, S.vertices(:,1),S.vertices(:,2),S.vertices(:,3),'EdgeColor', 'k', 'FaceAlpha', 0.5, 'FaceColor', 'r');
S=SurfaceB; trisurf(S.faces, S.vertices(:,1),S.vertices(:,2),S.vertices(:,3),'EdgeColor', 'k', 'FaceAlpha', 0.5, 'FaceColor', 'b');
S=SurfAB;   trisurf(S.faces, S.vertices(:,1),S.vertices(:,2),S.vertices(:,3),'EdgeColor', 'k', 'FaceAlpha', 0.5, 'FaceColor', 'm');
legend({'Surface #1', 'Surface #2', 'Intersection'});
axis equal
title('Coplanar surfaces with 100 points each')
```

## Compare runs with and without intersection calculation: test #1

```% Define two surfaces
Surface1 = surf2patch(membrane(1,30),'triangles');
Surface2 = surf2patch(membrane(3,30),'triangles');
% add a bit of perturbation to x and y coordinates of surface 2
Surface2.vertices(:,1:2) = Surface2.vertices(:,1:2) + 1E-3*randn(size(Surface2.vertices(:,1:2)));
fprintf('25%% coplanar faces.\n# faces in surface 1 = %i; # faces in surface 2 = %i\n', ...
size(Surface1.faces,1), size(Surface2.faces,1));
% Plot two surfaces
clf; hold on
subplot(2,1,1)
patch(Surface1, 'FaceColor', 'b','EdgeColor', 'none', 'FaceAlpha', 0.8);
patch(Surface2, 'FaceColor', 'r','EdgeColor', 'none', 'FaceAlpha', 0.8);
title('Two surfaces')
view(-195, 44)
% Run and time the intersection function
tic; [intersect_a] = SurfaceIntersection(Surface1, Surface2);
fprintf('Run time without intersection calculation is %f\n', toc)
tic; [intersect_b, Surf12] = SurfaceIntersection(Surface1, Surface2);
fprintf('Run time with    intersection calculation is %f\n', toc)
% Plot the results
subplot(2,1,2)
S=Surf12; trisurf(S.faces, S.vertices(:,1),S.vertices(:,2),S.vertices(:,3), 'FaceColor', 'g');
view(-195, 44)
title('Surface intersection')
fprintf('Number of face pairs is %i and number of intersections is %i\n', ...
size(Surface1.faces,1)*size(Surface2.faces,1), nnz(intersect_a));
fprintf('Number of differences: %i\n', nnz(intersect_a~=intersect_b));
```
```25% coplanar faces.
# faces in surface 1 = 7200; # faces in surface 2 = 7200
Run time without intersection calculation is 12.934015
Run time with    intersection calculation is 19.862958
Number of face pairs is 51840000 and number of intersections is 11023
Number of differences: 0
```

## Compare runs with and without intersection calculation: test #2

```% Define two surfaces
Surface1 = surf2patch(membrane(1,30),'triangles');
Surface2 = surf2patch(membrane(3,30),'triangles');
% add a bit of perturbation to the coordinates of surface 2
Surface2.vertices = Surface2.vertices + 1E-3*randn(size(Surface2.vertices));
fprintf('No coplanar faces.\n# faces in surface 1 = %i; # faces in surface 2 = %i\n', ...
size(Surface1.faces,1), size(Surface2.faces,1));
% Plot two surfaces
clf; hold on
subplot(2,1,1)
patch(Surface1, 'FaceColor', 'b','EdgeColor', 'none', 'FaceAlpha', 0.8);
patch(Surface2, 'FaceColor', 'r','EdgeColor', 'none', 'FaceAlpha', 0.8);
title('Two surfaces')
view(-195, 44)
% Run and time the intersection function
tic; [intersect_a] = SurfaceIntersection(Surface1, Surface2);
fprintf('Run time without intersection calculation is %f\n', toc)
tic; [intersect_b, Surf12] = SurfaceIntersection(Surface1, Surface2);
fprintf('Run time with    intersection calculation is %f\n', toc)
subplot(2,1,2)
S=Surf12; trisurf(S.faces, S.vertices(:,1),S.vertices(:,2),S.vertices(:,3), 'FaceColor', 'g');
% Plot the results
view(-195, 44)
title('Surface intersection')
fprintf('Number of face pairs is %i and number of intersections is %i\n', ...
size(Surface1.faces,1)*size(Surface2.faces,1), nnz(intersect_a));
fprintf('Number of differences: %i\n', nnz(intersect_a~=intersect_b));
```
```No coplanar faces.
# faces in surface 1 = 7200; # faces in surface 2 = 7200
Run time without intersection calculation is 5.702677
Run time with    intersection calculation is 6.239134
Number of face pairs is 51840000 and number of intersections is 3405
Number of differences: 0
```

## Draw contour lines of a sphere

```% Create Surface #1: the sphere
[x,y,z] = sphere(50);
DT = delaunayTriangulation(z(:), y(:), x(:));
[Surface1.faces, Surface1.vertices] = freeBoundary(DT);
% Create Surface #2: 11 equally spaced parallel planes
Surface2=[];
for i=0:10
z = -0.95 + i/5;
Surface2.vertices(3*i+(1:3),:) = [2, 0, z; -1, 1.7, z; -1, -1.7, z];
Surface2.faces(i+1,:) = 3*i+(1:3);
end
[~, Surf12] = SurfaceIntersection(Surface1, Surface2);
Surf12.vertices(:,3) = -1.5; % project the contour lines on a single plane
% plot the results
figure(1); clf; hold on
S=Surface1; trisurf(S.faces, S.vertices(:,1),S.vertices(:,2),S.vertices(:,3),'FaceAlpha', 0.5, 'FaceColor', 'r')
S=Surface2; trisurf(S.faces, S.vertices(:,1),S.vertices(:,2),S.vertices(:,3),'FaceAlpha', 0.5, 'FaceColor', 'b')
S=Surf12;   line(...
[S.vertices(S.edges(:,1),1), S.vertices(S.edges(:,2),1)]',...
[S.vertices(S.edges(:,1),2), S.vertices(S.edges(:,2),2)]',...
[S.vertices(S.edges(:,1),3), S.vertices(S.edges(:,2),3)]',...
'Color', 'r');
axis equal
view(170, 20)
```
```Warning: Duplicate data points have been detected and removed.
The Triangulation indices are defined with respect to the unique set of points
in delaunayTriangulation property X.
```

## Draw contour lines of a cone or conics

```% Create Surface #1: the double cone
XYZ=[];
XYZ(1,:)=[0 0 0];
for z = 1:100
n=z+5;
[x,y] = pol2cart((1:n)'*2*pi/n,z);
XYZ = [XYZ; [x y z*x./x]];
end
DT = delaunayTriangulation(XYZ(:,1),XYZ(:,2));
Surface1.faces = [DT.ConnectivityList; DT.ConnectivityList+size(XYZ,1)];
Surface1.vertices = [XYZ(:,3),XYZ(:,2),XYZ(:,1); -XYZ(:,3),XYZ(:,2),XYZ(:,1)]/100;
S=Surface1; trisurf(S.faces, S.vertices(:,1),S.vertices(:,2),S.vertices(:,3),'FaceAlpha', 0.5, 'FaceColor', 'r');
% Create Surface #2: 11 equally spaced parallel planes
Surface2=[];
for i=0:10
z = -0.95 + i/5;
Surface2.vertices(3*i+(1:3),:) = [2, 0, z; -1, 1.7, z; -1, -1.7, z];
Surface2.faces(i+1,:) = 3*i+(1:3);
end
[~, Surf12] = SurfaceIntersection(Surface1, Surface2);
Surf12.vertices(:,3) = -1.5; % project the contour lines on a single plane
% plot the results
figure(1); clf; hold on
S=Surface1; trisurf(S.faces, S.vertices(:,1),S.vertices(:,2),S.vertices(:,3),'FaceAlpha', 0.5, 'FaceColor', 'r')
S=Surface2; trisurf(S.faces, S.vertices(:,1),S.vertices(:,2),S.vertices(:,3),'FaceAlpha', 0.5, 'FaceColor', 'b')
S=Surf12;   line(...
[S.vertices(S.edges(:,1),1), S.vertices(S.edges(:,2),1)]',...
[S.vertices(S.edges(:,1),2), S.vertices(S.edges(:,2),2)]',...
[S.vertices(S.edges(:,1),3), S.vertices(S.edges(:,2),3)]',...
'Color', 'r');
axis equal
view(-190, 15)
```