MATLAB Examples

# Point in polyhedron (PIP) problem in 3-dimensional space

The point-in-polyhedron (PIP) problem asks whether each point of an arbitrary point set (query points) lies inside, outside, or on the boundary of the convex hull of another given point set (polyhedron in space).

## Initial data

Set the random number generator.

```rng(1); ```

Set the points the convex hull of which defines the polyhedron.

```points=rand(100,3); ```

Set the query points, i.e. the points for which it will be determined if they lie inside or outside the polyhedron formed by the convex hull of .

```querypoints=0.3+rand(10,3); ```

Plot the points the convex hull of which defines the polyhedron (in blue) and the query points (in orange).

```figure('Name','Point set and query points','NumberTitle','off') scatter3(points(:,1),points(:,2),points(:,3),... 'marker','o','MarkerEdgeColor',[0 0 1],'LineWidth',2) hold on scatter3(querypoints(:,1),querypoints(:,2),querypoints(:,3),... 'marker','o','MarkerEdgeColor',[1 0.5 0],'LineWidth',2) xlabel('x','FontSize',13); ylabel('y','FontSize',13); title('Point set and query points','FontSize',13) axis equal ``` ## Processing

Find the plane coefficients of the convex hull of the initial point set.

```[chull,cf,df]=convhull_nd(points); ```

Determine if the query points are inside or outside the polyhedron formed by the convex hull of . If is true, then the corresponding point is inside the polyhedron.

```inpolyhedron=~any(cf*querypoints'+df(:,ones(1,size(querypoints,1)))>0,1); ```

Find the coordinates of the query points which are inside the polyhedron.

```inpolyhedron_points=querypoints(inpolyhedron,:); ```

## Plots

Plot the convex hull of the initial point set (points in filled orange and edges in blue), the query points (in blue circles) and the query points which lie inside the convex hull (blue circles with orange asterisks).

```figure('Name','Query points in convex hull','NumberTitle','off') trisurf(chull,points(:,1),points(:,2),points(:,3),... 'FaceColor',[1 0.5 0],'EdgeColor',[0 0 1],'FaceAlpha',0.3); hold on scatter3(querypoints(:,1),querypoints(:,2),querypoints(:,3),... 'marker','o','MarkerEdgeColor',[0 0 1],'LineWidth',2) scatter3(inpolyhedron_points(:,1),inpolyhedron_points(:,2),... inpolyhedron_points(:,3),'marker','*','MarkerEdgeColor',[1 0.5 0],... 'LineWidth',2); xlabel('x','FontSize',13); ylabel('y','FontSize',13); title('Query points in convex hull','FontSize',13) ``` ## Contact author

```(c) 2014 by George Papazafeiropoulos
First Lieutenant, Infrastructure Engineer, Hellenic Air Force
Civil Engineer, M.Sc., Ph.D. candidate, NTUA```

Website: http://users.ntua.gr/gpapazaf/