quickhull3d
Class QuickHull3D

java.lang.Object
  quickhull3d.QuickHull3D

public class QuickHull3D
extends java.lang.Object

Computes the convex hull of a set of three dimensional points.

The algorithm is a three dimensional implementation of Quickhull, as described in Barber, Dobkin, and Huhdanpaa, ``The Quickhull Algorithm for Convex Hulls'' (ACM Transactions on Mathematical Software, Vol. 22, No. 4, December 1996), and has a complexity of O(n log(n)) with respect to the number of points. A well-known C implementation of Quickhull that works for arbitrary dimensions is provided by qhull.

A hull is constructed by providing a set of points to either a constructor or a build method. After the hull is built, its vertices and faces can be retrieved using getVertices and getFaces. A typical usage might look like this:

   // x y z coordinates of 6 points
   Point3d[] points = new Point3d[] 
    { new Point3d (0.0,  0.0,  0.0),
      new Point3d (1.0,  0.5,  0.0),
      new Point3d (2.0,  0.0,  0.0),
      new Point3d (0.5,  0.5,  0.5),
      new Point3d (0.0,  0.0,  2.0),
      new Point3d (0.1,  0.2,  0.3),
      new Point3d (0.0,  2.0,  0.0),
    };

   QuickHull3D hull = new QuickHull3D();
   hull.build (points);

   System.out.println ("Vertices:");
   Point3d[] vertices = hull.getVertices();
   for (int i = 0; i < vertices.length; i++)
    { Point3d pnt = vertices[i];
      System.out.println (pnt.x + " " + pnt.y + " " + pnt.z);
    }

   System.out.println ("Faces:");
   int[][] faceIndices = hull.getFaces();
   for (int i = 0; i < faceIndices.length; i++)
    { for (int k = 0; k < faceIndices[i].length; k++)
       { System.out.print (faceIndices[i][k] + " ");
       }
      System.out.println ("");
    }
 

As a convenience, there are also build and getVertex methods which pass point information using an array of doubles.

Robustness

Because this algorithm uses floating point arithmetic, it is potentially vulnerable to errors arising from numerical imprecision. We address this problem in the same way as qhull, by merging faces whose edges are not clearly convex. A face is convex if its edges are convex, and an edge is convex if the centroid of each adjacent plane is clearly below the plane of the other face. The centroid is considered below a plane if its distance to the plane is less than the negative of a distance tolerance. This tolerance represents the smallest distance that can be reliably computed within the available numeric precision. It is normally computed automatically from the point data, although an application may set this tolerance explicitly.

Numerical problems are more likely to arise in situations where data points lie on or within the faces or edges of the convex hull. We have tested QuickHull3D for such situations by computing the convex hull of a random point set, then adding additional randomly chosen points which lie very close to the hull vertices and edges, and computing the convex hull again. The hull is deemed correct if check returns true. These tests have been successful for a large number of trials and so we are confident that QuickHull3D is reasonably robust.

Merged Faces

The merging of faces means that the faces returned by QuickHull3D may be convex polygons instead of triangles. If triangles are desired, the application may triangulate the faces, but it should be noted that this may result in triangles which are very small or thin and hence difficult to perform reliable convexity tests on. In other words, triangulating a merged face is likely to restore the numerical problems which the merging process removed. Hence is it possible that, after triangulation, check will fail (the same behavior is observed with triangulated output from qhull).

Degenerate Input

It is assumed that the input points are non-degenerate in that they are not coincident, colinear, or colplanar, and thus the convex hull has a non-zero volume. If the input points are detected to be degenerate within the distance tolerance, an IllegalArgumentException will be thrown.

Author:

John E. Lloyd, Fall 2004

Field Summary

static double	AUTOMATIC_TOLERANCE Specifies that the distance tolerance should be computed automatically from the input point data.
static int	CLOCKWISE Specifies that (on output) vertex indices for a face should be listed in clockwise order.
static int	INDEXED_FROM_ONE Specifies that (on output) the vertex indices for a face should be numbered starting from 1.
static int	INDEXED_FROM_ZERO Specifies that (on output) the vertex indices for a face should be numbered starting from 0.
static int	POINT_RELATIVE Specifies that (on output) the vertex indices for a face should be numbered with respect to the original input points.

Constructor Summary

QuickHull3D()
Creates an empty convex hull object.

QuickHull3D(double[] coords)
Creates a convex hull object and initializes it to the convex hull of a set of points whose coordinates are given by an array of doubles.

QuickHull3D(Point3d[] points)
Creates a convex hull object and initializes it to the convex hull of a set of points.

Method Summary

void	build(double[] coords) Constructs the convex hull of a set of points whose coordinates are given by an array of doubles.
void	build(double[] coords, int nump) Constructs the convex hull of a set of points whose coordinates are given by an array of doubles.
void	build(Point3d[] points) Constructs the convex hull of a set of points.
void	build(Point3d[] points, int nump) Constructs the convex hull of a set of points.
boolean	check(java.io.PrintStream ps) Checks the correctness of the hull using the distance tolerance returned by getDistanceTolerance; see check(PrintStream,double) for details.
boolean	check(java.io.PrintStream ps, double tol) Checks the correctness of the hull.
boolean	getDebug() Returns true if debugging is enabled.
double	getDistanceTolerance() Returns the distance tolerance that was used for the most recently computed hull.
double	getExplicitDistanceTolerance() Returns the explicit distance tolerance.
int[][]	getFaces() Returns the faces associated with this hull.
int[][]	getFaces(int indexFlags) Returns the faces associated with this hull.
int	getNumFaces() Returns the number of faces in this hull.
int	getNumVertices() Returns the number of vertices in this hull.
int[]	getVertexPointIndices() Returns an array specifing the index of each hull vertex with respect to the original input points.
Point3d[]	getVertices() Returns the vertex points in this hull.
int	getVertices(double[] coords) Returns the coordinates of the vertex points of this hull.
void	print(java.io.PrintStream ps) Prints the vertices and faces of this hull to the stream ps.
void	print(java.io.PrintStream ps, int indexFlags) Prints the vertices and faces of this hull to the stream ps.
void	setDebug(boolean enable) Enables the printing of debugging diagnostics.
void	setExplicitDistanceTolerance(double tol) Sets an explicit distance tolerance for convexity tests.
void	triangulate() Triangulates any non-triangular hull faces.

Methods inherited from class java.lang.Object

equals, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Field Detail

CLOCKWISE

public static final int CLOCKWISE

Specifies that (on output) vertex indices for a face should be listed in clockwise order.

See Also:

Constant Field Values

INDEXED_FROM_ONE

public static final int INDEXED_FROM_ONE

Specifies that (on output) the vertex indices for a face should be numbered starting from 1.

See Also:

Constant Field Values

INDEXED_FROM_ZERO

public static final int INDEXED_FROM_ZERO

Specifies that (on output) the vertex indices for a face should be numbered starting from 0.

See Also:

Constant Field Values

POINT_RELATIVE

public static final int POINT_RELATIVE

Specifies that (on output) the vertex indices for a face should be numbered with respect to the original input points.

See Also:

Constant Field Values

AUTOMATIC_TOLERANCE

public static final double AUTOMATIC_TOLERANCE

Specifies that the distance tolerance should be computed automatically from the input point data.

See Also:

Constant Field Values

Constructor Detail

QuickHull3D

public QuickHull3D()

Creates an empty convex hull object.

QuickHull3D

public QuickHull3D(double[] coords)
            throws java.lang.IllegalArgumentException

Creates a convex hull object and initializes it to the convex hull of a set of points whose coordinates are given by an array of doubles.

Parameters:

coords - x, y, and z coordinates of each input point. The length of this array will be three times the the number of input points.

Throws:

java.lang.IllegalArgumentException - the number of input points is less than four, or the points appear to be coincident, colinear, or coplanar.

QuickHull3D

public QuickHull3D(Point3d[] points)
            throws java.lang.IllegalArgumentException

Creates a convex hull object and initializes it to the convex hull of a set of points.

Parameters:

points - input points.

Throws:

java.lang.IllegalArgumentException - the number of input points is less than four, or the points appear to be coincident, colinear, or coplanar.

Method Detail

getDebug

public boolean getDebug()

Returns true if debugging is enabled.

Returns:

true is debugging is enabled

See Also:

setDebug(boolean)

setDebug

public void setDebug(boolean enable)

Enables the printing of debugging diagnostics.

Parameters:

enable - if true, enables debugging

getDistanceTolerance

public double getDistanceTolerance()

Returns the distance tolerance that was used for the most recently computed hull. The distance tolerance is used to determine when faces are unambiguously convex with respect to each other, and when points are unambiguously above or below a face plane, in the presence of numerical imprecision. Normally, this tolerance is computed automatically for each set of input points, but it can be set explicitly by the application.

Returns:

distance tolerance

See Also:

setExplicitDistanceTolerance(double)

setExplicitDistanceTolerance

public void setExplicitDistanceTolerance(double tol)

Sets an explicit distance tolerance for convexity tests. If AUTOMATIC_TOLERANCE is specified (the default), then the tolerance will be computed automatically from the point data.

Parameters:

tol - explicit tolerance

See Also:

getDistanceTolerance()

getExplicitDistanceTolerance

public double getExplicitDistanceTolerance()

Returns the explicit distance tolerance.

Returns:

explicit tolerance

See Also:

setExplicitDistanceTolerance(double)

build

public void build(double[] coords)
           throws java.lang.IllegalArgumentException

Constructs the convex hull of a set of points whose coordinates are given by an array of doubles.

Parameters:

coords - x, y, and z coordinates of each input point. The length of this array will be three times the number of input points.

Throws:

java.lang.IllegalArgumentException - the number of input points is less than four, or the points appear to be coincident, colinear, or coplanar.

build

public void build(double[] coords,
                  int nump)
           throws java.lang.IllegalArgumentException

Constructs the convex hull of a set of points whose coordinates are given by an array of doubles.

Parameters:

coords - x, y, and z coordinates of each input point. The length of this array must be at least three times nump.

nump - number of input points

Throws:

java.lang.IllegalArgumentException - the number of input points is less than four or greater than 1/3 the length of coords, or the points appear to be coincident, colinear, or coplanar.

build

public void build(Point3d[] points)
           throws java.lang.IllegalArgumentException

Constructs the convex hull of a set of points.

Parameters:

points - input points

Throws:

java.lang.IllegalArgumentException - the number of input points is less than four, or the points appear to be coincident, colinear, or coplanar.

build

public void build(Point3d[] points,
                  int nump)
           throws java.lang.IllegalArgumentException

Constructs the convex hull of a set of points.

Parameters:

points - input points

nump - number of input points

Throws:

java.lang.IllegalArgumentException - the number of input points is less than four or greater then the length of points, or the points appear to be coincident, colinear, or coplanar.

triangulate

public void triangulate()

Triangulates any non-triangular hull faces. In some cases, due to precision issues, the resulting triangles may be very thin or small, and hence appear to be non-convex (this same limitation is present in qhull).

getNumVertices

public int getNumVertices()

Returns the number of vertices in this hull.

Returns:

number of vertices

getVertices

public Point3d[] getVertices()

Returns the vertex points in this hull.

Returns:

array of vertex points

See Also:

getVertices(double[]), getFaces()

getVertices

public int getVertices(double[] coords)

Returns the coordinates of the vertex points of this hull.

Parameters:

coords - returns the x, y, z coordinates of each vertex. This length of this array must be at least three times the number of vertices.

Returns:

the number of vertices

See Also:

getVertices(), getFaces()

getVertexPointIndices

public int[] getVertexPointIndices()

Returns an array specifing the index of each hull vertex with respect to the original input points.

Returns:

vertex indices with respect to the original points

getNumFaces

public int getNumFaces()

Returns the number of faces in this hull.

Returns:

number of faces

getFaces

public int[][] getFaces()

Returns the faces associated with this hull.

Each face is represented by an integer array which gives the indices of the vertices. These indices are numbered relative to the hull vertices, are zero-based, and are arranged counter-clockwise. More control over the index format can be obtained using getFaces(indexFlags).

Returns:

array of integer arrays, giving the vertex indices for each face.

See Also:

getVertices(), getFaces(int)

getFaces

public int[][] getFaces(int indexFlags)

Returns the faces associated with this hull.

Each face is represented by an integer array which gives the indices of the vertices. By default, these indices are numbered with respect to the hull vertices (as opposed to the input points), are zero-based, and are arranged counter-clockwise. However, this can be changed by setting POINT_RELATIVE, INDEXED_FROM_ONE, or CLOCKWISE in the indexFlags parameter.

Parameters:

indexFlags - specifies index characteristics (0 results in the default)

Returns:

array of integer arrays, giving the vertex indices for each face.

See Also:

getVertices()

print

public void print(java.io.PrintStream ps)

Prints the vertices and faces of this hull to the stream ps.

This is done using the Alias Wavefront .obj file format, with the vertices printed first (each preceding by the letter v), followed by the vertex indices for each face (each preceded by the letter f).

The face indices are numbered with respect to the hull vertices (as opposed to the input points), with a lowest index of 1, and are arranged counter-clockwise. More control over the index format can be obtained using print(ps,indexFlags).

Parameters:

ps - stream used for printing

See Also:

print(PrintStream,int), getVertices(), getFaces()

print

public void print(java.io.PrintStream ps,
                  int indexFlags)

Prints the vertices and faces of this hull to the stream ps.

This is done using the Alias Wavefront .obj file format, with the vertices printed first (each preceding by the letter v), followed by the vertex indices for each face (each preceded by the letter f).

By default, the face indices are numbered with respect to the hull vertices (as opposed to the input points), with a lowest index of 1, and are arranged counter-clockwise. However, this can be changed by setting POINT_RELATIVE, INDEXED_FROM_ZERO, or CLOCKWISE in the indexFlags parameter.

Parameters:

ps - stream used for printing

indexFlags - specifies index characteristics (0 results in the default).

See Also:

getVertices(), getFaces()

check

public boolean check(java.io.PrintStream ps)

Checks the correctness of the hull using the distance tolerance returned by getDistanceTolerance; see check(PrintStream,double) for details.

Parameters:

ps - print stream for diagnostic messages; may be set to null if no messages are desired.

Returns:

true if the hull is valid

See Also:

check(PrintStream,double)

check

public boolean check(java.io.PrintStream ps,
                     double tol)

Checks the correctness of the hull. This is done by making sure that no faces are non-convex and that no points are outside any face. These tests are performed using the distance tolerance tol. Faces are considered non-convex if any edge is non-convex, and an edge is non-convex if the centroid of either adjoining face is more than tol above the plane of the other face. Similarly, a point is considered outside a face if its distance to that face's plane is more than 10 times tol.

If the hull has been triangulated, then this routine may fail if some of the resulting triangles are very small or thin.

Parameters:

ps - print stream for diagnostic messages; may be set to null if no messages are desired.

tol - distance tolerance

Returns:

true if the hull is valid

See Also:

check(PrintStream)