# Test for Point Containment in 3D Polyhedron

### Description

Tests whether points are contained within a three-dimensional polyhedron.

### Usage

1 | ```
pip3d(Vertices,Faces,Queries)
``` |

### Arguments

`Vertices` |
N by 3 matrix containing the XYZ coordinates of N vertices of the polyhedron |

`Faces` |
M by 3 matrix containing the indices of the three vertices defining the M triangular faces of the polyhedron |

`Queries` |
P by 3 matrix containing the XYZ coordinates of P points to be tested for containment in the polyhedron defined by 'Vertices' and 'Faces' |

### Details

The values in the `Faces`

matrix must be integers with values running from 1 to N,
where N is the number of vertices. A value of '1' in this matrix, for example,
represents the 1st vertex, i.e., the vertex defined by the first row in the
matrix `Vertices`

.

### Value

Returns a vector containing P values, one for each of the P points listed in
the `Queries`

matrix.

'1' indicates that the point is contained in the polyhedron.

'0' indicates that the point lies exactly on the surface of the polyhedron.

'-1' indicates that the point lies outside the polyhedron.

'-2' (error) indicates that the polyhedron was topologically defective (e.g., had a hole)

'-3' (error) indicates that the `Vertices`

matrix didn't have three columns

'-4' (error) indicates that the `Faces`

matrix didn't have three columns

'-5' (error) indicates that the `Faces`

matrix was 0- rather than 1-offset

'-6' (error) indicates that the `Queries`

matrix didn't have three columns

'-7' (error) indicates that two faces in the polyhedron were too close to one another

'-8' (error) indicates computational error not otherwise specified. A possible cause is when two faces of the polygon are extremely close to one another (imagine bending a cylindrical balloon until the two ends meet). Adjusting the spatial smoothness of the data may fix this problem.

### Note

The polyhedron defined by `Vertices`

and `Faces`

*must* be "non-leaky";
i.e., it must define an "inside" versus "outside" and must not contain any holes.

For an example of external software that could potentially be used to fix defective polyhedra, see, e.g., PolyMender (http://www1.cse.wustl.edu/~taoju/code/polymender.htm).

Previous versions of this function would hang when there were more than two vertices very close
to one another; this problem was discovered with a polyhedron in which there were multip;le
duplicate vertices and one triplicate vertex. The triplicate vertex fulfilled the case of
"more than two vertices very close to one another", and caused the code to hang.
The threshold for vertices that are very close to one another has been increased to three.
It is advisable to make sure that a polyhedron does not have more than three vertices
that are "very close to one another", and to make sure that there are no duplicate vertices.
Similarly, it is advisable to make sure that a polyhedron does not have *faces* that
that are extremely close to one another.

### References

W.P. Horn and D.L. Taylor, *A theorem to determine the spatial containment of a point in a planar polygon*, Computer Vision, Graphics and Image Processing, vol. 45, pp. 106-116,1989.

S. Nordbeck and B. Rysedt, *Computer cartography point-in-polygon programs*, BIT, vol. 7, pp. 39-64, 1967.

J.A. Baerentzen and H. Aanaes, *Signed distance computation using the angle weighted pseudo-normal*, IEEE Trans. Visualization and Computer Graphics, vol. 11, no. 3, pp. 243-253, May/June 2005.

J. Liu, Y.Q. Chen, J.M. Maisog, G. Luta, *A new point containment test algorithm for polyhedron composed of huge number of triangles*, Computer-Aided Design, Volume 42, Issue 12, December 2010, Pages 1143-1150.

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 | ```
#-------------------------------------------
# Simple Cube example.
# Load sample data defining a simple cube.
data(verts)
data(faces)
# Also load sample data for five test points.
data(queries)
# Test whether each point in 'queries' is contained in
# the simple cube defined by 'verts' and 'faces'.
# This should return "1 0 0 0 -1".
containment <- pip3d(verts,faces,queries);
#-------------------------------------------
# Torus example.
# Make a donut-shaped polyhedron.
torus <- parametric3d(fx = function(u,v) (1+0.25*cos(v))*cos(u),
fy = function(u,v) (1+0.25*cos(v))*sin(u),
fz = function(u,v) 0.25*sin(v),
u = seq(0,2*pi,length.out=10),
v = seq(0,2*pi,length.out=10),
engine = "none", color="orange", alpha=0.25)
# If desired, this torus can be rendered for visualization, e.g.:
# library(geometry)
# library(rgl)
# drawScene.rgl(torus)
# Convert the torus to vertices-faces representation.
ve <- misc3d:::t2ve(torus)
Vertices <- t(ve$vb)
Faces <- t(ve$ib)
# Generate 3333 random test points.
set.seed(1902)
n <- 3333
x1 <- rnorm(n) ; x2 <- rnorm(n) ; x3 <- rnorm(n)
X <- cbind(x1,x2,x3)
Queries <- as.matrix(X)
# Check whether test points are contained in the torus.
# Most of these points will lie outside the torus.
containment <- pip3d(Vertices,Faces,Queries);
#-------------------------------------------
# If you remove one of the faces of the cube, the resulting cube
# becomes "leaky". Running 'pip3d' on the resulting defective
# polyhedron will return -2.
# NOT RUN
#
# badcube <- faces[1:11,]
# containment <- pip3d(verts,badcube,queries);
``` |

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