ashape3d: 3D $α$ -shape computation

Description

This function calculates the 3D $α$ -shape of a given sample of points in the three-dimensional space for $α > 0$ .

Usage

ashape3d(x, alpha, pert = FALSE, eps = 1e-09)

Value

An object of class "ashape3d" with the following components (see Edelsbrunner and Mucke (1994) for notation):

tetra: For each tetrahedron of the 3D Delaunay triangulation, the matrix tetra stores the indices of the sample points defining the tetrahedron (columns 1 to 4), a value that defines the intervals for which the tetrahedron belongs to the $α$ -complex (column 5) and for each $α$ a value (1 or 0) indicating whether the tetrahedron belongs to the $α$ -shape (columns 6 to last).
triang: For each triangle of the 3D Delaunay triangulation, the matrix triang stores the indices of the sample points defining the triangle (columns 1 to 3), a value (1 or 0) indicating whether the triangle is on the convex hull (column 4), a value (1 or 0) indicating whether the triangle is attached or unattached (column 5), values that define the intervals for which the triangle belongs to the $α$ -complex (columns 6 to 8) and for each $α$ a value (0, 1, 2 or 3) indicating, respectively, that the triangle is not in the $α$ -shape or it is interior, regular or singular (columns 9 to last). As defined in Edelsbrunner and Mucke (1994), a simplex in the $α$ -complex is interior if it does not belong to the boundary of the $α$ -shape. A simplex in the $α$ -complex is regular if it is part of the boundary of the $α$ -shape and bounds some higher-dimensional simplex in the $α$ -complex. A simplex in the $α$ -complex is singular if it is part of the boundary of the $α$ -shape but does not bounds any higher-dimensional simplex in the $α$ -complex.
edge: For each edge of the 3D Delaunay triangulation, the matrix edge stores the indices of the sample points defining the edge (columns 1 and 2), a value (1 or 0) indicating whether the edge is on the convex hull (column 3), a value (1 or 0) indicating whether the edge is attached or unattached (column 4), values that define the intervals for which the edge belongs to the $α$ -complex (columns 5 to 7) and for each $α$ a value (0, 1, 2 or 3) indicating, respectively, that the edge is not in the $α$ -shape or it is interior, regular or singular (columns 8 to last).
vertex: For each sample point, the matrix vertex stores the index of the point (column 1), a value (1 or 0) indicating whether the point is on the convex hull (column 2), values that define the intervals for which the point belongs to the $α$ -complex (columns 3 and 4) and for each $α$ a value (1, 2 or 3) indicating, respectively, if the point is interior, regular or singular (columns 5 to last).
x: A 3-column matrix with the coordinates of the original sample points.
alpha: A single value or vector of values of $α$ .
xpert: A 3-column matrix with the coordinates of the perturbated sample points (only when the input points are not in general position and pert is set to TRUE).

Arguments

x: A 3-column matrix with the coordinates of the input points. Alternatively, an object of class "ashape3d" can be provided, see Details.
alpha: A single value or vector of values for $α$ .
pert: Logical. If the input points are not in general position and pert it set to TRUE the observations are perturbed by adding random noise, see Details.
eps: Scaling factor used for data perturbation when the input points are not in general position, see Details.

Details

If x is an object of class "ashape3d", then ashape3d does not recompute the 3D Delaunay triangulation (it reduces the computational cost).

If the input points x are not in general position and pert is set to TRUE, the function adds random noise to the data. The noise is generated from a normal distribution with mean zero and standard deviation eps*sd(x).

References

Edelsbrunner, H., Mucke, E. P. (1994). Three-Dimensional Alpha Shapes. ACM Transactions on Graphics, 13(1), pp.43-72.

Examples

Run this code


T1 <- rtorus(1000, 0.5, 2)
T2 <- rtorus(1000, 0.5, 2, ct = c(2, 0, 0), rotx = pi/2)
x <- rbind(T1, T2)
# Value of alpha
alpha <- 0.25
# 3D alpha-shape
ashape3d.obj <- ashape3d(x, alpha = alpha)
plot(ashape3d.obj)

# For new values of alpha, we can use ashape3d.obj as input (faster)
alpha <- c(0.15, 1)
ashape3d.obj <- ashape3d(ashape3d.obj, alpha = alpha)
plot(ashape3d.obj, indexAlpha = 2:3)

Run the code above in your browser using DataLab