This function calculates the 3D \(\alpha\)-shape of a given sample of points in the three-dimensional space for \(\alpha>0\).
ashape3d(x, alpha, pert = FALSE, eps = 1e-09)An object of class "ashape3d" with the following components
(see Edelsbrunner and Mucke (1994) for notation):
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
\(\alpha\)-complex (column 5) and for each \(\alpha\) a value (1 or 0)
indicating whether the tetrahedron belongs to the \(\alpha\)-shape
(columns 6 to last).
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
\(\alpha\)-complex (columns 6 to 8) and for each \(\alpha\) a value (0,
1, 2 or 3) indicating, respectively, that the triangle is not in the
\(\alpha\)-shape or it is interior, regular or singular (columns 9 to
last). As defined in Edelsbrunner and Mucke (1994), a simplex in the
\(\alpha\)-complex is interior if it does not belong to the boundary of
the \(\alpha\)-shape. A simplex in the \(\alpha\)-complex is regular if
it is part of the boundary of the \(\alpha\)-shape and bounds some
higher-dimensional simplex in the \(\alpha\)-complex. A simplex in the
\(\alpha\)-complex is singular if it is part of the boundary of the
\(\alpha\)-shape but does not bounds any higher-dimensional simplex in the
\(\alpha\)-complex.
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
\(\alpha\)-complex (columns 5 to 7) and for each \(\alpha\) a value (0,
1, 2 or 3) indicating, respectively, that the edge is not in the
\(\alpha\)-shape or it is interior, regular or singular (columns 8 to
last).
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 \(\alpha\)-complex (columns 3 and 4) and
for each \(\alpha\) a value (1, 2 or 3) indicating, respectively, if the
point is interior, regular or singular (columns 5 to last).
A 3-column matrix with the coordinates of the original sample points.
A single value or vector of values of \(\alpha\).
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).
A 3-column matrix with the coordinates of the input points.
Alternatively, an object of class "ashape3d" can be provided, see
Details.
A single value or vector of values for \(\alpha\).
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.
Scaling factor used for data perturbation when the input points are not in general position, see 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).
Edelsbrunner, H., Mucke, E. P. (1994). Three-Dimensional Alpha Shapes. ACM Transactions on Graphics, 13(1), pp.43-72.
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)
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