# alpha-shape calculation This function calculates the $\alpha$-shape of a given sample for $\alpha>0$. ```r ashape(x, y = NULL, alpha) ``` ## Arguments - `x, y`: The `x` and `y` coordinates of a set of points. Alternatively, a single argument `x` can be provided, see Details. - `alpha`: Value of $\alpha$. ## Details An attempt is made to interpret the arguments x and y in a way suitable for computing the $\alpha$-shape, see `xy.coords`. The $\alpha$-shape is defined for any finite number of points. However, since the algorithm is based on the Delaunay triangulation, at least three non-collinear points are required. If `y` is NULL and `x` is an object of class `"delvor"`, then the $\alpha$-shape is computed without invoking again the function `delvor` (it reduces the computational cost). The function `ashape` returns (among other values) the matrix `edges`. The structure of `edges` is that of matrix `mesh` returned by the function `delvor`. Note that the $\alpha$-shape is a subgraph of the Delaunay triangulation and, therefore, `edges` is a submatrix of `mesh`. ## Returns A list with the following components: - **edges**: A **n.seg**-row matrix with the coordinates and indexes of the edges of the Delaunay triangulation that form the $\alpha$-shape. The number of rows **n.seg** coincides with the number of segments of the $\alpha$-shape. The matrix also includes information of the Voronoi extremes corresponding to each segment. - **length**: Length of the $\alpha$-shape. - **alpha**: Value of $\alpha$. - **alpha.extremes**: Vector with the indexes of the sample points that are $\alpha$-extremes. See Edelsbrunnner **et al.** (1983). - **delvor.obj**: Object of class `"delvor"` returned by the `delvor` function. - **x**: A 2-column matrix with the coordinates of the set of points. ## References Edelsbrunner, H., Kirkpatrick, D.G. and Seidel, R. (1983). On the shape of a set of points in the plane. **IEEE Transactions on Information Theory**, 29(4), pp.551-559. ## See Also `plot.ashape`, `delvor`. ## Examples ```r ## Not run: # Uniform sample of size n=300 in the annulus B(c,0.5)\B(c,0.25), # with c=(0.5,0.5). n <- 300 theta<-runif(n,0,2*pi) r<-sqrt(runif(n,0.25^2,0.5^2)) x<-cbind(0.5+r*cos(theta),0.5+r*sin(theta)) # Value of alpha alpha <- 0.1 # alpha-shape ashape.obj <- ashape(x, alpha = alpha) # If we change the value of alpha there is no need to compute # again the Delaunay triangulation and Voronoi Diagram alpha <- 0.4 ashape.obj.new <- ashape(ashape.obj$delvor.obj, alpha = alpha) # Random sample from a uniform distribution on a Koch snowflake # with initial side length 1 and 3 iterations x <- rkoch(2000, side = 1, niter = 3) # Value of alpha alpha <- 0.05 # alpha-shape ashape.obj <- ashape(x, alpha = alpha) ## End(Not run) ```

ashape function