blank space in alpha shape

using the alphavol function i wrote the following code:

%create crater alpha shape. 
clear;clc;

ddr=0.12; %depth diameter ratio
a=1/ddr/2; %semi major axis. horizontal axes are equal 
n=31; %number of points for x y vectors. 
x = linspace(-a,a,n);
y = linspace(-a,a,n);
[X,Y] = meshgrid(x,y);
Z =real(sqrt(1-(X.^2)/a^2-(Y.^2)/a^2))*-1+1; % the plus for Z>0
[V,S] = alphavol([X(:),Y(:),Z(:)],1,1);
trisurf(S.bnd,X,Y,Z,'FaceColor','blue','FaceAlpha',1)

xlabel('x');ylabel('y');zlabel('z'); 
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as the picture shows, i get blank space in the bottom of the shape. i know i can resolve this by assigning greater values to n, but it is still only an approximation. also, i dont want to set up R=inf, because this will give a surface on the X-Y plane(where z=1).

thank you for your help

[John D'Errico]

So what is your question? The result from an alpha shape is a consequence of the diameter alpha ball you use in the creation of the shape. Sufficiently large alpha (approaching infinity), and you get a convex hull. Too small alpha and you get holes that you won't like.

If you don't like the behavior of an alpha shape, then don't use it. So what is the problem? What is your question?

[Image Analyst]

Do you want to fill in the hole? If so try regionfill().

[John D'Errico]

Of course, if you fill in the hole, then what is the purpose of the alpha shape?

As for something being an approximation, ANYTHING you do on a computer is an approximation. The skill of working on a computer, in any discipline, be it applied mathematics, statistics, image processing, any engineering discipline, any physical science, etc., is in knowing what approximations are viable to use, and which ones will kill you.