# alphaTriangulation

Triangulation that fills alpha shape

## Description

example

tri = alphaTriangulation(shp) returns a triangulation that defines the domain of the alpha shape. Each row in tri specifies a triangle or tetrahedron defined by vertex IDs (the row numbers of the shp.Points matrix).

tri = alphaTriangulation(shp,RegionID) returns a triangulation for a region of the alpha shape. RegionID is the ID for the region and 1RegionIDnumRegions(shp).

[tri,P] = alphaTriangulation(___) also returns a matrix of vertex coordinates, P, using any of the previous syntaxes.

## Examples

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Create a set of 3-D points.

[x1, y1, z1] = sphere(24);
x1 = x1(:);
y1 = y1(:);
z1 = z1(:);
x2 = x1+5;
P = [x1 y1 z1; x2 y1 z1];
P = unique(P,'rows');

Create and plot an alpha shape for the point cloud using an alpha radius of 1.

shp = alphaShape(P,1);
plot(shp)

Use alphaTriangulation to recover the triangulation that defines the domain of the alpha shape.

tri = alphaTriangulation(shp);

Find the total number of tetrahedra that make up the alpha shape.

numtetrahedra = size(tri,1)
numtetrahedra = 3760

## Input Arguments

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Alpha shape, specified as an alphaShape object. For more information, see alphaShape.

Example: shp = alphaShape(x,y) creates a 2-D alphaShape object from the (x,y) point coordinates.

ID number for region in alpha shape, specified as a positive integer scalar between 1 and numRegions(shp).

An alpha shape can contain several smaller regions, depending on the point set and parameters. Each of these smaller regions is assigned a unique RegionID, which numbers the regions from the largest area or volume to the smallest. For example, consider a 3-D alpha shape with two regions. The region with the largest volume has a RegionID of 1, and the smaller region has a RegionID of 2.

Example: shp.RegionThreshold = area(shp,numRegions(shp)-2); suppresses the two smallest regions in 2-D alpha shape shp.

Data Types: double

## Output Arguments

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Triangulation, returned as a matrix. tri is of size mtri-by-nv, where mtri is the number of triangles or tetrahedra in the alpha shape and nv is the number of vertices. The value of nv is 3 for 2-D alpha shapes and 4 for 3-D alpha shapes.

The number of outputs you specify with alphaTriangulation can change the vertex indexing used in tri.

Vertex coordinates, returned as a matrix. P is of size N-by-dim, where N is the number of points in the alpha shape and dim is either 2 or 3 (for either a 2-D or 3-D alpha shape).

## Version History

Introduced in R2014b