# CVTM_1D Centroidal Voronoi Tessellation in Mirror-Periodic Interval [0,1]

CVTM_1D is a MATLAB program which estimates a mirror-periodic centroidal Voronoi Tessellation (CVTP) in the periodic interval [0,1], using a version of Lloyd's iteration.

The determination of the Voronoi regions is carried out using sampling. This means that the convergence of the iteration is influenced by the accuracy of the estimates provided by sampling.

For n generators, a solution set is known in advance:

x(i) = i / ( n - 1 ), i = 0 : n - 1
which includes a generator at both endpoints. Lloyd's algorithm starts from an arbitrary vector x, however, so it is interesting to see how the approximate solution evolves toward a correct solution, whose fundamental properties are that the generators are equally spaced within the periodic domain and include the endpoints.

### Usage:

cvtm_1d ( g_num, it_num, sample_num )
where
• g_num is the number of generators (try 11 to start);
• it_num is the number of iterative steps to take (try 10 initially).
• sample_num is the number of sample points in [0,1] used to estimate the Voronoi regions. (A value between 1,000 and 10,000 is typical).

### Languages:

CVTM_1D is available in a MATLAB version.

### Related Data and Programs:

CCVT_BOX, a MATLAB program which constructs a modified CVT in which some points are forced to lie on the boundary.

CCVT_REFLECT, a MATLAB program which tries to construct a modified CVT in which some points are forced to lie on the boundary, using a reflection idea.

CVT, a MATLAB library which computes CVT's.

CVT, a dataset directory which contains a variety of examples of CVT datasets.

CVT_1D_LLOYD, a MATLAB program which computes an N-point Centroidal Voronoi Tessellation (CVT) within the interval [0,1], under a uniform density, using Lloyd's method to compute the Voronoi regions exactly.

CVT_1D_NONUNIFORM, a MATLAB program which computes an N-point Centroidal Voronoi Tessellation in 1 dimension, under a nonuniform density, and plots the evolution of the locations of the generators during the iteration;

CVT_1D_SAMPLING, a MATLAB program which computes an N-point Centroidal Voronoi Tessellation (CVT) within the interval [0,1], under a uniform density, using sampling to estimate the Voronoi regions.

CVT_2D_SAMPLING, a MATLAB program which computes an N-point Centroidal Voronoi Tessellation (CVT) within the unit square [0,1]x[0,1], under a uniform density, using sampling to estimate the Voronoi regions.

CVT_3D_SAMPLING, a MATLAB program which computes an N-point Centroidal Voronoi Tessellation (CVT) within the unit cube [0,1]x[0,1]x[0,1], under a uniform density, using sampling to estimate the Voronoi regions.

CVT_CIRCLE_NONUNIFORM, a MATLAB program which calculates a nonuniform Centroidal Voronoi Tessellation (CVT) over a circle.

CVT_CIRCLE_UNIFORM, a MATLAB program which calculates a Centroidal Voronoi Tessellation (CVT) over a circle with uniform density.

CVT_CORN, a MATLAB program which studies a 2D model of the growth of a corn kernel, by treating the surface and interior biological cells as points to be organized by a Centroidal Voronoi Tessellation (CVT) with a nonuniform density; during a sequence of growth steps, new biological cells are randomly added to the surface and interior.

CVT_DATASET, a MATLAB program which can create a CVT dataset.

CVT_DEMO, a MATLAB program which demonstrates a CVT calculation.

CVT_ELLIPSE_UNIFORM, a MATLAB program which iteratively calculates a Centroidal Voronoi Tessellation (CVT) over an ellipse, with a uniform density.

CVT_METRIC, a MATLAB program which computes a Centroidal Voronoi Tessellation (CVT) under a spatially varying metric;

CVTP, a MATLAB library which creates a CVTP, that is, a Centroidal Voronoi Tessellation on a periodic domain.

CVTP_1D, a MATLAB program which estimates a periodic centroidal Voronoi Tessellation (CVTP) in the periodic interval [0,1], using a version of Lloyd's iteration.

FLORIDA_CVT_GEO, MATLAB programs which explore the creation of a centroidal Voronoi Tessellation (CVT) of the state of Florida, based solely on geometric considerations.

LCVT, a MATLAB library which computes a "Latinized" Centroidal Voronoi Tessellation.

### Reference:

1. Franz Aurenhammer,
Voronoi diagrams - a study of a fundamental geometric data structure,
ACM Computing Surveys,
Volume 23, Number 3, pages 345-405, September 1991.
2. Qiang Du, Vance Faber, Max Gunzburger,
Centroidal Voronoi Tessellations: Applications and Algorithms,
SIAM Review, Volume 41, 1999, pages 637-676.

### Source Code:

Last revised on 15 December 2018.