7.3 Barycentric subdivision

Barycentric subdivision is a powerful technique for refining simplicial complexes. It adds new vertices at the center of each face, creating smaller simplices that preserve the original structure. This process allows for more detailed representations of geometric shapes and topological spaces.

In the context of simplicial complexes, barycentric subdivision plays a crucial role in creating finer approximations. It's essential for proving key theorems in algebraic topology and has practical applications in computer graphics and numerical simulations. Understanding this process is fundamental to grasping more advanced concepts in the field.

Barycentric Subdivision Process

Fundamentals of Barycentric Subdivision

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• Barycentric subdivision refines a simplicial complex by introducing new vertices and dividing each simplex into smaller simplices
• Process begins by adding a new vertex at the barycenter (center of mass) of each non-empty face of the original simplicial complex
• For each simplex in the original complex, new simplices form by connecting the barycenters of its faces in a specific order
• Order of connecting barycenters determined by the dimension of the faces, starting from lowest dimension (vertices) to highest (entire simplex)
• Each new simplex in the subdivision forms by selecting one vertex from each dimension, maintaining order from lowest to highest dimension
• Number of new simplices created for each original n-simplex equals $(n+1)!$
• Barycentric subdivision process applies recursively to create increasingly fine subdivisions of the original complex

Examples and Applications

• For a 2-simplex (triangle), barycentric subdivision creates 6 smaller triangles
• In a tetrahedron (3-simplex), barycentric subdivision results in 24 smaller tetrahedra
• Barycentric subdivision of a square (two triangles) produces 8 smaller triangles
• Applications in computer graphics for mesh refinement (3D modeling)
• Used in finite element analysis to improve accuracy of numerical simulations (structural engineering)

Performing Barycentric Subdivision

Identifying and Marking Barycenters

• Identify all non-empty faces of the given simplicial complex (vertices, edges, triangles, higher-dimensional simplices)
• Calculate and mark the barycenter of each identified face
  • For edges barycenter is the midpoint
  • For triangles barycenter is the centroid
  • For higher dimensions use the average of the coordinates of the face's vertices
• Example: In a triangle ABC, barycenters include vertices A, B, C, midpoints of edges AB, BC, AC, and centroid of triangle ABC

Creating New Simplices

• For each original simplex, create a list of all possible combinations of barycenters
  • Ensure each combination includes one point from each dimension in ascending order
• Connect the points in each combination to form new simplices, maintaining proper orientation
• Verify newly created simplices cover the entire original complex without overlaps or gaps
• Update complex data structure to reflect new vertices, edges, and higher-dimensional simplices created by subdivision
• Ensure boundary of subdivided complex matches subdivision of boundary of original complex
• Example: For a tetrahedron ABCD, new simplices include ABCD (centroid), ABC (face centroid), AB (edge midpoint), A (vertex)

Properties of Barycentric Subdivision

Topological and Structural Properties

• Barycentric subdivision of a simplicial complex itself forms a simplicial complex, preserving topological structure of original
• Process decreases size of each simplex while increasing total number of simplices in complex
• Creates more uniform and regular structure, as all new simplices have comparable size and shape
• Preserves dimension of original complex without introducing or removing dimensions
• Star of each vertex in barycentric subdivision corresponds to closed simplex of original complex containing that vertex
• Repeated barycentric subdivisions of a complex converge to a limiting shape, each iteration refining approximation of underlying space
• Barycentric subdivision demonstrates functoriality, commuting with simplicial maps between complexes

Mathematical and Computational Aspects

• Barycentric subdivision increases number of vertices exponentially
  • For an n-simplex, number of new vertices after k subdivisions $= (n+1)((n+2)^k - 1)$
• Process preserves orientability of the original complex
• Barycentric subdivision of boundary of a simplex equals boundary of barycentric subdivision of the simplex
• Useful in simplifying computation of persistent homology (topological data analysis)
• Facilitates proof of excision theorem in homology theory (algebraic topology)

Barycentric Subdivision for Refinement

Topological Space Triangulation

• Barycentric subdivision creates finer approximations of continuous topological spaces by refining existing triangulations
• Process allows creation of homeomorphisms between topological spaces and geometric realizations of simplicial complexes with arbitrarily fine granularity
• Particularly useful in simplicial homology computations, simplifying calculation of homology groups
• Technique improves accuracy of piecewise linear approximations of smooth manifolds or other topological spaces
• In computational topology, refines meshes for more precise numerical simulations and geometric modeling
• Refinement process helps detect and resolve singularities or complex features in topological space being triangulated
• Plays crucial role in proving equivalence between singular and simplicial homology theories for triangulable spaces

Applications in Mathematics and Engineering

• Used in finite element analysis to increase mesh resolution for improved accuracy (structural engineering)
• Applies to computer graphics for adaptive mesh refinement in 3D modeling and animation
• Facilitates construction of Morse functions on simplicial complexes (differential topology)
• Enables more accurate approximation of minimal surfaces (differential geometry)
• Improves discretization of partial differential equations on complex domains (numerical analysis)
• Aids in constructing triangulations of algebraic varieties (algebraic geometry)