boundary_word_square

boundary_word_square, a MATLAB code which describes the outline of an object on a grid of squares, using a string of symbols that represent the sequence of steps tracing out the boundary.

The symbols might be 'u', 'd', 'r', and 'l', indicating that the boundary is to be drawn by a succession of movements of unit length, that are up, down, right or left.

In image processing, boundary words are known as "chain codes".

Licensing:

The computer code and data files made available on this web page are distributed under the MIT license

Languages:

boundary_word_square is available in a MATLAB version and an Octave version and a Python version.

Related Data and Programs:

boundary_word_square_test

boundary_word_drafter, a MATLAB code which describes the outline of an object on a grid of drafters, or 30-60-90 triangles, using a string of symbols that represent the sequence of steps tracing out the boundary.

boundary_word_equilateral, a MATLAB code which describes the outline of an object on a grid of equilateral triangles, using a string of symbols that represent the sequence of steps tracing out the boundary.

boundary_word_hexagon, a MATLAB code which describes the outline of an object on a grid of hexagons, using a string of symbols that represent the sequence of steps tracing out the boundary.

boundary_word_right, a MATLAB code which describes the outline of an object on a grid of isoceles right triangles, using a string of symbols that represent the sequence of steps tracing out the boundary.

eternity, a MATLAB code which considers the eternity puzzle, which considers an irregular dodecagon shape that is to be tiled by 209 distinct pieces, each formed by 36 contiguous 30-60-90 triangles, known as polydrafters.

pariomino, a MATLAB code which considers pariominoes, which are polyominoes with a checkerboard parity.

polyomino_parity, a MATLAB code which uses parity considerations to determine whether a given set of polyominoes can tile a specified region.

polyominoes, a MATLAB code which defines, solves, and plots a variety of polyomino tiling problems, which are solved by a direct algebraic approach involving the reduced row echelon form (RREF) of a specific matrix, instead of the more typical brute-force or backtracking methods.

Reference:

1. George Bell,
   The dynamics of spinning polyominoes,
   Gathering for Gardner, G4G13.
2. Srecko Briek, Gilbert Labelle, Ariane Lacasse,
   Algorithms for polyominoes based on the discrete Green theorem,
   Discrete Applied Mathematics,
   Volume 147, pages 187-205, 2005.

Source Code:

• boundary_word_area.m, returns the area of a polyomino defined by a boundary word.
• boundary_word_boundary.m, returns the boundary of a polyomino defined by a boundary word.
• boundary_word_boundary_plot.m, plots the boundary of a polyomino defined by a boundary word.
• boundary_word_centroid.m, returns the centroid of a polyomino defined by a boundary word.
• boundary_word_check.m, checks that a boundary word is valid.
• boundary_word_is_convex.m, is TRUE if the boundary word describes a convex shape.
• boundary_word_moment.m, determines the moment of inertia of a polyomino defined by a boundary word.
• boundary_word_parity.m, determines the parity of a polyomino defined by a boundary word.
• boundary_word_perimeter.m, determines the perimeter of a polyomino defined by a boundary word.
• boundary_word_plot.m, plots a polyomino defined by a boundary word.
• boundary_word_range.m, determines the range of a polyomino defined by a boundary word.
• boundary_word_representative.m, returns the representative for a boundary word, the lexically first member of the equivalence class.
• boundary_word_sense.m, determines the sense of a polyomino defined by a boundary word, which is +1 for a counterclockwise boundary, -1 otherwise.
• boundary_word_transform.m, returns the boundary word of a polyomino after reflections and rotations.
• chvec_eq.m, is TRUE if c1=c2, where c1 and c2 are character strings.
• chvec_lt.m, is TRUE if c1
• chvec_reduce.m, reduces a string of repeated characters to a single occurrence.
• is_octave.m, is TRUE if Octave is executing.
• pentomino_index.m, returns the symbol of any of the 12 pentominoes.
• pentomino_moment.m, returns the moment of inertia of any of the 12 pentominoes.
• pentomino_symbol.m, returns the symbol of the i-th pentomino.
• pentomino_to_boundary_word.m, returns the boundary_word describing any of the 12 pentominoes.
• polygon_contains_point.m, is TRUE if a point is contained in a polygon.
• polyomino_boundary_word.m, returns the boundary_word of a polyomino.
• polyomino_boundary_step.m, takes one step along the boundary of a polyomino, to compute the boundary word.
• polyomino_print.m, prints a polyomino.
• r8mat_print.m, prints an R8MAT.
• r8mat_print_some.m, prints some of an R8MAT.
• r8vec2_print.m, prints a pair of R8VEC's side by side.