Now that we have indexed our cells and lines, we can now index the borders, the edges, and the corners. A border is simply a side of a square. An edge pertains to an end of a line. A corner is the common cell of two adjacent lines. In essence, boundaries set limits to lines and cells with regards to the base.
The borders represent the four sides of the square. A border is a kind of a line that defines the outermost cells of the square. In a square, there are two horizontal borders and two vertical borders. The horizontal borders are top border and the bottom border. The vertical borders are left border and the right border.
l_hborders = 2
l_vborders = 2
l_borders = l_corners = l_hborders + l_vborders = 4
A border is always a line.
The top border is always the first horizontal border.
B_top = r_1
where B_top is top border and r_1 is row index 1.
The bottom border is always the last horizontal border.
B_bottom = r_b
where B_bottom is bottom border and r_b is a row with its index equal to the base.
The left border is always the first vertical border.
B_left = c_1
where B_left is left border and c_1 is column index 1.
The right border is always the last vertical border. (See 1.1.3)
B_right = c_b
where B_right is right border and c_b is a column with its index equal to the base.
To see which cell belongs in which border, see if it belongs to either row index 1, column index 1, row index b, or column index b. (see 1.3.5 and 1.3.6.)
r(1) = B_top
c(1) = B_left
r(b) = B_bottom
c(b) = B_right
The edges are the ends of a line. It can also pertain to cells found in a border. There are two kinds of edges: horizontal edges and vertical edges. Horizontal edges are cells from horizontal borders whilst vertical edges are cells from vertical borders.
To count all of the edges of a square, one must mind that squares have four sides(borders) and four corners.
l_edges = (b*l_borders) - l_corners
where l_edges are the number of the edges; l_borders, number of all borders; l_corners number of all corners; and b the base.
Cells of the top border. All integers from 1 to b constitute the indices of edges. All of these cells belong to first row, row index 1.
1 <= n <= b
where n is the cell index and b is the base
Cells of the bottom border. It always contains the cell with index S. All of these cells belong to the last row: row index b.
S-(b-1) <= n <= S
where n is the cell index, S is the square size, and b is the base
Cells of the left border. The topmost left edge is always 1. All of these cells belong to the first column: column index 1.
edges_left(r) = 1 + (b*r) - b
1 <= r <= b
1 <= n <= S-(b-1)
where S-(b-1) is the limit. S is the square size, b the base, r the row index, and n the cell index.
Cells of the right border. The topmost right edge is always b. All of these cells belong to the last column: column index b.
edges_right(r) = b*r
1 <= r <= b
1 <= n <= S
where S is the limit and the square size, b the base, r the row index, and n the cell index.
The corners are special kinds of edges that belongs to two borders. There are four corners because there are four possible combinations of two adjacent borders out of four.
The top-left corner always has a value of 1. This edge belongs to both top border and left border.
corner_tl = 1
The top-right corner always has a value of b. This edge belongs to both top border and right border.
corner_tr = b
The bottom-left corner is somewhat related to the square size and the base. This edge belongs to both bottom border and left border.
corner_bl = S - (b - 1)
The bottom-right corner is always the edge of the last row and the last column. This edge belongs to both bottom border and right border.
corner_br = S
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