Three operations that apply to the eleven region-region relations R in S2 as shown in the Point-Set Topological Spatial Relations are as follows:

• converse (conv)
• leftDual (ld)
• rightDual (rd)

Each of these three operations given will also produce a relation as a result, and the operations can also be applied recursively.

This program verifies exhaustively (for each of the eleven relations {R} ) that:

• a: conv(ld(R)) is equivalent to ld(conv(R))

• b: conv(rd(R)) is equivalent to rd(conv(R))

• c: ld(rd (R)) is equivalent to rd(ld(R))

This program also verifies exhaustively (for each of the eleven relations {R} ) that:

• d: rd(conv(ld(R))) is equivalent to rd(ld(conv(R)))

• e: ld(conv(rd(R))) is equivalent to ld(rd(conv(R)))

• f: conv(ld(rd(R))) is equivalent to conv(rd(ld(R)))

Conclusions

• The results in (a, b, c) and (d, e, f)

Conclusions

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The eleven region-region relations R in S2 and their intersection matrix:

embrace = np.array([[1, 1, 1], [1, 0, 0], [1, 0, 0]])

attach = np.array([[0, 0, 1], [0, 1, 0], [1, 0, 0]])

disjoin = np.array([[0, 0, 1], [0, 0, 1], [1, 1, 1]])

entwined = np.array([[1, 1, 1], [1, 1, 0], [1, 0, 0]])

meet = np.array([[0, 0, 1], [0, 1, 1], [1, 1, 1]])

overlap = np.array([[1, 1, 1], [1, 1, 1], [1, 1, 1]])

coveredBy = np.array([[1, 0, 0], [1, 1, 0], [1, 1, 1]])

covers = np.array([[1, 1, 1], [0, 1, 1], [0, 0, 1]])

inside = np.array([[1, 0, 0], [1, 0, 0], [1, 1, 1]])

equals = np.array([[1, 0, 0], [0, 1, 0], [0, 0, 1]])

contains = np.array([[1, 1, 1], [0, 0, 1], [0, 0, 1]])

Operations

Left Dual

def lft_dual(matrix):
    """Returns the left dual of a given matrix."""
    temp = [0, 0, 0]
    temp[0] = matrix[0][0]
    temp[1] = matrix[0][1]
    temp[2] = matrix[0][2]

    matrix[0][0] = matrix[2][0]
    matrix[0][1] = matrix[2][1]
    matrix[0][2] = matrix[2][2]

    matrix[2][0] = temp[0]
    matrix[2][1] = temp[1]
    matrix[2][2] = temp[2]

    return matrix

Right Dual

def rt_dual(matrix):
    """Returns the right dual of a given matrix."""
    temp = [0, 0, 0]
    temp[0] = matrix[0][0]
    temp[1] = matrix[1][0]
    temp[2] = matrix[2][0]

    matrix[0][0] = matrix[0][2]
    matrix[1][0] = matrix[1][2]
    matrix[2][0] = matrix[2][2]

    matrix[0][2] = temp[0]
    matrix[1][2] = temp[1]
    matrix[2][2] = temp[2]

    return matrix

Converse

def converse(matrix):
    """Returns the converse of a given matrix."""
    if (matrix[1][0] == matrix[0][1]) and (matrix[2][0] == matrix[0][2]) and (matrix[2][1] == matrix[1][2]):
        # symmetrical matrices will return the matrix
        return matrix
    elif (matrix[1][0] != matrix[0][1]) and (matrix[2][0] != matrix[0][2]) and (matrix[2][1] != matrix[1][2]):
        # converse matrices will return the converse matrix. For example, meet will
        # return a meet matrix, but inside will return contains, and contains will return inside.
        temp = [0, 0, 0]
        temp[0] = matrix[0][1]
        temp[1] = matrix[0][2]
        temp[2] = matrix[1][2]

        matrix[0][1] = matrix[1][0]
        matrix[0][2] = matrix[2][0]
        matrix[1][2] = matrix[2][1]

        matrix[1][0] = temp[0]
        matrix[2][0] = temp[1]
        matrix[2][1] = temp[2]

        return matrix

Double Dual

def double_dual(matrix):
    """Returns the double dual of a given matrix."""
    x = lft_dual(matrix)
    y = rt_dual(x)
    return y