Spatial Topologies Analysis

🌍 Advanced Geospatial Analysis

This program implements and verifies three fundamental operations that apply to the eleven region-region relations R in S2 as defined in Point-Set Topological Spatial Relations: converse (conv), leftDual (ld), and rightDual (rd).

Repository: https://github.com/ecorey/Spatial-Topologies

The spatial relations are represented as 3x3 intersection matrices in S2 space:

embrace

[[1, 1, 1], [1, 0, 0], [1, 0, 0]]

attach

[[0, 0, 1], [0, 1, 0], [1, 0, 0]]

disjoin

[[0, 0, 1], [0, 0, 1], [1, 1, 1]]

entwined

[[1, 1, 1], [1, 1, 0], [1, 0, 0]]

meet

[[0, 0, 1], [0, 1, 1], [1, 1, 1]]

overlap

[[1, 1, 1], [1, 1, 1], [1, 1, 1]]

coveredBy

[[1, 0, 0], [1, 1, 0], [1, 1, 1]]

covers

[[1, 1, 1], [0, 1, 1], [0, 0, 1]]

inside

[[1, 0, 0], [1, 0, 0], [1, 1, 1]]

equals

[[1, 0, 0], [0, 1, 0], [0, 0, 1]]

contains

[[1, 1, 1], [0, 0, 1], [0, 0, 1]]

Fundamental Operations

Three fundamental operations apply to all eleven region-region relations in S2 space:

converse (conv)

Reverses the spatial relationship by transposing the intersection matrix.

  • • Swaps matrix elements across the diagonal
  • • Effectively reverses the direction of the spatial relationship
  • • Symmetrical matrices return unchanged

leftDual (ld)

Applies left dual transformation by swapping the first and third rows.

  • • Exchanges row[0] with row[2]
  • • Preserves middle row structure
  • • Creates complementary spatial relationship

rightDual (rd)

Applies right dual transformation by swapping the first and third columns.

  • • Exchanges column[0] with column[2]
  • • Preserves middle column structure
  • • Creates complementary spatial relationship

Verification Tests

The program verifies six fundamental equivalence relationships between operations:

Primary Equivalence Tests

a: conv(ld(R)) ≡ ld(conv(R))
b: conv(rd(R)) ≡ rd(conv(R))
c: ld(rd(R)) ≡ rd(ld(R))

Extended Equivalence Tests

d: rd(conv(ld(R))) ≡ rd(ld(conv(R)))
e: ld(conv(rd(R))) ≡ ld(rd(conv(R)))
f: conv(ld(rd(R))) ≡ conv(rd(ld(R)))

Code Implementation

Left Dual Operation

Swaps the first and third rows of the intersection matrix:

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 Operation

Swaps the first and third columns of the intersection matrix:

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 Operation

Transposes the intersection matrix to reverse the spatial relationship:

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
        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

Installation & Setup

Prerequisites

  • • Python: Version 3.6 or later
  • • NumPy: For matrix operations (optional but recommended)
  • • Git: For repository management

Clone Repository

git clone https://github.com/ecorey/Spatial-Topologies.git
cd Spatial-Topologies

Install Dependencies (Optional)

pip install numpy  # For enhanced matrix operations

Usage Examples

Basic Operation Example

Testing Spatial Relationship Operations

# Define a spatial relation matrix (example: "embrace")
embrace_matrix = [[1, 1, 1], [1, 0, 0], [1, 0, 0]]

# Apply operations
left_dual_result = lft_dual(embrace_matrix.copy())
right_dual_result = rt_dual(embrace_matrix.copy())
converse_result = converse(embrace_matrix.copy())

print("Original:", embrace_matrix)
print("Left Dual:", left_dual_result)
print("Right Dual:", right_dual_result)
print("Converse:", converse_result)

Verification Testing

Testing Equivalence Relationships

# Test equivalence: conv(ld(R)) ≡ ld(conv(R))
original = [[1, 1, 1], [1, 0, 0], [1, 0, 0]]

# Left path: conv(ld(R))
left_path = converse(lft_dual(original.copy()))

# Right path: ld(conv(R))
right_path = lft_dual(converse(original.copy()))

# Verify equivalence
if left_path == right_path:
    print("✅ Equivalence test passed!")
else:
    print("❌ Equivalence test failed!")

💡 Key Features

  • • Exhaustively verifies six fundamental equivalence relationships
  • • Implements all eleven region-region spatial relations in S2
  • • Tests converse, left dual, and right dual operations
  • • Validates matrix transformation equivalences
  • • Demonstrates point-set topological spatial relations theory