Overview

When working with 2D grids, a common computational geometry problem involves determining exactly which cells a specific vector or line segment traverses. Unlike visual inspection, which is intuitiev, calculating this programmatically requires handling the intersection of the continuous line segment with the discrete boundaries of the grid. This process involves calculating the exact coordinates where the line crosses vertical and horizontal grid lines, sorting these intersection points, and mapping them to their respective grid indices.

Algorithmic Approach

Instead of iterating through every cell in the grid to check for intersection (which is inefficient for large grids), the optimal approach involves algebraic calculation:

1. Define the Line: Calculate the slope ($m$) and y-intercept ($b$) of the line segment connecting the two points.
2. Find Intersections: Solve for the coordinates where this line intersects the vertical lines ($x = k$) and horizontal lines ($y = k$) that constitute the grid.
3. Filter and Sort: Discard any intersection points that fall outside the bounds of the line segment. Sort the remaining points (inlcuding the start and end points) by their distance from the segment's origin to ensure correct order.
4. Map to Cells: The line segment between any two consecutive points in this sorted list lies entirely within a single grid cell. The midpoint of this sub-segment can be used to identify the cell coordinates.

Implementation in MATLAB

The following script implements this logic. It defines a grid, establishes the line parameters, and calculates the precise traversal path through the cells. It avoids brute-force checking by focusing only on relevant grid lines.

% Define the Grid Resolution
xGrid = 0:5;
yGrid = 0:5;

% Define the Line Segment Endpoints
startNode = [0.35, 2.65];
endNode   = [4.20, 4.73];

% 1. Derive Line Equation: y = mx + c
deltaX = endNode(1) - startNode(1);

% Check for vertical lines to prevent division by zero
if abs(deltaX) < 1e-10
    error('Vertical line detected. This implementation requires non-vertical slopes.');
end

slope = (endNode(2) - startNode(2)) / deltaX;
intercept = startNode(2) - slope * startNode(1);

% 2. Calculate Intersections with Vertical Grid Lines
% Identify x-grid lines strictly between the segment endpoints
minX = min(startNode(1), endNode(1));
maxX = max(startNode(1), endNode(1));
validVerticalX = xGrid(xGrid > minX & xGrid < maxX);

% Calculate y coordinates for these x values
yAtVerticalX = slope * validVerticalX + intercept;
verticalCrossings = [validVerticalX', yAtVerticalX'];

% 3. Calculate Intersections with Horizontal Grid Lines
% Identify y-grid lines strictly between the segment endpoints
minY = min(startNode(2), endNode(2));
maxY = max(startNode(2), endNode(2));
validHorizontalY = yGrid(yGrid > minY & yGrid < maxY);

% Calculate x coordinates for these y values
xAtHorizontalY = (validHorizontalY - intercept) / slope;
horizontalCrossings = [xAtHorizontalY', validHorizontalY'];

% 4. Aggregate and Sort Intersection Points
% Combine endpoints and internal crossings
allCrossings = [startNode; endNode; verticalCrossings; horizontalCrossings];

% Sort points based on their proximity to the start point to maintain topology
distancesFromStart = hypot(allCrossings(:,1) - startNode(1), ...
                          allCrossings(:,2) - startNode(2));
[~, sortIndices] = sort(distancesFromStart);
sortedCrossings = allCrossings(sortIndices, :);

% 5. Remove Duplicates and Outliers
% Small tolerance for floating point arithmetic
tolerance = 1e-9;
uniqueMask = [true; any(abs(diff(sortedCrossings)) > tolerance, 2)];
cleanPath = sortedCrossings(uniqueMask, :);

% 6. Calculate Segment Lengths and Cell IDs
numSegments = size(cleanPath, 1) - 1;
segmentLengths = zeros(numSegments, 1);
cellIndices = zeros(numSegments, 2);

disp('Cell ID | Segment Length');
disp('------------------------');

for i = 1:numSegments
    pA = cleanPath(i, :);
    pB = cleanPath(i+1, :);
    
    % Euclidean distance for the segment within the cell
    currentLen = norm(pB - pA);
    segmentLengths(i) = currentLen;
    
    % Calculate midpoint to find containing cell
    midPoint = (pA + pB) / 2;
    
    % Determine grid indices (assuming 1-based indexing starts at x(1), y(1))
    colIdx = find(midPoint(1) <= xGrid(2:end), 1, 'first');
    rowIdx = find(midPoint(2) <= yGrid(2:end), 1, 'first');
    
    cellIndices(i, :) = [colIdx, rowIdx];
    
    fprintf('  [%d,%d]  |    %.4f\n', colIdx, rowIdx, currentLen);
end

% Visualization (Optional)
figure;
hold on;
grid on;
% Draw Grid
xLimits = [min(xGrid), max(xGrid)];
yLimits = [min(yGrid), max(yGrid)];
for k = 1:length(xGrid)
    plot([xGrid(k), xGrid(k)], yLimits, 'k-');
end
for k = 1:length(yGrid)
    plot(xLimits, [yGrid(k), yGrid(k)], 'k-');
end
% Draw Line Segment
plot(cleanPath(:,1), cleanPath(:,2), 'r-', 'LineWidth', 2);
plot(cleanPath(:,1), cleanPath(:,2), 'bo', 'MarkerFaceColor', 'b');
axis equal;
hold off;

This script outputs the coordinates of the line as it passes through the grid, effectively discretizing the continuous line into segments corresponding to the grid cells. The midpoint logic ensures accurate cell identification without needing to check every cell in the grid manually.