The collinearity check and line parameterization tripped me up more than I expected.
First, normalize each segment by identifying its infinite line (using a canonical representation like slope-intercept or normalized coefficients) and projecting its points onto a 1D parameter along that line. Then group segments by line, sort their intervals, and merge overlapping or touching intervals while collecting all unique points; finally, output the sorted unique points for each merged interval.
Pro tip: Emphasize the importance of a canonical line representation to avoid floating-point issues and ensure segments on the same line are grouped correctly; also mention that using exact rational arithmetic or integer cross-products can prevent precision errors.
For each segment, compute a canonical representation of its infinite line (e.g., normalized coefficients A, B, C such that Ax+By+C=0) and use it as a key to group segments that lie on the same line.
For each segment, project all its points onto a 1D parameter along the line (e.g., using dot product with a direction vector) to obtain a set of scalar values representing the segment's extent.
For each line group, sort the intervals by their start parameter, then merge intervals that overlap or touch (i.e., next.start <= current.end) while maintaining a set of unique points from all contributing segments.
For each merged interval, gather all unique points from the contributing segments, sort them by their 1D parameter, and output the ordered list of points.
Consider edge cases like vertical lines, duplicate points, and floating-point precision; discuss time complexity (O(N log N) due to sorting) and space complexity.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.