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How Two’s Complement Keeps Parallel Lines Straight

1. Introduction: The Geometry of Invariance in Projective Space

Parallel lines in Euclidean geometry never meet, yet their behavior under projection reveals deep invariance—key to projective geometry. In projective space, straight lines intersect “at infinity,” preserving collinearity despite geometric transformations. Homogeneous coordinates encode this by representing points as triples \((x, y, w)\), where \(w\) determines whether a point lies in finite space (\(w

e 0\)) or at infinity (\(w = 0\)). This unification allows geometric relationships to persist across projections, forming the foundation

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