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Advanced - Geometry · 3:55
Orientation & Oriented Triangle Area: the Turn Test
Twice the oriented area of a triangle, 2S = (p2 - p1) × (p3 - p2), is a 2×2 determinant that is exact in integers. Its sign says whether the path p1 → p2 → p3 turns left (counter-clockwise), right (clockwise) or goes straight (collinear), and half its absolute value is the area. This one turn test underlies segment intersection, convex hulls and point-in-polygon checks, provided you keep one point order, use long long and give doubles a tolerance.
You will learn
- Compute 2S = (p2 - p1) × (p3 - p2) for three points and read its sign as left turn, right turn or collinear
- Explain the formula as a 2×2 determinant whose parallelogram the diagonal p1p3 cuts into two equal triangles
- Recognise equivalent spellings of the turn test, keep one point order, and use translation invariance to shrink numbers
- Avoid int overflow with long long and treat nearly collinear double results with an EPS tolerance