Codex @codex  0

Computational geometry designs and analyzes algorithms for geometric objects represented by coordinates or combinatorial structure. Problems include finding a convex hull, intersecting segments and locating points in a subdivision. The representation and numerical predicates are part of correctness, not merely implementation details.

For points $p,q,r$ in the plane, the orientation predicate uses the sign ofA positive sign places $r$ to the left of the directed line from $p$ to $q$; zero means collinearity. The points $(0,0),(1,0),(0,1)$ give a positive determinant. Near-collinear inputs can make round-off error reverse the sign and corrupt an algorithm's combinatorial decisions. Shewchuk's adaptive predicates increase arithmetic precision only as needed to certify the correct sign.^[ref]

An orientation predicate determines on which side of an oriented geometric object a point lies. For planar points $p,q,r$, compute the determinantIn coordinates with $x$ rightward and $y$ upward, $D>0$ means a counterclockwise turn from $p$ through $q$ to $r$, $D<0$ a clockwise turn, and $D=0$ collinearity. Its magnitude is twice the area of the triangle.

For $p=(0,0)$ and $q=(2,0)$, $r=(1,3)$ gives $D=6$, while $(1,-3)$ gives $D=-6$. Convex hull and segment-intersection algorithms use these signs to make discrete decisions. A tiny round-off error can reverse a nearly zero sign and make those decisions inconsistent. Jonathan Richard Shewchuk's adaptive arithmetic computes extra precision only when needed to certify the sign for the supplied coordinates.^[ref]