Jack Snoeyink & Andrea Mantler Computer Science, UNC Chapel Hill

Polygon overlay in double precision arithmeticOne example of why robust geometric code is hard to write Jack Snoeyink & Andrea Mantler Computer Science, UNC Chapel Hill

Outline • Motivating problems • Clipping, polygon ops, overlay, arrangement • Study precision required by algorithm • Quick summary of algorithms • Test pairs, sweep, topological sweep • A double-precision sweep algorithm • “Spaghetti” segments • Conclusions

Three problems in the plane • Polygon clipping (graphics) • Boolean operations (CAD) • Map overlay (GIS)

Build red/blue arrangement • Build the arrangement of: nred and nblue line segments in plane, specified by their endpoint coordinates & having no red/red or blue/blue crossings.

Why precision is an issue • Algorithms find geometric relationships from coordinate computations. • Efficient algorithms compute only a few relationships and derive the rest.

Assumptions & Goal • Assumptions for correctness • Solve the exact problem given by the input • Work for any distribution of the input • Goal: Input+output sensitive algorithm • Demand least precision possible • O(n log n + k) for n segs, k intersections

Algorithms to build line segment arrangements • Brute force: test all pairs • Sweep the plane with a line [BO79,C92] • Topological sweep [CE92] • Divide & Conquer [B95] • Trapezoid sweep [C94]

Four geometric tests • Orientation/Intersection test • Intersection-in-Slab • Order along line • Order by x coordinate

Algorithms to build line segment arrangements • Brute force: test all pairs • Sweep the plane with a line [BO79,C92] • Topological sweep [CE92] • Divide & Conquer [B95]

Brute force • Test all pairs for intersection

Brute force • Test all pairs for intersection • Sort along lines • Break&rejoin segs

Plane sweep [BO79,C92] • Maintain order along sweep line

Plane sweep [BO79,C92] • Maintain order along sweep line • Know all intersections behind • Nextevent queue

Topological sweep [CE92] • Maintain order along sweep curve • Know all intersections behind

Topological sweep [CE92] • Maintain order along sweep curve • Know all intersections behind • “20 easy pieces”

Divide and Conquer [B95] • Find intersections in slab with staircase

Divide and Conquer [B95] • Find intersections in slab with staircase • Remove staircase

Divide and Conquer [B95] • Find intersections in slab with staircase • Remove staircase • Partition & repeat

Degrees of predicates • Orientation/Intersection test (deg. 2) • Intersection-in-Slab (deg. 3) • Order along line (deg. 4) • Order by x coordinate (deg. 5)

Degree computations • Point p = (1,px,py) = ((0),(1),(1)) • Line equation through points p, q:

Degree computations • Point p = (1,px,py) = ((0),(1),(1)) • Line equation: (2)W+(1)X+(1)Y • Point at intersection of two lines:

Degree computations • Point p = (1,px,py) = ((0),(1),(1)) • Line equation: (2)W+(1)X+(1)Y • Point at intersection ((2),(3),(3)) • Orientation Test:(2)(0)+(1)(1)+(1)(1) = (2) • In Slab: (1) < (3)/(2) or (2)(1) < (3) = (3) • Same Line: (2)(2) + (1)(3) + (1)(3) = (4) • x Order: (3)/(2) < (3)/(2) or (5)<(5) = (5)

Restrict to double precision Degree of algorithms to build line segment arrangements • Brute force (2/4) • Sweep with a line (5) • Topological sweep (4) • Divide & Conquer (3/4) • Trapezoid sweep (3)

Restricted predicates imply... • Restricted to double precision: • Can’t test where an intersection is • Can’t sort on lines • Can’t sort by x

Spaghetti lines • Restricted to double precision: • Can’t test where an intersection is • Can’t sort on lines • Can’t sort by x

Spaghetti lines • Restricted to double precision: • Push segments as far right as possible

Spaghetti lines • Restricted to double precision: • Push segments as far right as possible • Endpoints witness intersections

A sweep for red/blue spaghetti • Maintain order along sweep consistent with pushing intersections to right • Detect an intersection when the sweep passes its witness

Data Structures • Sweep line:

Data Structures • Sweep line: • Alternate bundles of red and blue segs

Data Structures • Sweep line: • Alternate bundles of red and blue segs • Bundles are in doubly-linked list

Data Structures • Sweep line: • Alternate bundles of red and blue segs • Bundles are in doubly-linked list • Blue bundles are in a balanced tree

Data Structures • Sweep line: • Alternate bundles of red and blue segs • Bundles are in doubly-linked list • Blue bundles are in a balanced tree • Each bundle is a small tree

Events • Sweep events • Now only at vertices • Processing

Event processing • Sweep events • Now only at vertices • Processing red • Use trees to locate point in blue bundle

Event processing • Sweep events • Now only at vertices • Processing red • Use trees to locate point in blue bundle • Use linked list to locate in red

Event processing • Sweep events • Now only at vertices • Processing red • Use trees to locate point in blue bundle • Use linked list to locate in red • Split/merge bundles to restore invariant

Event processing • Process blue the same • Time: O(n log n + k) • Tree operations prop to # of vertices • Bundle operations prop to # of intersections • Degeneracies can easily be handled

Handling degeneracies • Shared endpoints • Endpoint on a line • Collinear segments

Jack Snoeyink & Andrea Mantler Computer Science, UNC Chapel Hill