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feat(geometry): Add Bentley-Ottmann algorithm for line segment intersection (#6871)
* feat(geometry): add Bentley-Ottmann line segment intersection algorithm - Implement sweep-line algorithm for finding all intersection points - Time complexity: O((n + k) log n) where n is segments, k is intersections - Uses event queue (PriorityQueue) and status structure (TreeSet) - Handles vertical/horizontal segments, collinear overlaps, and touching endpoints - Includes comprehensive Javadoc with examples and references * test(geometry): add comprehensive tests for Bentley-Ottmann algorithm - 19 test cases covering typical, edge, and degenerate cases - Tests include: single/multiple intersections, parallel segments, grid patterns - Performance test with 100 random segments - All tests validate correctness of intersection detection * style(geometry): fix code style * test(geometry): Achieve 100% test coverage for BentleyOttmann * style(geometry): fix code style again * fix: correct import order in BentleyOttmann * fix(geometry): Resolve SpotBugs and PMD static analysis warnings * Reorder import statements in BentleyOttmannTest --------- Co-authored-by: Deniz Altunkapan <deniz.altunkapan@outlook.com>
This commit is contained in:
Indolyn Yi
423
src/main/java/com/thealgorithms/geometry/BentleyOttmann.java
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423
src/main/java/com/thealgorithms/geometry/BentleyOttmann.java
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package com.thealgorithms.geometry;
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import java.awt.geom.Point2D;
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import java.util.ArrayList;
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import java.util.Comparator;
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import java.util.HashMap;
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import java.util.HashSet;
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import java.util.List;
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import java.util.Map;
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import java.util.NavigableSet;
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import java.util.Objects;
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import java.util.PriorityQueue;
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import java.util.Set;
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import java.util.SortedSet;
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import java.util.TreeSet;
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/**
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* Implementation of the Bentley–Ottmann algorithm for finding all intersection
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* points among a set of line segments in O((n + k) log n) time.
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*
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* <p>Uses a sweep-line approach with an event queue and status structure to
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* efficiently detect intersections in 2D plane geometry.</p>
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*
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* @see <a href="https://en.wikipedia.org/wiki/Bentley%E2%80%93Ottmann_algorithm">
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* Bentley–Ottmann algorithm</a>
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*/
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public final class BentleyOttmann {
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private BentleyOttmann() {
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}
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private static final double EPS = 1e-9;
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private static double currentSweepX;
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/**
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* Represents a line segment with two endpoints.
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*/
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public static class Segment {
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final Point2D.Double p1;
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final Point2D.Double p2;
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final int id; // Unique identifier for each segment
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Segment(Point2D.Double p1, Point2D.Double p2) {
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this.p1 = p1;
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this.p2 = p2;
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this.id = segmentCounter++;
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}
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private static int segmentCounter = 0;
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/**
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* Computes the y-coordinate of this segment at a given x value.
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*/
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double getY(double x) {
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if (Math.abs(p2.x - p1.x) < EPS) {
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// Vertical segment: return midpoint y
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return (p1.y + p2.y) / 2.0;
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}
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double t = (x - p1.x) / (p2.x - p1.x);
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return p1.y + t * (p2.y - p1.y);
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}
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Point2D.Double leftPoint() {
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return p1.x < p2.x ? p1 : p1.x > p2.x ? p2 : p1.y < p2.y ? p1 : p2;
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}
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Point2D.Double rightPoint() {
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return p1.x > p2.x ? p1 : p1.x < p2.x ? p2 : p1.y > p2.y ? p1 : p2;
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}
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@Override
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public String toString() {
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return String.format("S%d[(%.2f, %.2f), (%.2f, %.2f)]", id, p1.x, p1.y, p2.x, p2.y);
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}
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}
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/**
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* Event types for the sweep line algorithm.
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*/
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private enum EventType { START, END, INTERSECTION }
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/**
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* Represents an event in the event queue.
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*/
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private static class Event implements Comparable<Event> {
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final Point2D.Double point;
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final EventType type;
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final Set<Segment> segments; // Segments involved in this event
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Event(Point2D.Double point, EventType type) {
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this.point = point;
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this.type = type;
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this.segments = new HashSet<>();
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}
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void addSegment(Segment s) {
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segments.add(s);
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}
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@Override
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public int compareTo(Event other) {
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// Sort by x-coordinate, then by y-coordinate
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int cmp = Double.compare(this.point.x, other.point.x);
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if (cmp == 0) {
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cmp = Double.compare(this.point.y, other.point.y);
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}
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if (cmp == 0) {
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// Process END events before START events at same point
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cmp = this.type.compareTo(other.type);
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}
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return cmp;
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}
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@Override
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public boolean equals(Object o) {
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if (!(o instanceof Event e)) {
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return false;
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}
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return pointsEqual(this.point, e.point);
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}
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@Override
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public int hashCode() {
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return Objects.hash(Math.round(point.x * 1e6), Math.round(point.y * 1e6));
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}
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}
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/**
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* Comparator for segments in the status structure (sweep line).
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* Orders segments by their y-coordinate at the current sweep line position.
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*/
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private static final class StatusComparator implements Comparator<Segment> {
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@Override
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public int compare(Segment s1, Segment s2) {
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if (s1.id == s2.id) {
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return 0;
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}
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double y1 = s1.getY(currentSweepX);
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double y2 = s2.getY(currentSweepX);
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int cmp = Double.compare(y1, y2);
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if (Math.abs(y1 - y2) < EPS) {
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// If y-coordinates are equal, use segment id for consistency
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return Integer.compare(s1.id, s2.id);
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}
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return cmp;
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}
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}
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/**
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* Finds all intersection points among a set of line segments.
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*
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* <p>An intersection point is reported when two or more segments cross or touch.
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* For overlapping segments, only actual crossing/touching points are reported,
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* not all points along the overlap.</p>
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*
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* @param segments list of line segments represented as pairs of points
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* @return a set of intersection points where segments meet or cross
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* @throws IllegalArgumentException if the list is null or contains null points
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*/
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public static Set<Point2D.Double> findIntersections(List<Segment> segments) {
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if (segments == null) {
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throw new IllegalArgumentException("Segment list must not be null");
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}
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Segment.segmentCounter = 0; // Reset counter
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Set<Point2D.Double> intersections = new HashSet<>();
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PriorityQueue<Event> eventQueue = new PriorityQueue<>();
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TreeSet<Segment> status = new TreeSet<>(new StatusComparator());
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Map<Point2D.Double, Event> eventMap = new HashMap<>();
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// Initialize event queue with segment start and end points
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for (Segment s : segments) {
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Point2D.Double left = s.leftPoint();
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Point2D.Double right = s.rightPoint();
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Event startEvent = getOrCreateEvent(eventMap, left, EventType.START);
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startEvent.addSegment(s);
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Event endEvent = getOrCreateEvent(eventMap, right, EventType.END);
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endEvent.addSegment(s);
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}
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// Add all unique events to the queue
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for (Event e : eventMap.values()) {
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if (!e.segments.isEmpty()) {
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eventQueue.add(e);
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}
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}
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// Process events
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while (!eventQueue.isEmpty()) {
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Event event = eventQueue.poll();
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currentSweepX = event.point.x;
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handleEvent(event, status, eventQueue, eventMap, intersections);
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}
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return intersections;
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}
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private static Event getOrCreateEvent(Map<Point2D.Double, Event> eventMap, Point2D.Double point, EventType type) {
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// Find existing event at this point
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for (Map.Entry<Point2D.Double, Event> entry : eventMap.entrySet()) {
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if (pointsEqual(entry.getKey(), point)) {
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return entry.getValue();
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}
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}
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// Create new event
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Event event = new Event(point, type);
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eventMap.put(point, event);
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return event;
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}
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private static void handleEvent(Event event, TreeSet<Segment> status, PriorityQueue<Event> eventQueue, Map<Point2D.Double, Event> eventMap, Set<Point2D.Double> intersections) {
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Point2D.Double p = event.point;
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Set<Segment> segmentsAtPoint = new HashSet<>(event.segments);
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// Check segments in status structure (much smaller than allSegments)
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for (Segment s : status) {
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if (pointsEqual(s.p1, p) || pointsEqual(s.p2, p) || (onSegment(s, p) && !pointsEqual(s.p1, p) && !pointsEqual(s.p2, p))) {
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segmentsAtPoint.add(s);
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}
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}
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// If 2 or more segments meet at this point, it's an intersection
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if (segmentsAtPoint.size() >= 2) {
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intersections.add(p);
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}
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// Categorize segments
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Set<Segment> upperSegs = new HashSet<>(); // Segments starting at p
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Set<Segment> lowerSegs = new HashSet<>(); // Segments ending at p
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Set<Segment> containingSegs = new HashSet<>(); // Segments containing p in interior
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for (Segment s : segmentsAtPoint) {
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if (pointsEqual(s.leftPoint(), p)) {
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upperSegs.add(s);
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} else if (pointsEqual(s.rightPoint(), p)) {
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lowerSegs.add(s);
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} else {
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containingSegs.add(s);
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}
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}
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// Remove ending segments and segments containing p from status
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status.removeAll(lowerSegs);
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status.removeAll(containingSegs);
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// Update sweep line position slightly past the event
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currentSweepX = p.x + EPS;
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// Add starting segments and re-add containing segments
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status.addAll(upperSegs);
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status.addAll(containingSegs);
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if (upperSegs.isEmpty() && containingSegs.isEmpty()) {
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// Find neighbors and check for new intersections
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Segment sl = getNeighbor(status, lowerSegs, true);
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Segment sr = getNeighbor(status, lowerSegs, false);
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if (sl != null && sr != null) {
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findNewEvent(sl, sr, p, eventQueue, eventMap);
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}
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} else {
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Set<Segment> unionSegs = new HashSet<>(upperSegs);
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unionSegs.addAll(containingSegs);
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Segment leftmost = getLeftmost(unionSegs, status);
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Segment rightmost = getRightmost(unionSegs, status);
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if (leftmost != null) {
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Segment sl = status.lower(leftmost);
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if (sl != null) {
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findNewEvent(sl, leftmost, p, eventQueue, eventMap);
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}
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}
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if (rightmost != null) {
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Segment sr = status.higher(rightmost);
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if (sr != null) {
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findNewEvent(rightmost, sr, p, eventQueue, eventMap);
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}
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}
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}
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}
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private static Segment getNeighbor(NavigableSet<Segment> status, Set<Segment> removed, boolean lower) {
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if (removed.isEmpty()) {
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return null;
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}
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Segment ref = removed.iterator().next();
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return lower ? status.lower(ref) : status.higher(ref);
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}
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private static Segment getLeftmost(Set<Segment> segments, SortedSet<Segment> status) {
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Segment leftmost = null;
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for (Segment s : segments) {
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if (leftmost == null || Objects.requireNonNull(status.comparator()).compare(s, leftmost) < 0) {
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leftmost = s;
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}
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}
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return leftmost;
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}
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private static Segment getRightmost(Set<Segment> segments, SortedSet<Segment> status) {
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Segment rightmost = null;
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for (Segment s : segments) {
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if (status.comparator() != null && (rightmost == null || status.comparator().compare(s, rightmost) > 0)) {
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rightmost = s;
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}
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}
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return rightmost;
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}
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private static void findNewEvent(Segment s1, Segment s2, Point2D.Double currentPoint, PriorityQueue<Event> eventQueue, Map<Point2D.Double, Event> eventMap) {
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Point2D.Double intersection = getIntersection(s1, s2);
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if (intersection != null && intersection.x > currentPoint.x - EPS && !pointsEqual(intersection, currentPoint)) {
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// Check if event already exists
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boolean exists = false;
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for (Map.Entry<Point2D.Double, Event> entry : eventMap.entrySet()) {
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if (pointsEqual(entry.getKey(), intersection)) {
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exists = true;
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Event existingEvent = entry.getValue();
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existingEvent.addSegment(s1);
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existingEvent.addSegment(s2);
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break;
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}
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}
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if (!exists) {
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Event newEvent = new Event(intersection, EventType.INTERSECTION);
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newEvent.addSegment(s1);
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newEvent.addSegment(s2);
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eventMap.put(intersection, newEvent);
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eventQueue.add(newEvent);
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}
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}
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}
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private static Point2D.Double getIntersection(Segment s1, Segment s2) {
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double x1 = s1.p1.x;
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double y1 = s1.p1.y;
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double x2 = s1.p2.x;
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double y2 = s1.p2.y;
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double x3 = s2.p1.x;
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double y3 = s2.p1.y;
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double x4 = s2.p2.x;
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double y4 = s2.p2.y;
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double denom = (x1 - x2) * (y3 - y4) - (y1 - y2) * (x3 - x4);
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if (Math.abs(denom) < EPS) {
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// Parallel or collinear
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if (areCollinear(s1, s2)) {
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// For collinear segments, check if they overlap
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// Return any overlapping point
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List<Point2D.Double> overlapPoints = new ArrayList<>();
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if (onSegment(s1, s2.p1)) {
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overlapPoints.add(s2.p1);
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}
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if (onSegment(s1, s2.p2)) {
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overlapPoints.add(s2.p2);
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}
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if (onSegment(s2, s1.p1)) {
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overlapPoints.add(s1.p1);
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}
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if (onSegment(s2, s1.p2)) {
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overlapPoints.add(s1.p2);
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}
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// Remove duplicates and return the first point
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if (!overlapPoints.isEmpty()) {
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// Find the point that's not an endpoint of both segments
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for (Point2D.Double pt : overlapPoints) {
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boolean isS1Endpoint = pointsEqual(pt, s1.p1) || pointsEqual(pt, s1.p2);
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boolean isS2Endpoint = pointsEqual(pt, s2.p1) || pointsEqual(pt, s2.p2);
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// If it's an endpoint of both, it's a touching point
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if (isS1Endpoint && isS2Endpoint) {
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return pt;
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}
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}
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// Return the first overlap point
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return overlapPoints.getFirst();
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}
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}
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return null;
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}
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double t = ((x1 - x3) * (y3 - y4) - (y1 - y3) * (x3 - x4)) / denom;
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double u = -((x1 - x2) * (y1 - y3) - (y1 - y2) * (x1 - x3)) / denom;
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if (t >= -EPS && t <= 1 + EPS && u >= -EPS && u <= 1 + EPS) {
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double px = x1 + t * (x2 - x1);
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double py = y1 + t * (y2 - y1);
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return new Point2D.Double(px, py);
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}
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return null;
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}
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private static boolean areCollinear(Segment s1, Segment s2) {
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double cross1 = crossProduct(s1.p1, s1.p2, s2.p1);
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double cross2 = crossProduct(s1.p1, s1.p2, s2.p2);
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return Math.abs(cross1) < EPS && Math.abs(cross2) < EPS;
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}
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private static double crossProduct(Point2D.Double a, Point2D.Double b, Point2D.Double c) {
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return (b.x - a.x) * (c.y - a.y) - (b.y - a.y) * (c.x - a.x);
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}
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private static boolean onSegment(Segment s, Point2D.Double p) {
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return p.x >= Math.min(s.p1.x, s.p2.x) - EPS && p.x <= Math.max(s.p1.x, s.p2.x) + EPS && p.y >= Math.min(s.p1.y, s.p2.y) - EPS && p.y <= Math.max(s.p1.y, s.p2.y) + EPS && Math.abs(crossProduct(s.p1, s.p2, p)) < EPS;
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}
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private static boolean pointsEqual(Point2D.Double p1, Point2D.Double p2) {
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return Math.abs(p1.x - p2.x) < EPS && Math.abs(p1.y - p2.y) < EPS;
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}
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}
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