plotboilerplate/src/ts/utils/algorithms/convexPolygonIncircle.ts

A simple javascript plotting boilerplate for 2d stuff.

/**
 * @author   mbostock, extended and ported to TypeScript by Ikaros Kappler
 * @date     2020-05-19
 * @modified 2021-02-08 Fixed a lot of es2015 compatibility issues.
 * @modified 2024-03-24 Fixing some unstrict types to match with the new Typescript settings.
 * @version  1.0.3
 * @file     contextPolygonIncircle
 * @public
 */

import { Circle } from "../../Circle";
import { geomutils } from "../../geomutils";
import { Line } from "../../Line";
import { Polygon } from "../../Polygon";
import { Triangle } from "../../Triangle";
import { Vertex } from "../../Vertex";

/**
 * For circle-polygon-intersection-count detection we need an epsilon to
 * eliminate smaller precision errors.
 */
const EPS: number = 0.000001;

export interface PolygonIncircle {
  circle: Circle;
  triangle: Triangle;
}

/**
 * Compute the max sized inlying circle in the given convex (!) polygon - also called the
 * convex-polygon incircle.
 *
 * The function will return an object with either: the circle, and the triangle that defines
 * the three tangent points where the circle touches the polygon.
 *
 * Inspired by
 *  https://observablehq.com/@mbostock/convex-polygon-incircle
 *  https://observablehq.com/@mbostock/circle-tangent-to-three-lines
 *
 *
 * @requires Circle
 * @requires Line
 * @requires Vertex
 * @requires Triangle
 * @requires nsectAngle
 * @requires geomutils
 *
 * @global
 * @name convexPolygonIncircle
 * @param {Polygon} convexHull - The actual convex polygon.
 * @return {PolygonIncircle} A pair of a circle (the incircle) and a triangle (the three points where the circle is touching the polygon).
 */
export const convexPolygonIncircle = (convexHull: Polygon): PolygonIncircle => {
  var n: number = convexHull.vertices.length;
  var bestCircle: Circle | null = null;
  var bestTriangle: Triangle | null = null;
  for (var a = 0; a < n; a++) {
    for (var b = a + 1; b < n; b++) {
      for (var c = b + 1; c < n; c++) {
        // As these lines are part of the convex hull, we know that
        //  * line a preceeds line b and
        //  * line b preceeds line c :)
        let lineA: Line = new Line(convexHull.vertices[a], convexHull.vertices[(a + 1) % n]);
        let lineB: Line = new Line(convexHull.vertices[b], convexHull.vertices[(b + 1) % n]);
        let lineC: Line = new Line(convexHull.vertices[c], convexHull.vertices[(c + 1) % n]);

        // Find intersections by expanding the lines
        let vertB: Vertex | null = lineA.intersection(lineB);
        let vertC: Vertex | null = lineB.intersection(lineC);

        if (!vertB || !vertC) {
          continue;
        }

        // An object: { center: Vertex, radius: number }
        let triangle: Triangle | null = getTangentTriangle4(lineA.a, vertB, vertC, lineC.b);
        if (!triangle) {
          continue;
        }

        // Workaround. There will be a future version where the 'getCircumCircle()' functions
        // returns a real Circle instance.
        let _circle: Circle = triangle.getCircumcircle();
        let circle: Circle = new Circle(_circle.center, _circle.radius);

        // Count the number of intersections with the convex hull:
        // If there are exactly three, we have found an in-lying circle.
        //  * Check if this one is better (bigger) than the old one.
        //  * Also check if the circle is located inside the polygon;
        //    The construction can, in some cases, produce an out-lying circle.
        if (!convexHull.containsVert(circle.center)) {
          continue;
        }
        var circleIntersections: Array<number> = findCircleIntersections(convexHull, circle);
        if (circleIntersections.length == 3 && (bestCircle == undefined || bestCircle.radius < circle.radius)) {
          bestCircle = circle;
          bestTriangle = triangle;
        }
      }
    }
  }
  return { circle: bestCircle as Circle, triangle: bestTriangle as Triangle };
};

/**
 * This function computes the three points for the inner maximum circle
 * lying tangential to the three subsequential lines (given by four points).
 *
 * Compute the circle from that triangle by using Triangle.getCircumcircle().
 *
 * Not all three lines should be parallel, otherwise the circle might have infinite radius.
 *
 * LineA := [vertA, vertB]
 * LineB := [vertB, vertC]
 * LineC := [vertC, vertD]
 *
 * @param {Vertex} vertA - The first point of the three connected lines.
 * @param {Vertex} vertB - The second point of the three connected lines.
 * @param {Vertex} vertC - The third point of the three connected lines.
 * @param {Vertex} vertD - The fourth point of the three connected lines.
 * @return {Triangle} The triangle of the circular tangential points with the given lines (3 or 4 of them).
 */
const getTangentTriangle4 = (vertA: Vertex, vertB: Vertex, vertC: Vertex, vertD: Vertex): Triangle | null => {
  const lineA: Line = new Line(vertA, vertB);
  const lineB: Line = new Line(vertB, vertC);
  const lineC: Line = new Line(vertC, vertD);

  const bisector1: Line = geomutils.nsectAngle(vertB, vertA, vertC, 2)[0]; // bisector of first triangle
  const bisector2: Line = geomutils.nsectAngle(vertC, vertB, vertD, 2)[0]; // bisector of second triangle
  const intersection: Vertex | null = bisector1.intersection(bisector2);

  if (!intersection) {
    return null;
  }

  // Find the closest points on one of the polygon lines (all have same distance by construction)
  const circleIntersA: Vertex = lineA.getClosestPoint(intersection);
  const circleIntersB: Vertex = lineB.getClosestPoint(intersection);
  const circleIntersC: Vertex = lineC.getClosestPoint(intersection);

  return new Triangle(circleIntersA, circleIntersB, circleIntersC);
};

/**
 * Find all intersecting lines (indices) on the polyon that intersect the circle.
 *
 * @param {Polygon} convexHull - The polygon to detect the intersections on (here this is the convex hull given).
 * @param {Circle} circle - The circle to detect the intersections with.
 * @return {Array<number>} The indices of those lines that intersect (or touch) the circle.
 **/
const findCircleIntersections = (convexHull: Polygon, circle: Circle): Array<number> => {
  const result: Array<number> = [];
  for (var i = 0; i < convexHull.vertices.length; i++) {
    var line: Line = new Line(convexHull.vertices[i], convexHull.vertices[(i + 1) % convexHull.vertices.length]);
    // Use an epsilon here because circle coordinates can be kind of unprecise in the detail.
    if (circle.lineDistance(line) < EPS) {
      result.push(i);
    }
  }
  return result;
};