LineCircleIntersection.js

// See live demo of this code: www.williamfiset.com/circlelineintersection

// Let EPS (epsilon) be a small value
var EPS = 0.0000001;

// Let a point be a pair: (x, y)
function Point(x, y) {
  this.x = x;
  this.y = y;
}

// Define a circle centered at (x,y) with radius r
function Circle(x,y,r) {
  this.x = x;
  this.y = y;
  this.r = r;
}

// Define a line with the equation: ax + by = c
function Line(a, b, c) {
  this.a = a; this.b = b; this.c = c;
  // Normalize line for good measure
  if (Math.abs(b) < EPS) {
    c /= a; a = 1; b = 0;
  } else { 
    a = (Math.abs(a) < EPS) ? 0 : a / b;
    c /= b; b = 1; 
  }
}

// Given a line in standard form: ax + by = c and a circle with 
// a center at (x,y) with radius r this method finds the intersection
// of the line and the circle (if any). 
function circleLineIntersection(circle, line) {
  
  var a = line.a, b = line.b, c = line.c;
  var x = circle.x, y = circle.y, r = circle.r;
  
  // Solve for the variable x with the formulas: ax + by = c (equation of line)
  // and (x-X)^2 + (y-Y)^2 = r^2 (equation of circle where X,Y are known) and expand to obtain quadratic:
  // (a^2 + b^2)x^2 + (2abY - 2ac + - 2b^2X)x + (b^2X^2 + b^2Y^2 - 2bcY + c^2 - b^2r^2) = 0
  // Then use quadratic formula X = (-b +- sqrt(a^2 - 4ac))/2a to find the 
  // roots of the equation (if they exist) and this will tell us the intersection points
  
  // In general a quadratic is written as: Ax^2 + Bx + C = 0
  // (a^2 + b^2)x^2 + (2abY - 2ac + - 2b^2X)x + (b^2X^2 + b^2Y^2 - 2bcY + c^2 - b^2r^2) = 0
  var A = a*a + b*b;
  var B = 2*a*b*y - 2*a*c - 2*b*b*x;
  var C = b*b*x*x + b*b*y*y - 2*b*c*y + c*c - b*b*r*r;
  
  // Use quadratic formula x = (-b +- sqrt(a^2 - 4ac))/2a to find the 
  // roots of the equation (if they exist).
  
  var D = B*B - 4*A*C;
  var x1,y1,x2,y2;
  
  // Handle vertical line case with b = 0
  if (Math.abs(b) < EPS) {
    
    // Line equation is ax + by = c, but b = 0, so x = c/a
    x1 = c/a;
    
    // No intersection
    if (Math.abs(x-x1) > r) return [];
    
    // Vertical line is tangent to circle
    if (Math.abs((x1-r)-x) < EPS || Math.abs((x1+r)-x) < EPS)
      return [new Point(x1, y)];
    
    var dx = Math.abs(x1 - x);
    var dy = Math.sqrt(r*r-dx*dx);
    
    // Vertical line cuts through circle
    return [
      new Point(x1,y+dy),
      new Point(x1,y-dy)
    ];
    
  // Line is tangent to circle
  } else if (Math.abs(D) < EPS) {
    
    x1 = -B/(2*A);
    y1 = (c - a*x1)/b;
    
    return [new Point(x1,y1)];
  
  // No intersection
  } else if (D < 0) {
    
    return [];
    
  } else {
    
    D = Math.sqrt(D);
    
    x1 = (-B+D)/(2*A);
    y1 = (c - a*x1)/b;

    x2 = (-B-D)/(2*A);
    y2 = (c - a*x2)/b;
    
    return [
      new Point(x1, y1),
      new Point(x2, y2)
    ];
    
  }
  
}