Question

I have an equation for a parabolic curve intersecting a specified point, in my case where the user clicked on a graph.

 // this would typically be mouse coords on the graph
 var _target:Point = new Point(100, 50);

 public static function plot(x:Number, target:Point):Number{
  return (x * x) / target.x * (target.y / target.x);
 }

This gives a graph such as this:

I also have a series of line segments defined by start and end coordinates:

startX:Number, startY:Number, endX:Number, endY:Number

I need to find if and where this curve intersects these segments (A):

If it’s any help, startX is always < endX

I get the feeling there’s a fairly straight forward way to do this, but I don’t really know what to search for, nor am I very well versed in “proper” math, so actual code examples would be very much appreciated.

UPDATE:

I’ve got the intersection working, but my solution gives me the coordinate for the wrong side of the y-axis.

Replacing my target coords with A and B respectively, gives this equation for the plot:

(x * x) / A * (B/A)

// this simplifies down to:
(B * x * x) / (A * A)

// which i am the equating to the line's equation
(B * x * x) / (A * A) =  m * x + b

// i run this through wolfram alpha (because i have no idea what i'm doing) and get:
(A * A * m - A * Math.sqrt(A * A * m * m + 4 * b * B)) / (2 * B)

This is a correct answer, but I want the second possible variation.
I’ve managed to correct this by multiplying m with -1 before the calculation and doing the same with the x value the last calculation returns, but that feels like a hack.

SOLUTION:

 public static function intersectsSegment(targetX:Number, targetY:Number, startX:Number, startY:Number, endX:Number, endY:Number):Point {
  // slope of the line
  var m:Number = (endY - startY) / (endX - startX);

  // where the line intersects the y-axis
  var b:Number = startY - startX * m;

  // solve the two variatons of the equation, we may need both
  var ix1:Number = solve(targetX, targetY, m, b);
  var ix2:Number = solveInverse(targetX, targetY, m, b);

  var intersection1:Point;
  var intersection2:Point;

  // if the intersection is outside the line segment startX/endX it's discarded
  if (ix1 > startX && ix1 < endX) intersection1 = new Point(ix1, plot(ix1, targetX, targetY));
  if (ix2 > startX && ix2 < endX) intersection2 = new Point(ix2, plot(ix2, targetX, targetY));

  // somewhat fiddly code to return the smallest set intersection
  if (intersection1 && intersection2) {
   // return the intersection with the smaller x value
   return intersection1.x < intersection2.x ? intersection1 : intersection2;
  } else if (intersection1) {
   return intersection1;
  }

  // this effectively means that we return intersection2 or if that's unset, null
  return intersection2;
 }

 private static function solve(A:Number, B:Number, m:Number, b:Number):Number {
  return (m + Math.sqrt(4 * (B / (A * A)) * b + m * m)) / (2 * (B / (A * A)));
 }

 private static function solveInverse(A:Number, B:Number, m:Number, b:Number):Number {
  return (m - Math.sqrt(4 * (B / (A * A)) * b + m * m)) / (2 * (B / (A * A)));
 }

 public static function plot(x:Number, targetX:Number, targetY:Number):Number{
  return (targetY * x * x) / (targetX * targetX);
 }

Answer 1

Or, more explicit yet.

If your parabolic curve is

y(x)= A x2+ B x + C (Eq 1)

and your line is

y(x) = m x + b (Eq 2)


The two possible solutions (+ and -) for x are

x = ((-B + m +- Sqrt[4 A b + B^2 - 4 A C - 2 B m + m^2])/(2 A))   (Eq 3)

You should check if your segment endpoints (in x) contains any of these two points. If they do, just replace the corresponding x in the y=m x + b equation to get the y coordinate for the intersection

Edit>

To get the last equation you just say that the “y” in eq 1 is equal to the “y” in eq 2 (because you are looking for an intersection!).
That gives you:


A x2+ B x + C = m x + b


and regrouping


A x2+ (B-m) x + (C-b) = 0

Which is a quadratic equation.

Equation 3 are just the two possible solutions for this quadratic.

Edit 2>

re-reading your code, it seems that your parabola is defined by

y(x) = A x2


where

A = (target.y / (target.x)2)

So in your case Eq 3 becomes simply

 x = ((m +- Sqrt[4 A b + m^2])/(2 A))   (Eq 3b)  

HTH!