How to detect when rotated rectangles are colliding each other

Question

After seeing this question many times and replying with old (and not usable) code I decided to redo everything and post about it.

Rectangles are defined by:

• center : x and y for his position (remember that 0;0 is TOP Left, so Y go down)
• size: x and y for his size
• angle for his rotation (in deg, 0 deg is following axis OX and turn clockwise)

The goal is to know if 2 rectangles are colliding or not.

Answer 1

I will use Javascript in order to demo this (and also provide code) but I can be done on every language following the process.

Links

• Final Demo on Codepen
• GitHub repository

Concept

In order to achieve this we'll use corners projections on the other rectangle 2 axis (X and Y). The 2 rectangles are only colliding when the 4 projections on one rectangles hit the others:

• Rect Blue corners on Rect Orange X axis
• Rect Blue corners on Rect Orange Y axis
• Rect Orange corners on Rect Blue X axis
• Rect Orange corners on Rect Blue Y axis

Process

1- Find the rects axis

Start by creating 2 vectors for axis 0;0 (center of rect) to X (OX) and Y (OY) then rotate both of them in order to get aligned to rectangles axis.

Wikipedia about rotate a 2D vector

const getAxis = (rect) => {
  const OX = new Vector({x:1, y:0});
  const OY = new Vector({x:0, y:1});
  // Do not forget to transform degree to radian
  const RX = OX.Rotate(rect.angle * Math.PI / 180);
  const RY = OY.Rotate(rect.angle * Math.PI / 180);

  return [
     new Line({...rect.center, dx: RX.x, dy: RX.y}),
     new Line({...rect.center, dx: RY.x, dy: RY.y}),
  ];
}

Where Vector is a simple x,y object

class Vector {
  constructor({x=0,y=0}={}) {
    this.x = x;
    this.y = y;
  }
  Rotate(theta) {
    return new Vector({
      x: this.x * Math.cos(theta) - this.y * Math.sin(theta),
      y: this.x * Math.sin(theta) + this.y * Math.cos(theta),
    });
  }
}

And Line represent a slop using 2 vectors:

• origin: Vector for Start position
• direction: Vector for unit direction

class Line {
  constructor({x=0,y=0, dx=0, dy=0}) {
    this.origin = new Vector({x,y});
    this.direction = new Vector({x:dx,y:dy});
  }
}

Step Result

2- Use Rect Axis to get corners

First we want extend our axis (we are 1px/unit size) in order to get the half of width (for X) and height (for Y) so by adding then (or inverse of them) we can get the all corners.

const getCorners = (rect) => {
  const axis = getAxis(rect);
  const RX = axis[0].direction.Multiply(rect.w/2);
  const RY = axis[1].direction.Multiply(rect.h/2);
  return [
    rect.center.Add(RX).Add(RY),
    rect.center.Add(RX).Add(RY.Multiply(-1)),
    rect.center.Add(RX.Multiply(-1)).Add(RY.Multiply(-1)),
    rect.center.Add(RX.Multiply(-1)).Add(RY),
  ]
}

Using this 2 news methods for Vector:

  // Add(5)
  // Add(Vector)
  // Add({x, y})
  Add(factor) {
    const f = typeof factor === 'object'
      ? { x:0, y:0, ...factor}
      : {x:factor, y:factor}
    return new Vector({
      x: this.x + f.x,
      y: this.y + f.y,
    })
  }
  // Multiply(5)
  // Multiply(Vector)
  // Multiply({x, y})
  Multiply(factor) {
    const f = typeof factor === 'object'
      ? { x:0, y:0, ...factor}
      : {x:factor, y:factor}
    return new Vector({
      x: this.x * f.x,
      y: this.y * f.y,
    })
  }

Step Result

3- Get corners projections

For every corners of a rectangle, get the projection coord on both axis of the other rectangle.

Simply by adding this function to Vector class:

  Project(line) {
    let dotvalue = line.direction.x * (this.x - line.origin.x)
      + line.direction.y * (this.y - line.origin.y);
    return new Vector({
      x: line.origin.x + line.direction.x * dotvalue,
      y: line.origin.y + line.direction.y * dotvalue,
    })
  }

(Special thank to Mbo for the solution to get projection.)

Step Result

4- Select externals corners on projections

In order to sort (along the rect axis) all the projected point and take the externals projected points we can:

• Create a vector to represent: Rect Center to Projected corner
• Get the distance using the Vector Magnitude function.

  get magnitude() {
    return Math.sqrt(this.x * this.x + this.y * this.y);
  }

• Use the dot product to know if the vector is facing the same direction of axis or inverse (when signed distance is negative)

getSignedDistance = (rect, line, corner) => {
  const projected = corner.Project(line);
  const CP = projected.Minus(rect.center);
  // Sign: Same directon of axis : true.
  const sign = (CP.x * line.direction.x) + (CP.y * line.direction.y) > 0;
  const signedDistance = CP.magnitude * (sign ? 1 : -1);
}

Then using a simple loop and test of min/max we can find the 2 externals corners. The segment between them is the projection of one Rect on the other one axis.

Step result

5- Final: Do all projections hit rect ?

Using simple 1D test along the axis we can know if they hit or not:

const isProjectionHit = (minSignedDistance < 0 && maxSignedDistance > 0
        || Math.abs(minSignedDistance) < rectHalfSize
        || Math.abs(maxSignedDistance) < rectHalfSize);

Done

Testing all 4 projections will give you the final result. =] !!

Hope this answer will help as many people as possible. Any comments are appreciated.