How do I restrict the movement of a point to within a radius of another?

Question

I have a situation where a sprite's center point needs to be constrained within a certain boundary in 2D space. The boundary is circular, so the sprite is constrained within a radius. This radius is defined as a distance from the center of a certain point.

I've got this code, but it only tells me when the center has overstepped the allowed radius:

float distance = Vector2.Distance(centerPosition, spritePosition));
if (distance > allowedDistance) {
    // Point is outside allowed range
}

I would like to also return the sprite's center to the closest point within that radius whenever it oversteps the bounds.

How can I do that?

Answer 1

Take the position of the object, minus the origin of the circle. This will give you a vector from the origin to the object. The length of this vector is the distance to the object. Multiply this vector by the circle's radius divided by the distance to the object. Set the object's position to the circle's origin plus this vector.

Psuedo-Code:

Vector FromCircleToObject = Object.Position - Circle.Origin;
FromCircleToObject *= Circle.Radius / FromCircleToObject.Length;
Object.Position = Circle.Origin + FromCircleToObject;

You can optimize this by using the vector's length squared. This saves you a square root. If you use the vector's length squared, you must use the circle's radius squared. Luckily, squaring is cheaper than square rooting, so there's still an optimization to be had.

Answer 2

What you need to do is reset the sprite's position back to it's greatest extent.

You can do this by taking the vector from the centre to the sprites position, normalising it and then multiplying that by the allowed distance. However, that involves a square root which may be too slow for your game.

Another approach is to limit the sprite to a box around the centre position so your test would become:

if (spritePosition.x > centrePosition.x + allowedDistance)
{
    spritePosition.x = centrePosition.x + allowedDistance;
}
else if (spritePosition.x < centrePosition.x - allowedDistance)
{
    spritePosition.x = centrePosition.x - allowedDistance;
}

and the same for the y.

This obviously doesn't give the same effect but may be quicker. You'd have to profile them both to see and also check the behaviour to see if was acceptable.

Answer 3

For anyone struggling to visualise it:

If the distance between A and B is less than the radius, we can quit early. B is already within radius of A, so everything is fine.

Otherwise, find the vector u, from A to B. Divide u by the distance to get the unit vector û. Multiply that by the radius to get a vector to the furthest point from A toward B that is still within radius of A.

Answer 4

So, your sprite has a rectangular shape? If so, you are effectively asking, "How do I move a rectangle that intersects with a circle inside the circle so that the rectangle no longer intersects". I assume then that the rectangle is in fact smaller than your circle and the sprite position represents the center of the sprite. If it doesn't, you can convert from top-left to center by simply adding 1/2 the width, height, that is:

center.X = spritePosition.x + spriteImageSize.x/2;
center.Y = spritePosition.y + spriteImageSize.y/2;

In that case, you need to consider the 4 points of the sprite. Check if the first one is outside, and if it is, move it so it isn't. I don't have a formal proof, but I conjecture that it is impossible for you to move a corner inside the circle such that it puts another corner outside that was previously inside. This depends strongly on the fact that the rectangle fits inside the circle.

The code would be something like this:

TL = sprite.getTopLeft();
if(outside(TL, circle)) {
newPoint = moveInside(TL); //see how to move a point inside 
delta = newPoint - TL; //figure out how much we moved
spritePosition += delta; //move the center of the sprite
}

//repeat for TR, BL, BR -- IF YOU MOVE THE SPRITE, YOU MUST RECOMPUTE THE CORNERS! 

If you don't need that level of detail and you'd rather treat the sprite as an infinitely small single point, then you simply need to detect when it is outside the circle and set it so that it is inside. See ChrisF's response on how to do that.