Find the closest point along a rectangle, given an another point and direction

Question

Given a rectangle, and a point with a vector direction towards the rectangle. How can I find the closest point on the outside of that rectangle to the point in question?

Answer 1

One technique which you could use is called "ray casting". It is commonly used for rendering graphics, but has other applications such as line-of-sight (as you are wanting to do) and path-finding. In general terms it works by finding the intersection of a ray and an object. In your example the ray is the vector for the character's direction.

A useful reference for ray/object intersections (and incidentally other object/object intersections) is www.realtimerendering.com/intersections.html (look under the references for ray/aabb and ray/obb).

Answer 2

The rectangle has four sides. Each side is a line segment.

Test each of the four sides for intersection with the ray. Track the closest hit.

Here's some code to find out where on the segment the ray hits:

bool intersect(const ray& ray, const segment& segment,point& hit) {
    // where do we intersect this line?
    float t = ((ray.direction.x * ray.origin.y + ray.direction.y *
        (segment[0].x - ray.origin.x)) -
        (ray.direction.x * segment[1].y)) /
        (ray.direction.y * (segment[0].x + segment[1].x) -
        ray.direction.x * (segment[0].y + segment[1].y));
    if(t >= 0.0 && t<=1.0) { // in the segment
        hit = segment[0] + (segment[1]-segment[0]*t);  // lerp
        return true;
    }
    return false; // no hit
}

Answer 3

Well, you can use just linear algebra (analytic geometry, to be more specific) to solve this. It depends on how you modeled the rectangle.

Here's a general case: http://paulbourke.net/geometry/lineline2d/