Shortest route to intersection

I asked this question about 2 months ago but found none of the answers to be helpful enough. So I am giving it another shot. I think it was my fault not describing it well enough. So lets try again.

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Here is a rough idea of what I am trying to accomplish.

The goal is to send a projectile from point T to intercept the object represented by point R.

What is known:

1. The location of object R
2. The direction object R is traveling
3. The speed at which object R is traveling
4. The location of object T
5. Speed at which object T will travel

I am looking for the direction object T should be sent and thus find the location they will collide at. Either one.

For example: If...

1. The 开发者_如何学运维location of R was (1,5)
2. R is traveling at a 45 degree angle (relative to d)
3. R is traveling at 1 unit per second
4. T is located at (1,1)
5. T also travels at 1 unit per second

_{(source: bja888.com)}

L makes the location of the collision at (3,3)

There are an infinite number of possibilities. Consider a concentric series of circles from the points R and T, representing the distance each could travel at increasing times. Where the circles intersect is the point of collision, a vector between the point of collision and T is where T should have been aimed at for that particular instant. If you are looking for the shortest path, you need to get the normal to R's path and fire T at the appropriate time by calculating the time it takes for T to traverse this distance so it will arrive at the same time as R.

Assume no air friction, gravity, external forces, near-speed-of-light missiles, violation of physical laws, etc.

Suppose initially R = (x, y) and T = (0, 0). (Change your coordinates system if T not at the origin).

Suppose the velocity of T is constantly (cos θ, sin θ), where θ is unknown but does not depend on time. (Change your coordinates system or unit of time if the speed if not 1.)

Suppose the velocity of R is constantly (v, 0). (Change your coordinates system if the velocity is not along the horizontal.)

Suppose at time t the two objects collide. So we have 2 equations and 2 unknowns:

• t cos θ = x + vt
• t sin θ = y

So obviously

• t² = (x + vt)² + y².

Solve this quadratic equation to get t. Plug it into the y equation to get θ. QED.