Question : Finding the closest intersection point for two moving objects

[TerranceN]

Since this is not language specific I figured I would put this here.


Explanation of Question

For my geometry wars clone, I am trying to create an enemy that can cut the player off instead of just following the player. Here is a diagram of the problem:



P is the position of the player
Vp is the velocity of the player
Ω is the angle the player is moving in standard position (this is why I drew the horizontal lines)

E is the position of the enemy
Ve is the velocity of the enemy
Θ is the angle the enemy is moving in standard position

I is where the interception will occur
t is the time it takes for the enemy to intercept the player

Things I know:
P
Vp
Ω
E
Ve

Things I don't know:
Θ
I
t

My question is how would I go about finding any of the unknowns (once I know one it is easy to plug that in to the equations I know (see below)).


What I Have Tried

I started off with some equations that I know are true (note that I split 2d points into their x and y components):


The next thing I did is try to remove an unknown by isolating t in two different equations (I'm only showing one cause the other is the same rearranging but it is y instead of x and sine instead of cosine):


then I set the two equations equal to each other:


Then I tried to solve for Θ using this equation solver, but it told me that it is not solvable.

I then tried the same precess, but isolating Θ in two different equations and trying to solve for t:



But once again this is unsolvable.

I don't know what to do from here, and this problem has stumped my MSIP teacher (who is a math teacher), any help will be rewarded with bits and a solution (or something that directly leads me to one) will be rewarded with karma.

[The_Bean]

Potentially found a solution for the unknown angle.
Everything else should be simple after that.

Starting with this diagram (messed up one variable from yours)


Using the sin law, t can be easily removed


where


finally


tan inverse has to be modified for quadrents.

and heres a very basic example in turing.
(black=player , red=enemy)

Turing:

type player :     record         x, y : real         v : real         a : real     end record var p, e : player function setAngle (x, y : real) : real     if x = 0 and y = 0 then         result 0     elsif x = 0 and y > 0 then         result 90     elsif x = 0 and y < 0 then         result 270     elsif y = 0 and x > 0 then         result 0     elsif y = 0 and x < 0 then         result 180     elsif x > 0 and y > 0 then         result arctand (y / x)     elsif x < 0 and y > 0 then         result 180 + arctand (y / x)     elsif x > 0 and y < 0 then         result 360 + arctand (y / x)     elsif x < 0 and y < 0 then         result 180 + arctand (y / x)     else         result 0     end if end setAngle for j : 1 .. 5     p.x := Rand.Int(50,maxx-50)     p.y := Rand.Int(50,maxy-50)     p.v := 1     p.a := Rand.Int(0,360)     e.x := Rand.Int(50,maxx-50)     e.y := Rand.Int(50,maxy-50)     e.v := 1.5     var beta := setAngle (e.x - p.x, e.y - p.y) - p.a     e.a := setAngle (p.x - e.x, p.y - e.y) + arcsind (sind (beta) * p.v / e.v)     for i : 1 .. 200         Draw.FillOval (round (p.x), round (p.y), 1, 1, 7)         Draw.FillOval (round (e.x), round (e.y), 1, 1, 12)         p.x += (cosd (p.a) * p.v)         p.y += (sind (p.a) * p.v)         e.x += (cosd (e.a) * e.v)         e.y += (sind (e.a) * e.v)         delay (10)     end for end for

[TerranceN]

Thanks! I will try this solution today, but from what I have read in your post and seen in your example it is exactly what I was looking for. +1 Karma +100 Bits