Circle Collision-Help Needed

[zwnage]

Hello. While looking up peoples sources to try and get ideas, I came across this neat snippet of code from a pool game:

code:

angle := arctand ((ballY (k) - ballY (i)) / ((ballX (k) - ballX (i)))) % angle of hit cosAngle := cosd (angle) sinAngle := sind (angle) speedAfterHit := (2 * ((ballXSpeed (i) * cosAngle + ballYSpeed (i) * sinAngle) - (ballXSpeed (k) * cosAngle + ballYSpeed (k) * sinAngle))) / (ballMass (i) + ballMass (k)) % new velocities of the balls ballXSpeed (i) -= speedAfterHit * ballMass (k) * cosAngle ballYSpeed (i) -= speedAfterHit * ballMass (k) * sinAngle ballXSpeed (k) += speedAfterHit * ballMass (i) * cosAngle ballYSpeed (k) += speedAfterHit * ballMass (i) * sinAngle

I tried a few test programs with that code, and it works a lot better than the code I'm trying to use, but I really want to learn how this code works, especially whats happening in speedAfterHit and why are we muiltiplying the trig ratios of the angle hit with the ball speeds. Can anyone please explain the code? Thanks.

[Cervantes]

First, have you checked this out? Tutorial on Circular Collision data. I admit it isn't much of a tutorial, but it's something.

zynage wrote:

why are we muiltiplying the trig ratios of the angle hit with the ball speeds

This is very common. See, using cosd gives you a number between 0 and 1 that represents the length of a horizontal line from the origin to directly below the point.

math:

/|     / |    /  |   <-- this line is given from sin(theta) where theta is the angle from the horizontal line to the diagonal line   /   |  /    | /_____|      ^--- that line is given from cos(theta) where theta is the angle from that line to the diagonal line

So if the speed of a ball is 1, then use cos to get the horizontal speed of the ball. However, if the speed of the ball is 5, simply using cos won't work. We need to extend the length of that horizontal line by 5, because the diagonal line was extended by a factor of 5. So we'd do 5*cos(theta).

I'm not too sure what's going on in speedAfterHit. I'm a bit skeptical of this code because it seems to assuming that the speed of a ball divided by the mass of a ball is a constant number--speedAfterHit. That's clearly not true. Take two pool balls for example. If I shoot one ball directly at the other ball, (ignoring spin) the first ball will transfer 100% energy into the second ball, so the second ball will continue at the speed of the first ball and the first ball will come to a stop. This is because they have the same mass.

I suggest you try to work out the physics of this on paper. You'll need to use conservation of momentum and conservation of kinetic energy (assume this, unless you want inelastic collisions--ones where the balls stick together somewhat or deform slightly). I haven't actually worked out the math of this myself, but I think that'll be all you need in two dimensions.

[zwnage]

Thanks for the help. I have tried to work this out on paper, but their code surpasses mine because the balls don't tend to overlap as much. I think I originally got it from thoughtful's tutorial.

Heres what I'm trying to use (pardon the lack of comments, I made it right now out of some notes I had on paper.)

code:

setscreen ("offscreenonly") var ballX, ballY : flexible array 1 .. 0 of real var ballXSpeed, ballYSpeed : flexible array 1 .. 0 of real var ballRadius : flexible array 1 .. 0 of int var ballAmount : int := 3 var ballVelocity, ballVelocity2, ballNewVelocity, ballNewVelocity2, ballVelocityAngle, ballVelocityAngle2, ballNewAngle, ballNewAngle2 : real := 0 for i : 1 .. ballAmount     new ballX, upper (ballX) + 1     new ballY, upper (ballY) + 1     new ballXSpeed, upper (ballXSpeed) + 1     new ballYSpeed, upper (ballYSpeed) + 1     new ballRadius, upper (ballRadius) + 1     ballRadius (i) := Rand.Int (30, 50)     ballX (i) := Rand.Int (ballRadius (i), maxx - ballRadius (i))     ballY (i) := Rand.Int (ballRadius (i), maxy - ballRadius (i))     ballXSpeed (i) := Rand.Int (1, 2)     ballYSpeed (i) := Rand.Int (1, 2) end for function distance (xInput1, yInput1, xInput2, yInput2 : real) : real % function that returns the distance between two points     result ((xInput2 - xInput1) ** 2 + (yInput2 - yInput1) ** 2) ** 0.5 end distance loop     for i : 1 .. ballAmount         for j : 1 .. ballAmount             if i ~= j then                 if distance (ballX (i), ballY (i), ballX (j), ballY (j)) <= ballRadius (i) + ballRadius (j) then                                         ballVelocity := ((ballXSpeed (i) ** 2 + ballYSpeed (i) ** 2) ** 0.5)                     ballVelocity2 := ((ballXSpeed (j) ** 2 + ballYSpeed (j) ** 2) ** 0.5)                     ballVelocityAngle := arctand (ballYSpeed (i) / ballXSpeed (i)) - arctand ((ballY (i) - ballY (j)) / (ballX (i) - ballX (j)))                     ballVelocityAngle2 := arctand (ballYSpeed (j) / ballXSpeed (j)) - arctand ((ballY (i) - ballY (j)) / (ballX (i) - ballX (j)))                     ballNewVelocity := ((((cosd (ballVelocityAngle)*ballVelocity - cosd (ballVelocityAngle2)*ballVelocity2) ** 2) + (sin (ballVelocityAngle)*ballVelocity) ** 2) ** 0.5)                     ballNewVelocity2 := ((((cosd (ballVelocityAngle)*ballVelocity - cosd (ballVelocityAngle2)*ballVelocity2) ** 2) + (sin (ballVelocityAngle2)*ballVelocity2) ** 2) ** 0.5)                     ballNewAngle := (arctand ((ballY (i) - ballY (j)) / (ballX (i) - ballX (j))) + arctand (ballVelocityAngle / ballNewVelocity))                     ballNewAngle2 := (arctand ((ballY (i) - ballY (j)) / (ballX (i) - ballX (j))) + arctand (ballVelocityAngle2 / ballNewVelocity2))                     ballXSpeed (i) := cos (ballNewAngle) * ballNewVelocity                     ballYSpeed (i) := sin (ballNewAngle) * ballNewVelocity                     ballXSpeed (j) := cos (ballNewAngle2) * ballNewVelocity2                     ballYSpeed (j) := sin (ballNewAngle2) * ballNewVelocity2                     ballX (i) += ballXSpeed (i)                     ballY (i) += ballYSpeed (i)                     ballX (j) += ballXSpeed (j)                     ballY (j) += ballYSpeed (j)                 end if             end if         end for     end for     for i : 1 .. ballAmount         if ballX (i) > maxx - ballRadius (i) or ballX (i) < ballRadius (i) then             ballXSpeed (i) *= -1         end if         if ballY (i) > maxy - ballRadius (i) or ballY (i) < ballRadius (i) then             ballYSpeed (i) *= -1         end if         ballX (i) += ballXSpeed (i)         ballY (i) += ballYSpeed (i)         Draw.Oval (round (ballX (i)), round (ballY (i)), ballRadius (i), ballRadius (i), black)     end for     delay (10)     View.Update     cls end loop

Since I havn't taken physics yet (its next sem), I might have mixed up a few terms. The velocities might be called speed or momentem or something.

Heres the jist of it:

find a hit

find the distance traveled each frame by both the balls before the hit (speed or velocity or whatever you call it)---ie : ballVelocity := ((ballXSpeed (i) ** 2 + ballYSpeed (i) ** 2) ** 0.5)

find the angle the ball was originally travelling at. -- ie : ballVelocity := ((ballXSpeed (i) ** 2 + ballYSpeed (i) ** 2) ** 0.5)

calculate the new velocities by subtracting the balls velocity by the other balls velocity, thus slowing the ball down or reversing the direction.-- ie :ballNewVelocity := ((((cosd (ballVelocityAngle)*ballVelocity - cosd (ballVelocityAngle2)*ballVelocity2) ** 2) + (sin (ballVelocityAngle)*ballVelocity) ** 2) ** 0.5)

calculate the new angle of bounce

use the angle of bounce and the new velocites to get the x and y speeds.


Now this code is far from perfect. The program cannot account for when both balls goes hits while going in the same direction (ie : one is faster than the other and hits it from behind). The velocity minus part will slow one of the balls down like its supposed to, but the other ball is slowed too (and probably will go in the opposite direction, towards the first ball). This causes an overlap, and causes them to stick or fly away wildly. The solution I thought of is to give the velocites a positive or negative value, but that would mean I would have to divide my velocites into a further set: the x and y values, I think.....

I dunno if anyone can do anything with that snippet of code, but your welcome to try.

[Ultrahex]

Just for your guys information, this is collision response, not collision detection correct?, so what you are trying to do is determine angular momentum, along with momentum, so you need to determine the new velocities along with the new direction.

For those who don't understand the collision detection:

The Collision of a Circle to a Circle is very simple, It is explained by the pictures below:

code:

type Ball :     record         x : real         y : real         v : real % magnitude of velocity         dir : real         r : real     end record var balls : array 1 .. 2 of Ball balls (1).x := 50.0 balls (1).y := 50.0 balls (1).v := 5 balls (1).dir := 180 balls (1).r := 40 balls (2).x := 150.0 balls (2).y := 150.0 balls (2).v := 5 balls (2).dir := 45 balls (2).r := 40 View.Set ("offscreenonly") loop     % Update Position     for i : 1 .. upper (balls)         % Wall Collision         %---Horizontal         if (balls (i).x + balls (i).r + cosd (balls (i).dir) * balls (i).v) >= maxx or (balls (i).x - balls (i).r + cosd (balls (i).dir) * balls (i).v) < 0 then             balls (i).dir := -balls (i).dir + 180             put balls (i).dir         end if         %---Vertical         if (balls (i).y + balls (i).r + sind (balls (i).dir) * balls (i).v) > maxy or (balls (i).y - balls (i).r + sind (balls (i).dir) * balls (i).v) < 0 then             balls (i).dir := -balls (i).dir + 360         end if         % Update x/y positon based on magnitude of velocity         balls (i).x += cosd (balls (i).dir) * balls (i).v         balls (i).y += sind (balls (i).dir) * balls (i).v     end for     % Drawing     for i : 1 .. upper (balls)         drawoval (round (balls (1).x), round (balls (1).y), round (balls (1).r), round (balls (1).r), red)         drawoval (round (balls (2).x), round (balls (2).y), round (balls (2).r), round (balls (2).r), blue)     end for     % Circle-To-Circle Collison     if sqrt ((balls (1).x - balls (2).x) ** 2 + (balls (1).y - balls (2).y) ** 2) <= (balls (1).r + balls (2).r) then         % Collison Has Occured         put "Collision!"         View.Update()         delay(100)     end if     View.Update ()     delay (20)     cls end loop

[zwnage]

Yes, I'm trying to find collision response, not collision detection. I did a little research overnight and found that the method I was trying to implement is based on vectors, yet I still have no idea how to fix it so that they don't stick so often. Is there an expample code with vectors out there?

And I'm still confused about the first piece of code. I looked over thoughtful's tutorial a few more times, but there wern't too many comments explaining the collision responce process. Anyone have any ideas?

[Skynet]

Yeah, I figured out how their code works.

There are two major points:
1) All of those cosAngle and sinAngle terms are performing a rotation on the x and y axis. When you have a collision between two circles, force is only applied in the direction from one center to the other. (or, in more complicated terms, the normal of the surfaces at the point of contact) So, if you have objects moving in two dimensions, force is going to be applied in both the x and the y direction. The programmer redefines the x and y direction and solves the physics in only one dimension - simplifying the collision response to something that acts like two balls moving directly towards each other. That's easier to solve.

2) There's a generalization the programmer derived - "speedAfterHit". It's not a speed, it's...something else, with the units of m/(s * kg). I didn't find it until I actively tried to make my equations I found match the code you posted. It's a clean way of reducing the amount of work the computer has to do.

If you'd like, I can clean up my work and post it this evening.
For the benefit of anyone else reading this, concepts you should be familiar with:
Trigonometry
Vectors/'Components'
Momentum
Coefficient of Restitution

[zwnage]

Thank you for your help. The "speedAfterHit" was originally named something else, but I renamed it while I was testing the formula in my code. I guess "speed" is not really a good way of describing it.

After thinking your post over a bit, I think my approach to collision responce is simular to the method in question, but as you said, it is much simpler and faster than what I was doing. Oh, and after a bit more tinkering, I think I found my mistake in the code I posted. Instead of subtracting the vectors, I think I should switch them between the two balls. Hopefully, this is the proper way of doing it; I will try to impliment it as soon as I have the time.