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Direction traveled in a circle
wallabeeX
Registered User regular
Registered User regular
I'm currently working on a script for a visual effects project I'm embarking on. This particular problem involves figuring out which way a bird would tip it's wings as it comes into a curve to bank. Essentially, we can assume the bird tips one of two ways, and I'm just trying to set up the algorithm that's capable of figuring that out.
This can be better boiled down to traveling in a circle - how do we mathmatically determine whether an object is traveling clockwise or counter-clockwise in a circle? I have access to most any statistic to determine this, though I'd imagine velocity and position, or perhaps just position, will do.
Can anyone shed some light on this?
This can be better boiled down to traveling in a circle - how do we mathmatically determine whether an object is traveling clockwise or counter-clockwise in a circle? I have access to most any statistic to determine this, though I'd imagine velocity and position, or perhaps just position, will do.
Can anyone shed some light on this?
wallabeeX on
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3clipse: The key to any successful marriage is a good mid-game transition.
EDIT: In fact, I think this would work for most arbitrary curves...no?
If I'm trying to get to 1,0 from 0,-.5 I'll have to take a right. If I'm traveling from 0,.5 and trying to get to the same place, I'll have to take a left.
I've written a script that procedurallly has a flock of seagulls chase a seagull someone's animated on their own. This means that the animator only has to animate ONE seagull and he gets the other 50, or 150, for free. To make it more believable, I'd love to have them bank dependent on the direction it's turned.
But I have no idea what direction it's GOING to change to - just the position it's at, the position it was at, and it's current and prior velocity. That's to say it's easy to figure out you need to take a left if you're triyng to get to 1,0 from 0,-.5, but I don't have that information. Only the past.
Edit: the "real-time" thing shouldn't even matter, as you would have to get a chance to process his movements in order to even display them on screen.
Double edit: I think I may be out of my depth here, sorry. I'm pretty interested in your problem though, and I would love to see what solution you end up with.
3clipse: The key to any successful marriage is a good mid-game transition.
The program I work with is capable of saying to a cloud of points (henceforth: particles) "follow that". I'm essentially doing that, although with some added complexity. These particles are capable of carrying the image of a seagull flapping - they can aim in the direction of their velocity, which they do, so the seagulls always point forward. I need a way to look at the values from the past frame, and the values from the current frame, and know whether it's heading in a straight line, or in a curve.
I'll let you know how it goes.
I'm pretty sure if you look at the velocity as an (X,Y) value, and acceleration as (A,B), and compare the signs of the four (X/A and Y/B) you can get a direction (as in, is the bird turning left or right) based on the comparative signs.
If that's not sufficient, you may need to go with looking at the angle between the two vectors ((X,Y) and (A,B)) to determine what direction that acceleration is turning the bird in (left or right, relative to the bird, would depend on whether the angle between the two was positive or negative).
If that's not sufficient, you may need to go with something like a cross product. But I'm pretty sure that won't be necessary, particularly if we're just talking about 2D coordinates.
EDIT: Actually, yeah...I think you're going to want to go with angles. Just find the angle of the current velocity, then the angle of the acceleration, subtract the two, and that'll give you clockwise or counterclockwise.
Is it always Y? If the bird goes 360 degrees, then a positive change in Y (assuming Z is height, and X and Y are directions of travel) may correlate to either a counterclockwise or clockwise turn depending on the direction of travel in the X axis.
EDIT: If X direction is positive, then a positive change in Y means a counterclockwise rotation. If X is negative, then a positive change in Y means counterclockwise. Though that just makes it a two-step check, instead of one. The main problem then is dealing with your zeros (straight "north or south" so to speak). But that can just be a third step (if X is zero, check if why is positive or negative, then act accordingly). I still say calculating angles would work better, though.
But you're correct. My current example scene DOES have the bird traveling in a 360 circle and the Y travels is entirely based on it's travel on the X axis. It's pretty worthless.
I'm looking into your suggestion above but it's a bit above my head. This is the type of stuff that tends to make me wish I was still taking math courses.
It's not too tough, though. To find the angle, it's just arcsine(X/Z). Do this for velocity and acceleration. Make sure to add 180 degrees if Z is negative (taking Z to be "up/down" and X to be "left/right"). If the angle is 0 (and Z will be 0), then determine between 0 and 180 degrees based on whether X is positive or negative (0 if positive, 180 if negative).
That's a pretty simple step-by-step process.
Then at any given moment, you've got two angles, velocity and acceleration. If the angle of acceleration is greater than velocity, it's going clockwise, otherwise counterclockwise.
Really appreciate your help.
Also, you should be able to (if it's something your app supports) use the magnitude of that difference in angles to determine how hard they bank; greater the angle, greater the bank.
From what I understand arcsine will only take real numbers, but you see the numbers I"m dealing with. From what I understand velocity is the difference in the current position and the last prior simulation time step (typically a frame), but these are hardly real numbers.
x = r cos (theta)
z = r sin (theta)
r ^2 = x ^ 2 + z ^ 2
Find the derivative of theta with respect to time, which you can do yourself because I'm lazy.
If dtheta/dt is positive, it's counterclockwise. If dtheta/dt is negative, clockwise. Again, only works if it's traveling in a circle, but any arc is part of some arbitrarily large circle. mcdermott's method should also work, I think.
My bad. Arcsine/arccosine can only take numbers up to 1. You can still use them, but you'd have to get a unit vector. Maybe arctan will work?
To get a unit vector, you simply find the length of the vector, then divide both values (X, Z) by that length...then you can use cos/sin (and arccos/arcsin). Cosine is adjacent/hypotenuse, but with a unit vector hypotenuse is 1, thus you can just use the Z value of the unit vector for your arccos. IIRC, we actually use arccos, because it's easier to work with for this.
So, in your example:
V= <<-553.492,0,1001.971>> A = <<-763.608,0,-99.551>>
Velocity:
length=(553^2+1001^2)^.5, or 1143.6
So your unit vector is <-553/1143.6, 0, 1001/1143.6>, or <-.484, 0, .875>
your angle will be arccos(-.484), or 118.9 deg.
Accel:
length=(763^2+99^2)^.5 or 769.4
Unit vector is <-763/769.4,0,-99/769.4> or <-.992,0,-.129>
angle is arccos(-.992), or 172.7 deg.
Because Z is negative, though, that's actually -172.7 deg, or rather 187.3 deg (you can check this by thinking about where those vector values put you on a circle).
Now, you've got your angles. I'm pretty sure it's not as easy as "greater than" or "less than" with those angles, though, because (duh) it's a circle. Forgot about that. But I know you can use those angles to determine which way you're rotating, I'll either leave it to you or somebody else to explain how. I forget.
EDIT: And I think converting to polar and working that way would be harder, depending on what environment he's doing it in.
I suppose this would be the case, yes. :P
So yeah, go with that.
Since we're just talking about turning, let's toss out the height so we're just left with vectors of two dimensions:
Velocity: v <a,b>
Acceleration: k <c,d> (i'm using k for acceleration to avoid confusion with the a component of v)
Now, both of these can be represented on the complex plane, where the vertical axis is complex and horizontal is real numbers.
v=a+bi
k=c+di
r=0+1i (reference, unit straight up)
Let's let z represent the vector we need to help rotate things.
z = v/r
z =
So here, we get:
z = b - ai
Now, we want to make z a unit vector so that we only rotate and not scale (in case you want to do proportional banking)
z' = z/|z| = z/sqrt(b^2+a^2)
If v' is the rotated (vertical) velocity and k' is the rotated acceleration
v' = v/z'
k' = k/z'
Now, all you have to do is check the real part (or the x portion of the vector) of the rotated acceleration, k'. If it's positive, you're turning right. If it's negative, you're turning left. Bam.
The scripting language I'm working with has a "unit" command that can turn my large numbers into a unit vector, which was what I was in the middle of before I had to leave for the day. I'm planning on going over these new details later today and hopefully find a solution. Comically, I've been moved to a different project, but my own curiosity will drive this towards a final solution.
Thanks again, I'll keep you all posted. I'm also on my way to the bookstore to pick up a text book one this kind of math. This is trig, right? I've got a linear algebra text-book but I think that's even a bit high for this.
Yup.
I loves me some complex numbers, but probably not necessary. Same basic idea as mcdermott and myself though. Moving things to a more convenient coordinate system. I think my solution is the most elegant. :P
Upon re-examination, I agree.
EDIT: I have no experience in these things, this is just the first thing that popped into my head. And I didn't read the more complicated posts above mine, so they may have said the same thing or something better.
EDIT2: Actually, I'm just plain wrong. If he's flying along the X axis negatively, a left turn would result in a negative product of the two accelerations. I'm a moron.