Javascript spring physics: Apply direction vector to spring - javascript

I'm implementing spring physics in Javascript, inspired by this blog post. My system uses Hooke's Law formula to calculate spring forces:
F = -k(|x|-d)(x/|x|) - bv
I made a CodePen that shows the implementation of a spring between two points. The CodePen has two points connected by a spring, and every 2 seconds the point positions are randomized. You can see the points bounce on the spring towards each other.
If you look at the source, you can see I've defined a direction vector for my spring:
var spring = {
length: 100,
direction: {
x: 1, y: 1
}
};
I'm trying to make it so that the spring always "resolves" in this direction. Put another way, I'd like the spring to always be "pointing" up and to the right. If this were implemented in the CodePen, it means the resting orientation of the points would always be the green point on the bottom left, and the blue point on the top right. No matter where the points start, they should end up in the following orientation (which matches the direction vector):
I've tried multiplying the normals by the spring vector:
norm1 = multiplyVectors( normalize( sub1 ), spring.direction ),
However this is a noop because the vector is (1,1). I've been hacking on this system for a few days now and I'm stuck. How can I constrain my 2d spring to a specific direction?

Spring forces are central just like gravity, which means that the total angular momentum of the system is conserved. Since you start with zero initial velocities, the angular momentum of the system is initially zero. The spring interaction keeps it zero, therefore the final orientation of the spring equals its initial orientation - the weights only move along the line connecting them.
To have the system rotate into the desired final position, you should also apply torque. The easiest way is to give the blue weight a positive charge and the green weight a negative one and then apply a constant external field in direction (1,1). That way the two charges will form a dipole and the interaction with the external field will generate the desired torque.
I don't get along with JavaScript, but I tried to write something based on your initial code here. The force that an external field with intensity E exerts on charge q is F = q * E, with both F and E being vectors. By adjusting q and E you can control how quickly the dipole will orient in the direction of the external field.
The force now becomes F = -k(|x|-d)(x/|x|) + qE - bv.
This has the probably undesired side effect that the final length of the spring will be slightly longer by delta, where delta = 2 * |q||E| / k. You can always adjust for that by reducing the length of the spring. Also, there is a little problem with that approach. Namely, there are two equilibrium states: one with the dipole facing the direction of the field (stable equilibrium) and one with the dipole facing the opposite direction (unstable equilibrium). A bit of random noise in the initial steps of the simulation will prevent the dipole from being trapped into the latter state.

Related

Acceleration over time

Scenario
I am working on a top-down view game in which enemies move towards certain positions. The destination being moved towards will often change drastically - sometimes while the enemy is still in motion towards the previous destination...
I want to achieve movement which is more realistic than linear movement, so there should be some acceleration and deceleration as enemies switch between the destinations.
Steering (direction) is not a factor. You may assume the sprites will move much like a hovercraft, drifting between the destinations as quickly as it can manage with respect to acceleration and deceleration.
For simplicity - lets assume there is only 1 dimension (Y) instead of X and Y... the movement should emulate a car which can only move north or south.
Since we are considering realistic movement in this scenario, you might not be surprised that a maximum speed over time is also a consideration. The enemy should never exceed it's own maximum speed; enemies store their own maximum speed in a variable.
One final consideration is that enemies will not only have a 'maximum speed' value, but it will also have a 'maximum acceleration' value - this would be the thing which guides how quickly each enemy can respond to moving in the opposite direction.
For simplicity, assume that the enemy does not have any movement friction... when it stops accelerating, it just keeps cruising at the current velocity forever.
Example
For context, lets elaborate on the car example. A particular car has:
Maximum speed: 10 meters per second
Maximum acceleration: can reach top speed in 2 seconds
(Other factors like destination y-pos, position y-pos, current velocity, etc)
Just like when I'm driving a car, I imagine all of these values are present, but I can't change them. All I can really change is how much I'm putting my foot on the acceleration/brake. Let's call this the 'throttle'. Like the acceleration pedal in a car, I can change this value to any amount at any time, in no time at all.
I can plant my foot down (throttle=1), let go of the pedal immediately (throttle=0), and even change into reverse and plant my foot down again (throttle=-1). Lets assume these changes in throttle are INSTANT (unlike speed or acceleration, which grow/shrink over TIME)
All that said, I imagine the only value that I really need to calculate is what the throttle should be, since thats the only thing I can control in the vehicle.
So, how then do I know how much throttle to use at any given moment, to arrive as quickly as possible to a destination (like some traffic lights) without overshooting my destination? I will need to know how much to push down the accelerator, briefly not accelerate at all, and then how hard decelerate as I'm closing in on the destination.
Preempted movement
This game will likely have an online component. That said, players will be transmitting their positions over a socket connection... however even the best connection could never achieve sending positions frequently enough to achieve smooth movement - you need interpolation. You need to send the 'velocity' (speed) along with the co-ordinates so that I can assume future positions in the time between packets being received.
For that reason, Tweens are a no-go. There would be no way to send a tween, then accurately inform the other parties at what point along each tween each entity currently is (I mean, I guess it is possible, but horrendously overcomplicated and probably involves rather large packet sends, and is probably also very exploitable in terms of the online component), then you need to consider aborting Tweens as the destinations change, etc.
Don't get me wrong, I could probably already model some very realistic movement with the ease-in/ease-out functionality of Tweens, and it would look great, but in an online setting that would be.. very messy.
How's it look so far?
So, essentially I have established that the primary aspect which needs to be calculated at any time is how much throttle to use. Let's work through my logic...
Imagine a very basic, linear movement over time formula... it would look like this:
Example 1 - position over time
currentDY = 5; // Current 'velocity' (also called Delta Y or 'DY')
currentY += currentDY * time // Current Y pos is increased by movement speed over time.
As you can see, at any given moment, the Y position increases over time due to the 'velocity' or DY (over time). Time is only a factor ONCE, so once we reach the destination we just set the DY to zero. Very sharp unrealistic movement. To smoothen the movement, the velocity ALSO needs to change over time...
Example 2 - velocity over time
throttle = -1
currentDY += throttle * time;
currentY += (currentDY * time);
//Throttle being -1 eventually decelerates the DY over time...
Here, the throttle is '-1' (maximum reverse!), so over time this will reduce the velocity. This works great for realistic acceleration, but provides no realistic anticipation or deceleration.
If we reach the destination this time, we can set the throttle to '0'... but it's too late to brake, so the resulting enemy will just keep moving past the target forever. We could throttle = '1' to go back, but we'll end up swinging back and forth forever.
(Also note that the maximum acceleration and speed isn't even a factor yet - it definately needs to be! The enemy cannot keep ramping up their speed forever; velocity delta AND ALSO acceleration delta need to have limits).
All that said, it's not enough to simply change velocity over time, but we also need be able to anticipate how much to decelerate (i.e. 'backwards throttle') BEFORE IT HAPPENS. Here's what I've got so far, and I'm practically certain this is the wrong approach...
Example 3 - throttle over time?? (I'm stuck)
guideY = currentY + (currentDY * (timeScale * 3000));
dist = endY - guideY;
throttle = Math.max(-1, Math.min(1, dist / 200));
currentDY += throttle * time;
currentY += (currentDY * time);
As you can see, this time we attempt to anticipate how much throttle to use by guessing where the enemies position will be in an arbitrary time in the future (i.e. 3 seconds). If the guideY went past the destination, we know that we have to start BRAKING (i.e. reducing speed to stop on top of the destination). By how much - depends on how far away the enemies future position is (i.e. throttle = dist / 200;)
Here's is where I gave up. Testing this code and changing the values to see if they scale properly, the enemy always swings way over the destination, or takes far too long to 'close in' on the destination.
I feel like this is the wrong approach, and I need something more accurate. It feels like I need an intersection to correctly anticipate future positions.
Am I simply using the wrong values for the 3 seconds and dist / 200, or am I not implementing a fully working solution here?
Presently, compared to the linear movement, it always seems to take 8 times longer to arrive at the target position. I haven't even reached the point of implementing maximum values for DeltaVelocity or DeltaAcceleration yet - the accepted solution MUST consider these values despite not being present in my JSFiddle below...
Test my logic
I've put all my examples in a working JSFiddle.
JSFiddle working testbed
(Click the 'ms' buttons below the canvas to simulate the passing of time. Click a button then press+hold Return for very fast repetition)
The sprite is initially moving in the 'wrong' direction - this is intended for testing robustness - It assumes an imaginary scenario where we just finished moving as fast as possible toward an old destination far lower on the screen, and now we need to suddenly start moving up...
As you can see, my third example (see the update function), the time for the sprite to 'settle' at the desired location takes far longer than it should. My math is wrong. I can't get my head around what is needed here.
What should the throttle be at any given time? Is using throttle even a correct approach? Your assistance is very appreciated.
Tiebreaker
Alright, I've been stuck on this for days. This question is going up for some phat bounty.
If a tiebreaker is required, the winner will need to prove the math is legit enough to be tested in reverse. Here's why:
Since the game also comprises of a multiplayer component, enemies will be transmitting their positions and velocities.
As hacking protection, I will eventually need a way to remotely 'check' if the velocity and position at between any two sample times is possible
If the movement was too fast based on maximum velocity and acceleration, the account will be investigated etc. You may assume that the game will know the true maximum acceleration and velocity values of the enemies ahead of time.
So, as well as bounty, you can also take satisfaction in knowing your answer will contribute to ruining the lives of filthy video game cheaters!
Edit 2: Fiddle added by answer author; adding into his answer in case anyone else finds this question: http://jsfiddle.net/Inexorably/cstxLjqf/. Usage/math explained further below.
Edit 1: Rewritten for clarification received in comments.
You should really change your implementation style.
Lets say that we have the following variables: currentX, currentY, currentVX, currentVY, currentAX, currentAY, jerk.
currentVX is what would have been your currentDX. Likewise, currentAX is the x component of your delta velocity value (accel is derivative of velocity).
Now, following your style, we're going to have a guideX and a guideY. However, there is another problem with how you're doing this: You are finding guideY by predicting the target's position in three seconds. While this is a good idea, you're using three seconds no matter how close you are to the target (no matter how small dist is). So when the sprite is 0.5 seconds from the target, it's going to still be moving towards the target's estimated position (three seconds into the future). This means it won't actually be able to hit the target, which seems to be the problem that you implied.
Moving on, recall the previous variables that I mentioned. Those are the current variables -- ie, they will be updated at every call after some seconds have been passed (like you have been doing before). You also mentioned a desire to have maxV, and maxA.
Note that if you have a currentVX of 5 and a currentVY of 7, the velocity is (5^2+7^2)^0.5. So, what you're going to want to do each time you're updating the 'current' archetype of variables is before updating the value, see if the magnitude (so sqrt(x^2+y^2) of those variables like I showed with velocity) will exceed the respective maxV, maxA, or jmax values that you have set as constants.
I'd also like to improve how you generate your guide values. I'm going to assume that the guide can be moving. In this case, the target will have the values listed above: x, y, vx, vy, ax, ay, jx, jy. You can name these however you'd like, I'm going to use targetX, targetY... etc to better illustrate my point.
From here you should be finding your guide values. While the sprite is more than three seconds away from the target, you may use the target's position in three seconds (note: I recommend setting this as a variable so it is simple to modify). For this case:
predictionTime = 3000*timescale; //You can set this to however many seconds you want to be predicting the position from.
If you really wanted to, you could smooth the curve using integration functions or loops for the guide values to get a more accurate guide value from the target values. However, this is not a good idea because if you ever implement multiple targets / etc, it could have a negative impact on performance. Thus, we will use a very simple estimation that is pretty accurate for being such low cost.
if (sprite is more than predictionTime seconds away from the target){
guideX = targetX + predictionTime * targetVX;
guideY = targetY + predictionTime * targetVY;
}
Note that we didn't account for the acceleration and jerk of the target in this, it's not needed for such a simple approximation.
However, what if the sprite is lese than predictionTime seconds away from the target? In this case we want to start increasingly lessening our predictionTime values.
else{
guideX = targetX + remainingTime * targetVX;
guideY = targetY + remainingTime * targetVY;.
}
Here you have three choices on finding the value of remaining time. You can set remainingTime to zero, making the guide coordinates the same as the targets. You can set remainingTime to be sqrt((targetX-currentX)^2+(targetY-currentY))/(sqrt(currentVX)^2+(currentVY)^2), which is basically distance / time in 2d, a cheap and decent approximation. Or you can use for loops as mentioned before to simulate an integral to account for the changing velocity values (if they are deviating from maxV). However, generally you will be remaining at or close to maxV, so this is not really worth the expense. Edit: Also, I'd also recommend setting remainingTime to be 0 if it is less than some value (perhaps about 0.5 or so. This is because you don't want hitbox issues because your sprite is following guide coordinates that have a slight offset (as well as the target moving in circles would give it a larger velocity value while changing direction / essentially a strong evasion tactic. Maybe you should add something in specifically for that.
We now have the guideX and guideY values, and account for getting very close to a moving target having an effect on how far from the target the guide coordinates should be placed. We will now do the 'current' value archetype.
We will update the lowest derivative values first, check to see if they are within bounds of our maximum values, and then update the next lowest and etc. Note that JX and JY are as mentioned before to allow for non constant acceleration.
//You will have to choose the jerk factor -- the rate at which acceleration changes with respect to time.
//We need to figure out what direction we're going in first. Note that the arc tangent function could be atan or whatever your library uses.
dir = arctan((guideY-currentY)/(guideX-currentX));
This will return the direction as an angle, either in radians or degree depending on your trig library. This is the angle that your sprite needs to take to go in the direction of guide.
t = time; //For ease of writing.
newAX = currentAX + jerk*t*cos(dir);
newAY = currentAY + jerk*t*sin(dir);
You may be wondering how the newAx value will ever decrease. If so, note that cos(dir) will return negative if guide is to the left of the target, and likewise sin(dir) will return negative if the sprite needs to go down. Thus, also note that if the guide is directly below the sprite, then newAx will be 0 and newAY will be a negative value because it's going down, but the magnitude of acceleration, in other words what you compare to maxA, will be positive -- as even if the sprite is moving downwards, it's not moving at negative speed.
Note that because cos and sin are of the same library as atan presumably, so the units will be the same (all degrees or all radians). We have a maximum acceleration value. So, we will check to make sure we haven't exceeded that.
magnitudeA = sqrt(newAX^2+newAY^2);
if (magnitudeA > maxA){
currentAX = maxA * cos(dir);
currentAY = maxA * sin(dir);
}
So at this point, we have either capped our acceleration or have satisfactory acceleration components with a magnitude less than maxA. Let us do the same for velocity.
newVX = currentVX + currentAX*t;
newVY = currentVY + magnitudeA*t*sin(dir);
Note that I have included two ways to find the velocity components here. Either one works, I'd recommend choosing one and using it for both x and y velocity values for simplicity. I just wanted to highlight the concept of magnitude of acceleration.
magnitudeV = sqrt(newVX^2+newVY^2);
if (magnitudeV > maxV){
currentVX = maxV * cos(dir);
currentVY = maxV * sin(dir);
}
We'd also like to stop boomeranging around our target. However, we don't want to just say slow down alot like you did in your JSFiddle, because then if the target is moving it will get away (lol). Thus, I suggest checking how close you are, and if you are in a certain proximity, reduce your speed linearly with distance with an offset of the target's speed. So set closeTime to something small like 0.3 or what ever feels good in your game.
if (remainingTime < closeTime){
//We are very close to the target and will stop the boomerang effect. Note we add the target velocity so we don't stall while it's moving.
//Edit: We should have a min speed for the enemy so that it doesn't slow down too much as it gets close. Lets call this min speed, relative to the target.
currentVX = targetVX + currentVX * (closeTime - remainingTime);
currentVY = targetVY + currentVY * (closeTime - remainingTime);
if (minSpeed > sqrt(currentVX^2+currentVY^2) - sqqrt(targetVX^2-targetVY^2)){
currentVX = minSpeed * cos(dir);
currentVY = minSpeed * sin(dir);
}
}
magnitudeV = sqrt(currentVX^2+currentVY^2);
At this point we have good values for velocity too. If you going to put in a speedometer or check the speed, you're interested in magnitudeV.
Now we do the same for position. Note you should include checks that the position is good.
newX = currentX + currentVX*t;
newY = currentY + currentVY*t;
//Check if x, y values are good.
current X = newX; currentY = newY;
Now everything has been updated with the good values, and you may write to the screen.

Implement a Kalman filter to smooth data from deviceOrientation API

I'm trying to smooth the data I'm getting from the deviceOrientation API to make a Google Cardboard application in the browser.
I'm piping the accelerometer data straight into the ThreeJs camera rotation but we're getting a lot of noise on the signal which is causing the view to judder.
Someone suggested a Kalman filter as the best way to approach smoothing signal processing noise and I found this simple Javascript library on gitHub
https://github.com/itamarwe/kalman
However its really light on the documentation.
I understand that I need to create a Kalman model by providing a Vector and 3 Matrices as arguments and then update the model, again with a vector and matrices as arguments over a time frame.
I also understand that a Kalman filter equation has several distinct parts: the current estimated position, the Kalman gain value, the current reading from the orientation API and the previous estimated position.
I can see that a point in 3D space can be described as a Vector so any of the position values, such as an estimated position, or the current reading can be a Vector.
What I don't understand is how these parts could be translated into Matrices to form the arguments for the Javascript library.
Well, I wrote the abhorrently documented library a couple of years ago. If there's interest I'm definitely willing to upgrade it, improve the documentation and write tests.
Let me shortly explain what are all the different matrices and vectors and how they should be derived:
x - this is the vector that you try to estimate. In your case, it's probably the 3 angular accelerations.
P - is the covariance matrix of the estimation, meaning the uncertainty of the estimation. It is also estimated in each step of the Kalman filter along with x.
F - describes how X develops according to the model. Generally, the model is x[k] = Fx[k-1]+w[k]. In your case, F might be the identity matrix, if you expect the angular acceleration to be relatively smooth, or the zero matrix, if you expect the angular acceleration to be completely unpredictable. In any case, w would represent how much you expect the acceleration to change from step to step.
w - describes the process noise, meaning, how much does the model diverge from the "perfect" model. It is defined as a zero mean multivariate normal distribution with covariance matrix Q.
All the variables above define your model, meaning what you are trying to estimate. In the next part, we talk about the model of the observation - what you measure in order to estimate your model.
z - this is what you measure. In your case, since you are using the accelerometers, you are measuring what you are also estimating. It will be the angular accelerations.
H - describes the relation between your model and the observation. z[k]=H[k]x[k]+v[k]. In your case, it is the identity matrix.
v - is the measurement noise and is assumed to be zero mean Gaussian white noise with covariance R[k]. Here you need to measure how noisy are the accelerometers, and calculate the noise covariance matrix.
To summarize, the steps to use the Kalman filter:
Determine x[0] and P[0] - the initial state of your model, and the initial estimation of how accurately you know x[0].
Determine F based on your model and how it develops from step to step.
Determine Q based on the stochastic nature of your model.
Determine H based on the relation between what you measure and what you want to estimate (between the model and the measurement).
Determine R based on the measurement noise. How noisy is your measurement.
Then, with every new observation, you can update the model state estimation using the Kalman filter, and have an optimal estimation of the state of the model(x[k]), and of the accuracy of that estimation(P[k]).
var acc = {
x:0,
y:0,
z:0
};
var count = 0;
if (window.DeviceOrientationEvent) {
window.addEventListener('deviceorientation', getDeviceRotation, false);
}else{
$(".accelerometer").html("NOT SUPPORTED")
}
var x_0 = $V([acc.x, acc.y, acc.z]); //vector. Initial accelerometer values
//P prior knowledge of state
var P_0 = $M([
[1,0,0],
[0,1,0],
[0,0,1]
]); //identity matrix. Initial covariance. Set to 1
var F_k = $M([
[1,0,0],
[0,1,0],
[0,0,1]
]); //identity matrix. How change to model is applied. Set to 1
var Q_k = $M([
[0,0,0],
[0,0,0],
[0,0,0]
]); //empty matrix. Noise in system is zero
var KM = new KalmanModel(x_0,P_0,F_k,Q_k);
var z_k = $V([acc.x, acc.y, acc.z]); //Updated accelerometer values
var H_k = $M([
[1,0,0],
[0,1,0],
[0,0,1]
]); //identity matrix. Describes relationship between model and observation
var R_k = $M([
[2,0,0],
[0,2,0],
[0,0,2]
]); //2x Scalar matrix. Describes noise from sensor. Set to 2 to begin
var KO = new KalmanObservation(z_k,H_k,R_k);
//each 1/10th second take new reading from accelerometer to update
var getNewPos = window.setInterval(function(){
KO.z_k = $V([acc.x, acc.y, acc.z]); //vector to be new reading from x, y, z
KM.update(KO);
$(".kalman-result").html(" x:" +KM.x_k.elements[0]+", y:" +KM.x_k.elements[1]+", z:" +KM.x_k.elements[2]);
$(".difference").html(" x:" +(acc.x-KM.x_k.elements[0])+", y:" +(acc.y-KM.x_k.elements[1])+", z:" +(acc.z-KM.x_k.elements[2]))
}, 100);
//read event data from device
function getDeviceRotation(evt){
// gamma is the left-to-right tilt in degrees, where right is positive
// beta is the front-to-back tilt in degrees, where front is positive
// alpha is the compass direction the device is facing in degrees
acc.x = evt.alpha;
acc.y = evt.beta;
acc.z = evt.gamma;
$(".accelerometer").html(" x:" +acc.x+", y:" +acc.y+", z:" +acc.z);
}
Here is a demo page showing my results
http://cardboard-hand.herokuapp.com/kalman.html
I've set sensor noise to a 2 scalar matrix for now to see if the Kalman was doing its thing but we have noticed the sensor has greater variance in the x axis when the phone is lying flat on the table. We think this might be an issue with Gimbal lock. We haven't tested but its possible the variance changes in each axis depending on the orientation of the device.

Tile Filling Algorithm for Game

Background:
I am working on a tile-based game in Javascript where a character freely moves around the map (no diagonal - Left/Right/Up/Down) and fills in tiles as he moves around the map. There are three tile types -- tiles you've filled (blue), your current path (red), and empty ones (black). There are also enemies (stars) that move around the map as well, but only in empty areas. The objective is to fill as much of the map as possible.
Map is sized as roughly 40x40 tiles. There is a 1 tile thick border around the entire outside of the map that is already "filled" (blue).
I have established that a flood-fill algorithm will work for filling up areas of tiles when needed. However, my problem is as follows:
PROBLEM STATEMENT:
I want to only fill a sectioned-off part of the map if there are no enemies in it.
My Question:
I could run flood-fill algorithm and stop it if it reaches a tile occupied by an enemy -- however, is this the most efficient approach (for a real time game)?
IF YES, how do I determine where to start the algorithm from in a systematic way since there are multiple areas to check and the character doesn't have to move in a perfectly straight line (can zig-zag up/down/right/left, but can't move diagonally).
Picture Example 1 (pics explain better):
Note: red areas turn blue (filled) once you reach another filled area. In example below, there are no enemies in the contained area, so the area is filled.
Picture Example 2:
In this second example, there is an enemy within the contained area (and on the outside area - not shown) so nothing but the line is filled.
Summary: What is the best approach for doing this type of filling? Is flood fill the best choice for determining whether to fill or not -- 40x40 makes for a pretty large calculation. If yes, how do I determine what tile do I start with?
Let me suggest a different way of looking at your problem.
Going by the description of your game, it seems like the user's main, perhaps only, "verb" (in game design terms) is to draw a line that divides the open area of the field into two sections. If either of these two sections is free of enemies, that section gets filled in; if neither section is free of enemies, the line remains but both sections remain open. There are no other conditions determining whether a section gets filled or not, right?
So the most efficient way to solve this problem, I would think, is simply to draw a continuous line, which may make corners but only moves in horizontal or vertical directions, from one of your enemies to every other enemy in turn. We'll call this line the "probe line". From here on, we're using the approach of Derek's suggested "Ray casting algorithm": We look at the number of times the "probe line" crosses the "border line", and if the number of crossings is ever odd, it means you have at least one enemy on each side of the line, and there's no filling.
Note, though, that there's a difference between the two lines coinciding and the two lines crossing. Picture a probe line that goes from the coordinates (0,10) to (39,10) , and a border line that goes down from (5,0) to (5,10) and then goes right to (13,10). If it goes down from there towards (13,39), the two lines are crossing; if instead it goes upwards toward (13,0), they're not.
After a lot of thought, I strongly suggest that you store the "border line", and construct the "probe line", in terms of line segments - rather than trying to determine from which cells are filled which line segments created them. That will make it much harder than it has to be.
Finally, one odd game design note to be aware of: unless you constrict the user's control so that he cannot bring the border line back to within one cell of itself, then a single border line drawn by a user might end up sectioning off the field into more than two sections - there could be sections created by the border line looping right back on itself. If you allow that, it could very drastically complicate the calculation of where to fill. Check the following diagram I made via Derek's fiddle (thank you, Derek!):
As you can see, one border line has actually created three sections: one on the upper side of the line, one below the line, and one formed by the line itself. I'm still thinking about how that would affect things algorithmically, for that to be possible.
EDIT: With a) time to think about the above creation-of-multiple-sections-by-loops, and b) the Simulation of Simplicity resource brought up by Derek, I think I can outline the simplest and most efficient algorithm that you're likely to get.
There's one subproblem to it which I'll leave to you, and that is determining what your new sections are after the player's actions have drawn a new line. I leave that to you because it's one that would have had to be solved before a solution to your original problem (how to tell if there are enemies within those sections) could have been called.
The solution, presented here as pseudocode, assumes you have the border of each section stored as line segments between coordinates.
Create a list of the sections.
Create a list of the enemies.
Continue as long as neither list is empty:
For each enemy in the enemy list:
Designate "Point A" as the coordinates of the enemy, PLUS 0.5 to both x and y.
For each section in the section list:
Designate "Point B" as the upper-left-most coordinate, PLUS 0.5 to both x and y.
Count how many of the section border segments cross a line between A and B.
If the answer is even:
remove this section from the section list
skip forward to the next enemy
If any sections remain in the list, they are free of enemies. Fill them in.
The addition of the 0.5 to the coordinates of the "probe line" are thanks to Derek's SoS resource; they eliminate the difficult case where the lines coincide rather than simply crossing or not crossing.
If you have the points of the border of your shape that lies on the same y as the enemy, then you can simply count the number of borders, starting from either left or right to the enemy. If it's odd then it's inside. If it's even then it's outside.
Since you are using a grid system this should be easy to implement (and very fast). This algorithm is called the Ray casting algorithm.
Here's a simple example I created: http://jsfiddle.net/DerekL/8QBz6/ (can't deal with degenerate cases)
function testInside(){
var passedBorder = 0,
passingBorder = false;
for(var x = 0; x <= enemy[0]; x++){
if(board[x][enemy[1]] === 1) passingBorder = true;
else if(board[x][enemy[1]] === 0 && passingBorder){
passingBorder = false;
passedBorder++;
}
}
return !!(passedBorder%2);
}
For example, you have this shape which you have determined:
removed
Guess what I found, (slightly modified)
//simple enough, it only needs the x,y of your testing point and the wall.
//no direction or anything else
function testInside3() {
var i, j, c = 0;
for (i = 0, j = wallList.length-1; i < wallList.length; j = i++) {
if ( ((wallList[i][1]>enemy[1]) ^ (wallList[j][1]>enemy[1])) &&
(enemy[0] < (wallList[j][0]-wallList[i][0]) * (enemy[1]-wallList[i][1]) / (wallList[j][1]-wallList[i][1]) + wallList[i][0]) )
c = !c;
}
return !!c;
}
http://jsfiddle.net/DerekL/NvLcK/
This is using the same ray casting algorithm I mentioned, but this time the "ray" is now mathematical using the following inequality for x and a condition for y:
(X2 - X1)(Py - Y1)
Px < ────────────────── + X1
Y2 - Y1
which is derived by combining these two:
Ray:
x(t) = Px + t, y(t) = Py, where t > 0 (the ray goes to the right)
Edge:
x(u) = (X2 - X1)u + X1, y(u) = (Y2 - Y1)u + Y1, where 0 <= u <= 1
And the condition for y:
(Y1 > Py) ⊕ (Y2 > Py)
which is equivalent to:
(Y1 ≥ Py > Y2) ∨ (Y2 ≥ Py > Y1)
and yadi yadi yada some other interesting technical stuff.
Seems like this is the default algorithm in many native libraries. The method used to dealing with degenerate cases is called Simulation of Simplicity, described in this paper (section 5.1).
Nevertheless, here's the result generated with the algorithm testing every coordinate:
If it's easy to determine where the borders of a region to possibly fill are, you can use the following approach:
Assign each edge a clockwise directionality. That is, construct a vector for each edge that starts on its corners and has a direction such that a clockwise path around the region is described by these vectors.
For each enemy:
Construct a vector starting from the enemy and ending on the closest edge. We'll call this an enemy_vector.
Calculate the cross product of the enemy_vector and the vector corresponding to the closest edge. The sign of the cross product will tell you whether the enemy is inside the region: if it's positive, the enemy is outside of it, and if it's negative it isn't!
EXAMPLE:
Suppose we have the following region and enemy to evaluate the inside-ness of.
We can encode the region as a series of vectors that give it a clockwise orientation, like so:
So how do we use that to determine the side of the region inhabited by the enemy? We draw a vector from it (which I've colored red) to the nearest edge (which I've colored green)...
...and take the cross product of the red vector and the green vector. Application of the right-hand rule tells us that (red) x (green) > 0, so the enemy must be outside the region!

The closer the x/y the more x/y velocity

I am writing a fairly simple script in JavaScript using the canvas. It draws a central node which pulls all of the surrounding nodes towards it. This works great, however I need each node to repel each other.
I am going to do this by increasing each nodes velocity away from each other so eventually they should level out and end up looking something like a flower. It needs to be enough force to stop them from hitting each other or sinking into the center node without flying off into the distance.
I just can not work out how I can have a higher number the closer they get.
So if two nodes where 10px away from each other it would add 5 in force to one of their x velocities. But if they where 1000px away from each other then it would add almost nothing to the force of one of the nodes.
Does anyone know of a mathematical equation I can use to work this kind of thing out, or maybe a nudge in the right direction?
TL;DR: Depending on how close two x values are, I need to increment the x velocity of one node so they move apart but eventually level out. It is just the maths I can not crack, I have pretty much all of the JavaScript done, including the implementation of velocity.
Thanks, and sorry it is a bit wordy.
You just need an inverse (or inverse square) relationship:
var increment = k / distance;
or:
var increment = k / (distance * distance);
You can determine k based on the actual values you want, for example, in the first case, if you wanted an increment of 5 for a distance of 10, you would set k = increment * distance = 50.
Look into the equations governing electrical point charges, have the velocity be based on the "force" each "charge" would feel based on its proximity.

2d parabolic projectile

I'm looking to create a basic Javascript implementation of a projectile that follows a parabolic arc (or something close to one) to arrive at a specific point. I'm not particularly well versed when it comes to complex mathematics and have spent days reading material on the problem. Unfortunately, seeing mathematical solutions is fairly useless to me. I'm ideally looking for pseudo code (or even existing example code) to try to get my head around it. Everything I find seems to only offer partial solutions to the problem.
In practical terms, I'm looking to simulate the flight of an arrow from one location (the location of the bow) to another. I have already simulated the effects of gravity on my projectile by updating its velocity at each logic interval. What I'm now looking to figure out is exactly how I figure out the correct trajectory/angle to fire my arrow at in order to reach my target in the shortest time possible.
Any help would be greatly appreciated.
Pointy's answer is a good summary of how to simulate the movement of an object given an initial trajectory (where a trajectory is considered to be a direction, and a speed, or in combination a vector).
However you've said in the question (if I've read you correctly) that you want to determine the initial trajectory knowing only the point of origin O and the intended point of target P.
The bad news is that in practise for any particular P there's an infinite number of parabolic trajectories that will get you there from O. The angle and speed are interdependent.
If we translate everything so that O is at the origin (i.e. [0, 0]) then:
T_x = P_x - O_x // the X distance to travel
T_y = P_y - O_y // the Y distance to travel
s_x = speed * cos(angle) // the X speed
s_y = speed * sin(angle) // the Y speed
Then the position (x, y) at any point in time (t) is:
x = s_x * t
y = s_y * t - 0.5 * g * (t ^ 2)
so at impact you've got
T_x = s_x * t
T_y = -0.5 * g * (t ^ 2) + s_y * t
but you have three unknowns (t, s_x and s_y) and two simultaneous equations. If you fix one of those, that should be sufficient to solve the equations.
FWIW, fixing s_x or s_y is equivalent to fixing either speed or angle, that bit is just simple trigonometry.
Some combinations are of course impossible - if the speed is too low or the angle too high the projectile will hit the ground before reaching the target.
NB: this assumes that position is evaluated continuously. It doesn't quite match what happens when time passes in discrete increments, per Pointy's answer and your own description of how you're simulating motion. If you recalculate the position sufficiently frequently (i.e. 10s of times per second) it should be sufficiently accurate, though.
I'm not a physicist so all I can do is tell you an approach based on really simple process.
Your "arrow" has an "x" and "y" coordinate, and "vx" and "vy" velocities. The initial position of the arrow is the initial "x" and "y". The initial "vx" is the horizontal speed of the arrow, and the initial "vy" is the vertical speed (well velocity really but those are just words). The values of those two, conceptually, depend on the angle your bowman will use when shooting the arrow off.
You're going to be simulating the progression of time with discrete computations at discrete time intervals. You don't have to worry about the equations for "smooth" trajectory arcs. Thus, you'll run a timer and compute updated positions every 100 milliseconds (or whatever interval you want).
At each time interval, you're going to add "vx" to "x" and "vy" to "y". (Thus, note that the initial choice of "vx" and "vy" is bound up with your choice of time interval.) You'll also update "vx" and "vy" to reflect the effect of gravity and (if you feel like it) wind. If "vx" doesn't change, you're basically simulating shooting an arrow on the moon :-) But "vy" will change because of gravity. That change should be a constant amount subtracted on each time interval. Call that "delta vy", and you'll have to tinker with things to get the values right based on the effect you want. (Math-wise, "vy" is like the "y" component of the first derivative, and the "delta vy" value is the second derivative.)
Because you're adding a small amount to "vy" every time, the incremental change will add up, correctly simulating "gravity's rainbow" as your arrow moves across the screen.
Now a nuance you'll need to work out is the sign of "vy". The initial sign of "vy" should be the opposite of "delta vy". Which should be positive and which should be negative depends on how the coordinate grid relates to the screen.
edit — See #Alnitak's answer for something actually germane to your question.

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