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How can I ensure that an object using a cosine equation to determine speed meets its lowest valley at the same time it hits a waypoint?

I'm working on a game for game jam and part of it is to make platforms that move smoothly. They slow down at the ends of their movement before turning around. The platforms simply move side to side or up and down between two waypoints(which are just empty transforms). I have code that uses cosine to determine the speed which works well except it doesn't align with the waypoints, the platforms tend to slow and change direction before ever reaching the waypoints. I need a way to use the distance between the waypoints as a variable in determining how the cosine equation changes speed so that the platforms slow and reverse direction exactly at the waypoints.
Here is what I have so far:
void Side_to_side()
{
if (waypointIndex < horWaypoints.Length)
{
platformSpeed = (1f * (float)Mathf.Cos(2f * (float)Mathf.PI * 1f * totalTime));
Vector3 targetPosition = horWaypoints[waypointIndex].position;
float delta = platformSpeed * Time.deltaTime;
transform.position = Vector2.MoveTowards(transform.position, targetPosition, delta);
if (transform.position.x == targetPosition.x && transform.position.y == targetPosition.y)
{
if (waypointIndex + 1 == horWaypoints.Length)
waypointIndex = 0;
else
waypointIndex++;
}
}
else
{
waypointIndex = 0;
}
//Translate platform back and forth between two waypoints
}
As I have said this code moves the platforms in the motions i want but they don't use the waypoints as turn around points. I understand I could do away with the waypoints and just calculate how far I would like each platform to go before turning around individually but that would take time to do it for each platform whereas I'd like to quickly put down waypoint pairs for them to use and the script calculates what the perfect values would be to match the waypoint locations.

If I understand you correctly you want to move an object forth and back between exactly two positions and apply some smoothing to the movement.
I would rather use a combination of Vector2.Lerp with a Mathf.PingPong as factor and you can then apply ease in and out using additionally Mathf.SmoothStep.
This could look like e.g.
public Transform startPoint;
public Transform endPoint;
// Desired duration in seconds to go exactly one loop startPoint -> endPoint -> startPoint
public float duration = 1f;
private void Update ()
{
// This will move forth and back between 0 and 1 within "duration" seconds
var factor = Mathf.PingPong(Time.time / (2f * duration), 1f);
// This adds additional ease-in and -out near to 0 and 1
factor = Mathf.SmoothStep(0, 1, factor);
// This interpolates between the given positions according to the given factor
transform.position = Vector2.Lerp(startPoint, endPoint, factor);
}
you could of course still use cosine if necessary, basically any function that returns a value between 0 and 1. You just have to use the correct multiplier in order to achieve the desired duration in seconds.
Note: Typed on the phone and not 100% sure on the math but I hope the idea gets clear

Using Vector3.slerp

I just want to know if I'm using vector3.slerp correctly as I have some issues.
I have set up a cube in a scene and I want it move smoothly from its current position to one being passed into by the program. Before I was trying to use the slerp I simply had my cube moving from its current point to its new point like this (as I said, the math all works):
cubeFour.transform.localPosition = new Vector3((B2*C1 - B1*C2)/delta,0,(A1*C2 - A2*C1)/delta);
But when I put it in a slerp call, my cube is no longer on the screen. This is how I'm calling it:
Vector3 targetPosition = new Vector3(lerpX, lerpY, lerpZ);
cubeFour.transform.localPosition = Vector3.Slerp(cubeFour.transform.localPosition, targetPosition, Time.deltaTime);
LerpX, LerpY & LerpZ are local variables I've set to contain the X, Y & Z of the first Vector3 I created in my first attempt.
Have I set up the slerp correctly or have I made a kerfuffle somewhere?

Slerp is best for directions, Lerp is best for positions. You should probably be using Lerp. And Time.deltaTime is almost certainly the wrong choice for t. You should be giving it a number that moves from 0 to 1 over the time that you want the cube to move, e.g.
float moveTimeInSeconds = 2;
cubeFour.transform.localPosition = Vector3.Lerp(cubeStartingPosition, targetPosition, (Time.time - startTime) / moveTimeInSeconds);
Or use MoveTowards if it makes more sense to define the speed of motion, and to move towards a target regardless of a starting point, instead of via starting and ending positions and times.
float step = speed * Time.deltaTime;
cubeFour.transform.localPosition = Vector3.MoveTowards(cubeFour.transform.localPosition, targetPosition, step);

Lerp uses a percentage to animate!
This means you must plug in a percentage, not a value near the same thing each frame.
Example Update():
var animationSpeedInSeconds=1;
var animationCounter:float=0;
function Update(){
if(animationCounter>=0 && animationCounter<=1){
animationCounter+=Time.deltaTime/animationSpeedInSeconds;
cubeFour.transform.localPosition = Vector3.MoveTowards(cubeFour.transform.localPosition, targetPosition, animationCounter);
}
}
I hope this helps!

Implement bouncing balls collision detection

I have a bouncing ball application and I have to extend it to prevent overlapping of the balls.
When ball overlaps another, they should move away as in real life.
I have to extend the given MoveBall method:
private void MoveBall()
{
prevX = x;
prevY = y;
x += xVelocity;
y += yVelocity;
// Is there too closed ball?
foreach (Ball ball in parentForm.balls)
{
distance = Math.Sqrt(Math.Pow((double)(ball.prevX - prevX), 2) +
Math.Pow((double)(ball.prevY- prevY), 2));
overlap = ((radius + ball.radius) - distance);// +ball.radius;
if (ball.id != this.id &&
ball.id != lastID &&
overlap > 0)
{
lastID = this.id;
if (xVelocity > 0) // roading right
{
xVelocity = -xVelocity;
x -= xVelocity - ball.xVelocity;
}
else if (xVelocity <= 0) // roading left
{
xVelocity = -xVelocity;
x += xVelocity + ball.xVelocity;
}
if (yVelocity > 0)
{ // going up
yVelocity = -yVelocity;
y -= yVelocity - ball.yVelocity;
}
else if (yVelocity <= 0) // down
{
yVelocity = -yVelocity;
y += yVelocity + ball.yVelocity;
}
}
}
// ***********************************************
// ***************** END MY CODE *****************
if (x > parentForm.Width - 10 - (radius) || x < 0)
{
if (x < 0) x = 0;
if (x > parentForm.Width - 10) x = parentForm.Width - 10 - radius;
xVelocity = -xVelocity;
}
if (y > parentForm.Height - 40 - (radius) || y < 0)
{
if (y < 0) y = 0;
if (y > parentForm.Height - 40) y = parentForm.Height - 40 - (radius);
yVelocity = -yVelocity;
}
}
x,y, xVelocity, yVelocity, radius, prevX, prevY declared as int.
overlap, distance as double.
When 2 overlap, they are getting stuck. Why?
Unfortunately, I can't upload all source code because there are lot of modules.
I'm using Visual C# Express 2010.

As no Question is asked explicitly, I will assume the question "Why are the balls sticking together?"
You have only shown one loop in source code, that's not enough ;-) To check all possible collisions, you need to check n*(n-1)/2 possible collisions. That is normally done with two loops. You have to put in careful measures to avoid handling the same collision twice.
The reason that your balls get stuck is that you handle the same collision multiple times. For example two balls colliding exactly horizontal: The left one has velocity 5 and x-position of 100. The other one shall have a position of 110 and velocity of -6. When the collision happens:
x is set to 105.
Collision detected: x is set to 104 and velocity to -5.
The other Ball handles the same collision:
He moves according to his velocity to position 104.
Collision handling: His velocity becomes 6 and position becomes 105.
The balls were at 100 and 110 resp. and have been moved to 104 and 105. While the velocities are now pointing away from each other, the collision handling in the following step will invert them again. So the positions are close together and the velocities are changing sign every frame. The balls seem "stuck to each other".
I hope the answer helps you to understand your problem. For a better implementation of an elastic collision (that handles each collision exactly once) look here: Ball to Ball Collision - Detection and Handling

Having stumbled upon similar issues when I made my first attempts at collision detection algorithms, I'll try to describe what I think is the problem here.
Maybe the balls move fast enough so that, before collision is even detected by your code, they are already partially "inside" each other. When collision detection comes and notices that, it does what it's supposed to do: change the planned trajectories of the objects according to the details of the collision that just happened. The problem is that, because these objects got sort-of "merged" before collision detection caught them, they can't get unstuck because collision detection is fired again, trapping them with each other.
If this is the source of the problem, then maybe the above code would work with a small enough velocity vector. Of course, that's not a real solution, but if it does work for very small velocities, it probably confirms my hypothesis and you have some idea regarding how to proceed.

Gravity with air-time, acceleration and speed gaining

I am trying to accomplish a gravity, where airtime is included, and also acceleration.I have tried using usual gravity, which looks something like this:
velocity += gravity * dt;
position += velocity * dt;
This would probably work good enough for a normal platformer game, but I am trying to make a game inspired by "The helicopter game", where you have to navigate through a tunnel, without touching the walls.what I want to do different, is that I want to be able to save up speed on the way down, which will be used on the way up again, so I will have some acceleration at the beginning.I also want some kind of airtime, so when you hit the top it would not force you down as fast as It would, if I had used the gravity from the code sample.This image illustrates the curve I would like to have:Please note that the whole controlling is done by one key, so for example you would fly up if you held down space, and dive if you released it.The character also never moves left or right, since it will have a static X position on the screen, so vectors can't be used.I have spent hours trying to make it work, but with no success. I have also tried searching on the internet, but without any luck.The game "Whale Trails" got the gravity I kind of want.Here is a link for a video of the game: http://www.youtube.com/watch?v=5OQ0OWcuDJsI'm not that big of a physics guy, so it would be cool if you could give an example of the actual code I hope anyone can help me figure this out.

Gravity is the force that pulls objects down. Your player is the force that pulls objects up. Accordingly your code must be:
if(keyPressed) {
velocity += POWER_OF_PLAYER;
}
velocity += G;
position += velocity;
This is enough to create a curve like you illustrated. Of course POWER_OF_PLAYER must be of a different sign and the absolute value must be greater to make this work.
G = -9.81
POWER_OF_PLAYER = 20
Saving power is then a simple check.
if(keyPressed) {
if(powerSaved > 0) {
velocity += POWER_OF_PLAYER;
powerSaved -= SOMETHING;
}
} else if (velocity >= SOME_MINIMUM_SPEED_BEFORE_GETTING_POWER) {
powerSaved += SOMETHING;
}
SOME_MINIMUM_SPEED_BEFORE_GETTING_POWER should be something less or equal 0.
P.S. I assumed your Y axis starts at ground and shoots into the sky. Signs put accordingly.

It looks like the velocity of the fish is constant.
Try the following:
velocity is fixed and should not change (unless the fish eats a speed power-up).
angle = 0 is equivalent to level flight.
angle -= gravity * dt;
if (angle < - Math.PI / 2) {
angle = Math.PI / 2;
}
if (keyPressed) {
angle += power * dt;
}
if (angle < - Math.PI / 2) {
// Stop the player from doing a looping
angle = - Math.PI / 2;
}
if (angle > Math.PI / 2) {
// Stop the player from doing an inverted looping
angle = Math.PI / 2;
}
// Move the fish (vertical component of velocity)
position += velocity * dt * Math.sin(angle);
// Move the background (horizontal component of velocity)
background_position -= velocity * dt * Math.sin(angle);

It sounds like incorporating 'lift' that is based on horizontal speed and have the button press trigger a 'noseup' movement would work sufficiently well.
So lift would be some constant k multiplied by the horizontal speed Vx and vertical speed Vy would be the difference of gravity and lift times the change in time dt
lift = k * Vx
Vy += ( lift - gravity ) * dt
def noseup
k = 0.01 #some small chunk
dx = k * Vx
dy = k * Vy
Vx -= dy
Vy += dx
When the plane (or whatever) noses up it basically lowers the velocity on one axis while increasing it on the other.
Probably wouldn't be a bad idea to throw drag in there somewhere as well now that I think about it, would have to be dependent on the absolute velocity V = ( Vx**2 + Vy**2 )**0.5...and weight is a better word than gravity in this case (less confusing, units wise) but it works I guess.
Not exactly "physics" but a close approximation that should work fairly well. Play around with the values of k and see if you can make it do what you want.
BTW sorry for the uber crappy psuedocode :-P

How to manage "speed" in a simple racing game?

I'm trying to develop a simple racing 2d game (view top-down) in C#, sdl.net.
Now, I'm trying to manage speed, acceleration and brakes of my car.
My problem is the algorithm.
I have the loop (Events_Tick) executed 50 times per seconds, where the position of my car is processed like the following:
private void Events_Tick(object sender, TickEventArgs e)
{
car.ProcessPosition();
}
and ProcessPosition is something like:
if (throttle)
{
speed += 1;
x += 1;
}
and finally, I draw my sprite on the new X.
The problem is that it is too fast!
So I'm asking you how to maintain 50 FPS (frames per second) and move my sprite (the car) only N pixels per second (based on its speed).
Thank you in advance! Regards!

First of all, because you have 2d not 1d game, you'll need your speed to be
class Vector
{
double x;
double y;
}
with this class, you should maintain position and speed.
Since you are 2d top-down, you'll have to implement some .RotateLeft() and .RotateRight() on the speed vector.
Rotation will be implemented like:
x' = cos(a) * x - sin(a) * y
y' = sin(a) * x + cos(a) * y
And you'll have to implement your .Move() method as follows:
void Move(Vector v)
{
x+=v.x;
y+=v.y;
}
EDIT: please ask away if clarification is needed, or some more in-depth discussion.
Also, you can use timer here, but try to calculate time spent from last timer event, and then multiply speed with that value when adding to the current position, you will get more accurate position that way.
a in sin() and cos() will be an angle in radians, and you will probably want degrees here.
Here is something to get you going regarding degrees and radians.

Use a double value for speed and x, then you can use Math.Round(x) for drawing.
if (throttle == true) {
speed+=0.05;
}
x+=speed; //move car even without acceleration
Follow Henk's advice to make your program more flexible and to be able to show the user something like 50mph instead of 0.05px per 0.02s:
double speedInMph = 50;
double screenSizeInMiles = .1;
double screenSizeInPx = 500;
if(throttle)
{
speedInMph += 1;
}
double speedInMpms = 50/(60*60*1000);//Miles per millisecond
double xInMiles = speedInMpms * 50;
double pxPerMile = screenSizeInPx/screenSizeInMiles;
x+= xInMiles * pxPerMile;

Acceleration will be constant, depending on user input (on a simple level the car is either accelerating, deccelerating or maintaining speed).
Speed will change each frame depending on that acceleration. Each frame, update the car's speed based on both the acceleration and the proportion of a second the frame covers (i.e. 1/50th of a second in your case).
So, if your car is accelerating at 10 units distance per second, in your example the speed will increase by 1/5th of a unit distance per frame.
Just thought it was worth elaborating a little on the other answers, instead of just posting more code :-)

You should add velocity according to current acceleration and handle just acceleration with your controls, limiting speed with a cap and increasing it according to time elapsed between two game loop iterations.
float accel = 10.0f; // units per squared second
float maxSpeed = 50.0f // units per second
float curSpeed = 0.0f;
//game init
while (1) {
long timeBefore = time();
// game loop
long timeElapsed = time() - timeBefore;
if (throttle) {
curSpeed += accel*(timeElapsed / 1000.0);
if (curSpeed > maxSpeed)
curSpeed = maxSpeed;
}
}
In this way you take into account the fact that a longer frame will increase your car speed more than a quicker ore, keeping it consistent with time.
This model implies a constant acceleration while you could hypotetically want a dynamic one, in that case you just move the update to the acceleration (taking into account the frame duration as for speed) and cap it to not go over a fixed threshold.

Henk is right. You will want to create a
Thread
and call its
Sleep(Int32)
method each time in your loop to slow the animation down, where the int parameter is the pause time in milliseconds.
In your case, you will want to move the car N/50 pixels each time looping, and sleep the thread for 20 milliseconds each time.