Implement hermite interpolation - multiplayer game

Question

I am trying to make a client-server architecture. I am stuck at the interpolation part. Right now, I have a very naive implementation of the interpolation algorithm. I have every player given a position history and whenever I receive a position data for other player from the server I push the position in to that array. Every client frame I use the oldest position history to interpolate to a new position with a constant speed.

// when new position for other player recieved 
p.stateHistory.push(data)

// Every client frame
if(p.stateHistory.length < 1)
            return false

    let deltaPosition = p.stateHistory[0].position.clone().sub(p.clientPosition)
    let direction = Math.atan2(deltaPosition.y, deltaPosition.x)
    let velocity = new Vector2(Math.cos(direction), Math.sin(direction)).scale(30/100)

    let threshold = 10
    if(deltaPosition.magnitude() < threshold) {
        p.clientPosition.x = p.stateHistory[0].position.x
        p.clientPosition.y = p.stateHistory[0].position.y
        p.stateHistory.shift()
    } else {
        p.clientPosition.add(velocity.clone().scale(deltaTime))
    }

I couldn't find way other to interpolate with a constant speed. I came to know about hermite interpolation from gafferongames. But it's sad that the article didn't have anything about its math and its implementation. I tried to go through the wikipedia article on hermite interpolation, but it didn't help. I know nothing about the math behind it. A pseudo code would be appreciated.

What I have been able to do so far: http://client-side-prediction-attempt.herokuapp.com/

Answer 1

Let's say say that your client receives a new position-velocity update at time currentTime. Then, you need to save the current position/velocity, the target position/velocity, the current time, and the time when you expect the next update:

function updateFromServer(position, velocity) {
    startP = currentPosition; //the current position of the player
    startV = currentVelocity;
    endP   = position;
    endV   = velocity;
    startT = currentTime; //the current time of the game
    endT   = startT + 0.1; //expect the next update in 100 ms
}

Once you have stored this data, you can do your frame update using interpolation. If you are outside of the [startT, endT] interval, you might just want to continue a uniform motion:

function frameUpdate(deltaT) {
    if(currentTime > endT)
        //uniform motion
        currentPosition += deltaT * currentVelocity;
    else {
        //cubic Hermite interpolation
        var t = (currentTime - startT) / (endT - startT); //interpolation parameter
        var t2 = t * t;
        var t3 = t2 * t;
        currentPosition = 
            (2 * t3 - 3 * t2 + 1) * startP + 
            (t3 - 2 * t2 + t)     * (endT - startT) * startV + 
            (-2 * t3 + 3 * t2)    * endP   + 
            (t3 - t2)             * (endT - startT) * endV;
        currentVelocity = 1 / (endT - startT) * (
            (6 * t2 - 6 * t)     * startP + 
            (3 * t2 - 4 * t + 1) * (endT - startT) * startV +
            (-6 * t2 + 6 * t)    * endP   +
            (3 * t2 - 2 * t)     * (endT - startT) * endV);
    }
}

Note that the formulas in this snippet are not valid JavaScript code. They must be translated to whatever library you use.