Showing trajectory indicator

Question

From the image you can see that the ball fired on the left that fire behind it, does not match the calculated trajectory. Im drawing the ball trajectory using an equation from a SO question, this is modified to take into consideration the box2d steps of 30 frames per second. This does calculate a valid trajectory but it does not match the actual trajectory of the ball, the ball has a smaller trajectory. I am applying a box2d force to the ball, this also has a density set and a shape. The shape radius varies depending on the type of ball. Im setting the start velocity in the touchdown event.

public class ProjectileEquation {  

    public float gravity;  
    public Vector2 startVelocity = new Vector2();  
    public Vector2 startPoint = new Vector2();  
    public Vector2 gravityVec = new Vector2(0,-10f);

    public float getX(float n) {  
        return startVelocity.x * (n * 1/30f) + startPoint.x;  
    }  

    public float getY(float n) {
        float t = 1/30f * n;
        return 0.5f * gravity * t * t + startVelocity.y * t + startPoint.y;  
    }  



} 

@Override  
    public void draw(SpriteBatch batch, float parentAlpha) {  
        float t = 0f;  
        float width = this.getWidth();  
        float height = this.getHeight();  

        float timeSeparation = this.timeSeparation;  

        for (int i = 0; i < trajectoryPointCount; i+=timeSeparation) {  
            //projectileEquation.getTrajectoryPoint(this.getX(), this.getY(), i);
            float x = this.getX() + projectileEquation.getX(i);  
            float y = this.getY() + projectileEquation.getY(i);  

            batch.setColor(this.getColor());  
            if(trajectorySprite != null) batch.draw(trajectorySprite, x, y, width, height);  

           // t += timeSeparation;  
        }  
    } 



public boolean touchDown (InputEvent event, float x, float y, int pointer, int button) {
                if(button==1 || world.showingDialog)return false;
                touchPos.set(x, y);
                float angle = touchPos.sub(playerCannon.position).angle();
                if(angle > 270 ) {
                    angle = 0;
                }
                else if(angle >70) {
                    angle = 70;
                }
                playerCannon.setAngle(angle);
                world.trajPath.controller.angle = angle;
                float radians =  (float) angle * MathUtils.degreesToRadians;
                float ballSpeed = touchPos.sub(playerCannon.position).len()*12;
                world.trajPath.projectileEquation.startVelocity.x = (float) (Math.cos(radians) * ballSpeed);
                world.trajPath.projectileEquation.startVelocity.y = (float) (Math.sin(radians) * ballSpeed);
                return true;
            }



public CannonBall(float x, float y, float width, float height, float damage, World world,  Cannon cannonOwner) {
        super(x, y, width, height, damage, world);
        active = false;
        shape = new CircleShape();
        shape.setRadius(width/2);

        FixtureDef fd = new FixtureDef();
        fd.shape = shape;
        fd.density = 4.5f;
        if(cannonOwner.isEnemy) { //Enemy cannon balls cannot hit other enemy cannons just the player
            fd.filter.groupIndex = -16;
        }
        bodyDef.type = BodyType.DynamicBody;
        bodyDef.position.set(this.position);

        body = world.createBody(bodyDef);
        body.createFixture(fd);
        body.setUserData(this);
        body.setBullet(true);
        this.cannonOwner = cannonOwner; 
        this.hitByBall = null;
        this.particleEffect = null;
    }

  private CannonBall createCannonBall(float radians, float ballSpeed, float radius, float damage)
    {
        CannonBall cannonBall =  new CannonBall(CannonEnd().x, CannonEnd().y, radius * ballSizeMultiplier, radius * ballSizeMultiplier, damage, this.world, this);
        cannonBall.velocity.x = (float) (Math.cos(radians) * ballSpeed);
        //cannonBall.velocity.x = (float) ((Math.sqrt(10) * Math.sqrt(29) *
            //  Math.sqrt((Math.tan(cannon.angle)*Math.tan(cannon.angle))+1)) / Math.sqrt(2 * Math.tan(cannon.angle) - (2 * 10 * 2)/29))* -1f;
        cannonBall.velocity.y = (float) (Math.sin(radians) * ballSpeed);
        cannonBall.active = true;
        //cannonBall.body.applyLinearImpulse(cannonBall.velocity, cannonBall.position);
        cannonBall.body.applyForce(cannonBall.velocity, cannonBall.position );
        return cannonBall;
    }


trajPath = new TrajectoryActor(-10f);
        trajPath.setX(playerCannon.CannonEnd().x);
        trajPath.setY(playerCannon.CannonEnd().y);
        trajPath.setWidth(10f);
        trajPath.setHeight(10f);
        stage.addActor(trajPath);

Answer 1

The problem is that your trajectory is built from an analytical form, while your physics engine is using an erroneous numerical approximation. Here are 3 solutions:

1. Make a kinematic projectile: I don't know Box2D, but I'd guess there's a way to make a rigid body "kinematic," which would allow you to specify its position/velocity as a function of time. You would then just use the same logic you're using to draw the trajectory to place the projectile.

2. Draw the trajectory using the same numerical integrator Box2D does: A simpler solution is the inverse of that, where you'd recreate the integration scheme Box2D is using to numerically solve for the drawn trajectory. Assuming Box2D uses symplectic Euler integration (I think all the physics engines do), it'd look something like this, in pseudo-code:

   while(ball.y > 0)
       ball.velocity += g * dt
       ball.position += ball.velocity * dt
       // draw ball

3. Find a closed form solution: The tidiest answer is to find a closed form solution to the Euler integration. I just ran through all the math and come up with:

   def get_pos_at_nth_step(n)
       return x0 + n * ((0.5 * g * dt^2) * (n + 1) + v0 * dt)

   where x0 and v0 are the initial position and velocity, respectively. This allows you to keep your code as it is, but end up with a trajectory equal to what your physics engine will find. Note that the solution won't work if Box2D doesn't use symplectic Euler, but you can take the same approach.