xna - Curved movement between two points

What is a good technique to enable an object to move between to points in a nice curved motion?

The end position could also be in motion, such as the trajectory of a homing missile.

Answer

Assuming you want this to be a frame-by-frame thing (where the target could even be moving), and not a precomputed trajectory, this is actually incredibly simple:

Every frame, compare the velocity-vector of the missile to the vector (position_target - position_missile); that is, the vector pointing from the missile to the target. Every frame, if the two vectors do not have the same direction, then rotate the veloity-vector a little bit towards the other vector, so every frame the missle gets a little bit closer to looking at its target.

You can determine whether to rotate clockwise or counter-clockwise by looking at the sign of the cross-product between the two vectors.

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[Edit] XNA-ish code (I don't have XNA to test this):

//Once a frame:

//Get vector spanning from missile to target
Vector2 vectorToTarget = target.Position - missile.Position;

//Convert to Vector3 to do cross-product
Vector3 vectorToTarget3 = new Vector3(vectorToTarget, 0);
Vector3 missileVelocity3 = new Vector3(missile.Velocity, 0);


//Rotate clockwise/counter-clockwise is determined by sign of cross-product
int crossProductSign = Vector3.Cross(missileVelocity3, vectorToTarget3).Z;

//Positive cross-product means rotate counter-clockwise, negative is clockwise
double rotationAngle = 0;
if(crossProductSign > 0)
    rotationAngle = -0.05;
else if(crossProductSign < 0)
    rotationAngle = 0.05;


//I'm not sure how to do rotation in XNA, but the internets tell me it's something like this:
missile.velocity = Vector2.Transform(missile.velocity, Matrix.CreateRotationZ(rotationAngle))

Note that, because rotationAngle has only three possible values, all possible values of Matrix.CreateRotationZ(rotationAngle) can be cached so you don't have to call it every frame.