如何从位置和地标获取（X，Y）向量

Question

I currently have an agent in a map, whose position is known as myPos=(myX,myY) , but whose orientation myOri=(oriX,oriY) is unknown. I also see a landmark at position lm=(lmX,lmY) and I have both Cartesian and polar coordinates from my point of view to the landmark, as relLM=(relX,relY) and polLM=(r,theta) , respectively.

My goal is to find how my orientation vector is related with the X and Y axis, as XX=(xX, xY) and YY=(yX, yY) . Assume for the following examples that X grows to the right and Y grows upwards, and that an agent with 0 rotation follows the X axis (so an agent looking right has XX=(1,0) and YY=(0,1) . This follows from the intuition where 0 angle rotation is on the X axis, PI/2 rotation is on the Y, PI is on -X, 3PI/2 is on -Y and 2PI is X.

Example) If myOri=(1,1) (agent is facing top right), then XX=(1, -1) (since the X axis is top right to him) and YY=(1, 1) (the Y axis is top left). In the picture below, X and Y are shown in red and green. My agent is in blue and the landmark in pink. Hence, our initial data are myPos=(0,-2) , lm=(0,-1) , relLM=(~0.7,~0.7) .

By knowing myPos and lmPos , as well as relLM , this should be possible. However, I'm having trouble finding the appropriate vectors. What is the correct algorithm?

bool someFunction(Landmark *lm, Vector2f myPos, Vector2f *xx, Vector2f *yy){
    // Vector from agent to landmark
    Vector2f agentToLandmark(lm->position.x - myPos.x,
            lm->position.y - myPos.y);

    // Vector from agent to landmark, regarding agent's orientation
    Vector2f agentFacingLandmark = lm->relPosition.toCartesian();        

    // Set the XX and YY values
    // how?
}

My problem is actually in 3D, but using 2D makes the problem easier to explain.

Answer 1

Finding myOri

Since relLM is lm relative to myOri , lm + relLM must be in myPos + µ * myOri . Thus lm + relLM - myPos = myOri * µ . Since µ > 0 must be given in this case, and myOri only needs to indicate a direction, it's sufficient to choose an arbitrary µ > 0 .

Finding xx and yy

I think your definition of xx is simply a vector representing the x-axis from the POV of the agent. And same for yy and the y-axis. This can easily be achieved. The angle between myOri and the x-axis is equal to the angle between the x-axis and xx , thus simply mirror myOri at the x-axis and you got xx . So xx = (myOri.x , myOri.y * (-1)) . The angle between myOri and the y-axis is equal to the angle between myOri and yy , so yy = myOri .

Note that this is only a guess on what you mean.
Might be that I've misunderstood something. Just notify me if that's the case.