C ++查找点集在其他两个点之间的位置

Question

I have three sets of 2d points. What i need to do is to find out where one sits in relation to the other two. Every set has the same points, in the same order. One is 'neutral', one is 'max', and the third is unknown. What I need is to return a single value, between 0 and 1, that illustrates the amount that the unknown set is between the other two.

For example, in the image:

I would somehow get the 'distance' or 'weight' between Set A and Set B, then find out where Set C sits between them. In this example, i would expect a value of around 75%, or 0.75.

I have looked at using point set registration algorithms that return a scale amount to match Set C to Set B, but i am not convinced that this is the best way. What approach would be suitable for this problem? What algorithms should I be searching for?

Answer 1

You could try to solve this with a simple linear interpolation between the two sets. This works if the transition between the sets is indeed nearly linear. If you know that it is something else, you can adapt the interpolation function.

Let us focus on a single point p . We know its coordinates in all sets p_A , p_B , and p_C . Then, we specify that p_C is more or less a linear interpolation between p_A and p_B with parameter t (where t=0 represents set A and t=1 represents set B):

      p_C = (1 - t) * p_A + t * p_B
          = p_A - t * p_A + t * p_B
          = p_A + t * (p_B - p_A)
p_C - p_A = t * (p_B - p_A)

The question now is to find a t that approximately holds for all your points.

We can solve this by stating the problem as a linear least squares problem. Ie we want to minimize the summed residuals (difference between left-hand sides and right-hand sides of the above equation) for all points:

arg min_t Σ_i (pi_C.x - pi_A.x - t * (pi_B.x - pi_A.x))^2
               + (pi_C.y - pi_A.y - t * (pi_B.y - pi_A.y))^2

The optimal t is then:

numX = Σ_i (pi_A.x^2 - pi_A.x * pi_B.x - pi_A.x * pi_C.x + pi_B.x * pi_C.x)
numY = Σ_i (pi_A.y^2 - pi_A.y * pi_B.y - pi_A.y * pi_C.y + pi_B.y * pi_C.y)
denX = Σ_i (pi_A.x^2 - 2 * pi_A.x * pi_B.x + pi_B.x^2)
denY = Σ_i (pi_A.y^2 - 2 * pi_A.y * pi_B.y + pi_B.y^2)
t = (numX + numY) / (denX + denY)

If your points have higher dimension, just add the new dimension with the same pattern.