Do you have a formula that you can show here that describes the paths of the two curves? Otherwise, multidimensional Newton-Raphsons method will be good enough if the only thing you want is a numerical value. Normally, the method works as follows:
But if you instead have a function operating on a vector
there is no derivative of f, so the original formula doesn't work. However, there are directional derivatives, and the function does have a gradient. The gradient will point in the direction in which the function will increase the fastest. Therefore, setting
will tell in which direction the function increases. Now, starting with the original formula, there are two things that have to be changed.
1. While becomes the vector , the second term in the subtraction is a scalar. It has to be made a vector too, before the subtraction be performed. Therefore, we will multiply it by the (normalized) vector d, since d is the direction in which the function increases. So
2. There is no . However, there is a directional derivative in the direction (d) we are heading, which should be used instead. Substitution will give us
Now, we can make the realization that , so we will get
Inserting this into the original formula yields