Shortest distance from a point to this curve

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广开言路
广开言路 2021-01-12 20:19

I need to find the distance of multiple points to a curve of the form: f(x) = a^(k^(bx))

My first option was using its derivative, using a line of the f

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  • 2021-01-12 20:39

    The distance between a point (c,d) and your curve is the minimum of the function

    sqrt((c-x)^2 + (d-a^(k^(bx)))^2)
    

    To find its minimum, we can forget about the sqrt and look at the first derivative. Find out where it's 0 (it has to be the minimal distance, as there's no maximum distance). That gives you the x coordinate of the nearest point on the curve. To get the distance you need to calculate the y coordinate, and then calculate the distance to the point (you can just calculate the distance function at that x, it's the same thing).

    Repeat for each of your points.

    The first derivative of the distance function, is, unfortunately, a kind of bitch. Using Wolfram's derivator, the result is hopefully (if I haven't made any copying errors):

    dist(x)/dx = 2(b * lna * lnk * k^(bx) * a^(k^(bx)) * (a^(k^(bx)) - d) - c + x)
    
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  • 2021-01-12 20:42

    To find distance from point to curve it's not a simple task, for that you need to find the global of function enter image description here where f(x) is the function which determine your curve.

    For that goal you could use:
    Simplex method
    Nelder_Mead_method
    gradient_descent

    This methods implemented in many libraries like Solver Foundation, NMath etc.

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