Shortest distance from a point to this curve

Question

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 form with the inverse of the derivative, giving it coordinates of the Point and intersecting it with the original curve. Finally, we calculate the distance between points with simple geometry.

That's the mathematical process that I usually follow. I need to save time (since I'm doing a genetic algorithms program) so I need an efficient way to do this. Ideas?

Answer 1

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)