CODEWITHSUNDEEP
c#geometrylinear-programming
OFRBG
OFRBG

Reputation: 1778

Shortest distance from a point to this curve

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?

Upvotes: 6

Views: 5067

Answers (2)

Ph0en1x
Ph0en1x

Reputation: 10067

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.

Upvotes: 0

zmbq
zmbq

Reputation: 39013

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)

Upvotes: 4

Related Questions