The smallest distance between two curves defined by functions

Question

I have two functions that describe two curves in 2D.

p1 = f1(t1)
p2 = f2(t2)

where p1and p2are vectors, t1and t2are scalars with a value from 0.0 to 1.0.

The curves are convex, and the "bellies" are facing each other. They can be rotated and new functions can be defined y = h(x), so that their derivatives on xwill monotonically increase / decrease.

Example:

I am trying to find an efficient algorithm to find the minimum distance between these curves.

A possible approach, I think, could be to define a distance function:

g(t1, t2) = |f1(t1) - f2(t2)|

and then use the generalization of Newton's method to solve the system of equations

0 = ∂g(t1, t2)/∂t1    // partial derivative of g for t1
0 = ∂g(t1, t2)/∂t2    // partial derivative of g for t2

, , , g, .

, ?

Answer 1

, , . , , , . ( " ", ).

, ( 10-20), /.

" -", .

---

2, 0,5 0,125 ( 21, 31 50 ).

( , 58, 120 248 ).

Answer 2

z 0<=x<=1 0<=y<=1. z | error | x - t1, y - t2.

, |f1(t1) - f2(t2)|, z= pow(f1(t1) - f2(t2),2) .

f1(t1) f2(t2), , , ,

fooobar.com/questions/759973/...
fooobar.com/questions/759973/...