How to write a function of distance between curves?

Question

Mathematics

Hello!
You need to find the smallest distance between two curves using the gradient descent method:
x^2+y^2−2*x+4*y+3 = 0 and x = (cosϕ)^3 − 1, y = 2 + (sinϕ)^3
The crux of the matter is that I cannot compose a function of the distance between points on these lines in order to minimize it later. Can you tell me how to do it? Thanks in advance

Answer 1

There is also a formula for the distance between two points:
Here we substitute x and y from the first and second equations into it, and in the first one we will have to express y through x.
We get the distance function between two arbitrary points of the curves:
ps
If we are talking about real numbers.