Is there any known relationship of the complex root of two non intersecting curve eqations with the minimal distance between the two curve equation?

Question

Suppose the curve lines do intersect in every possible ways. Then all the roots will be purely real and the minimal distance between them will be zero. Now suppose the curve lines don't intersect at all. Then the solution of the two curve lines intersection point will be complex with non zero imaginary part and the distance in between them will be non zero too.

For example lets take two simple curves.

Example 1

1. y = 5
2. y = $(x-4)^2$ + 7

the distance between the two curves in this case is 2. and the imaginary part of the complex root of the two equations is $\sqrt{2}$. You can change the values of constant but as long as they don't intersect the distance will be square of the imaginary part of the root. When they do intersect the imaginary part of the root will become zero.
There seems to be a relationship between the two quantities, (i) the physical minimal distance btween the two curve in real space and (ii) The imaginary part of the complex solution of the two equation defining them.

But the relationship is not so easy. For the above example it was easy, but now lets take another example.

Example 2

1. y = 5
2. $(y-8)^2$ + $(x-4)^2$ = 1 The physical minimal distance between these two curves in real space is 2. But the solution of the two equations are $4 \pm i 2\sqrt{2}$ Some relationship between two of them is still there but its not so simple like the 1st example.

So my question is if whats the realtion ship between the two quantities (i)the physical minimal distance between the two cuves in real space and (ii)the imaginary part of the complex root? How is the relationship defined?

Please let me know If I need to add more text. Any reference or explanation will be highly appreciated.

Answer 1

Parabolas

If you have a parabola of the form $y=(x-h)^2+k$ and a non-intersecting line $y=c$ parallel to its directrix, then the distance between them is simply $k-c$. However, the complex intersection point has $x$ coordinate equal to a solution to $c=(x-h)^2+k$. The solutions are $h\pm\sqrt{c-k}=h\pm i\sqrt{k-c}$, so the relationship to the distance is clear.

Circles

If you have a circle of the form $(x-h)^2+(y-k)^2=r^2$ and a non-intersecting line $y=c$ (let's assume below the circle), then the distance between them is the difference between the $y$ coordinate of the lowest point of the circle and $c$: $(k-r)-c$.

However, the complex points of intersection have $x$-coordinates that are solutions to $(x-h)^2+(c-k)^2=r^2$: $x=h\pm i\sqrt{k-r-c}\sqrt{k+r-c}$. In other words, both the shortest distance between the line and the circle ($k-r-c$) and the longest vertical distance ($k+r-c$) appear here. The shortest distance isn't enough information; you also need the radius $r$.

Ellipses

Ellipses are similar to circles, but with a bit more complication. If you have an ellipse of the form $(x-h)^2/a^2+(y-k)^2/b^2=1$ and a non-intersecting line $y=c$ parallel to an axis (let's assume below the ellipse), then the distance between them is the difference between the $y$ coordinate of the lowest point of the ellipse and $c$: $(k-b)-c$.

However, the complex points of intersection have $x$-coordinates that are solutions to $(x-h)^2/a^2+(c-k)^2/b^2=1$: $x=h\pm i\frac{a}{b}\sqrt{k-b-c}\sqrt{k+b-c}$. In other words, both the shortest distance between the line and the ellipse ($k-b-c$) and the longest vertical distance ($k+b-c$) appear here. The shortest distance isn't enough information; you also need the semiaxes $a$ and $b$.

Hyperbolas

If you have a vertical hyperbola of the form $(y-k)^2/a^2-(x-h)^2/b^2=1$ and a non-intersecting line $y=c$ perpendicular to the transverse axis (let's assume closer to the higher vertex), then the distance between them is the difference between the $y$ coordinate of the lowest point of upper half and $c$: $(k+a)-c$ (I'm assuming this is less than $c-(k-a)$.)

However, the complex points of intersection have $x$-coordinates that are solutions to $(c-k)^2/a^2-(x-h)^2/b^2=1$: $x=h\pm i\frac{b}{a}\sqrt{(k+a)-c}\sqrt{c-(k-a)}$. In other words, the distances to both vertices (not just the closest one) appear here. The shortest distance isn't enough information; you also need the semiaxes (or whatever you call them for a hyperbola) $a$ and $b$.

Answer 2

Partial answer.

By variational calculus if we find minimum distance between non-intersecting curves $ f(x,y)=0, g(x,y)=0 $ to be positive then the roots are complex.

The simplest example perhaps is like $ y=0, y= x^2+1, x_{min}= \pm i . $

So partial derivatives of $(f-g)$ on $(x ,y) $ determine the complex roots.