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In some intersection problems, like the 2D circle-circle intersection, there are two possible solutions that arise from a quadratic equation. If the circles do not intersect on the cartesian plane, the intersection points become complex numbers, because the discriminant of the quadratic is negative.

Can these complex intersection points be visualized spatially? And what is the significance of the magnitudes and phases of these complex numbers?

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If the circles do not intersect on the cartesian plane, the intersection points become complex numbers

To be precise: the coordinates of the points of intersection become complex numbers. More precisely, they will form a conjugate pair of complex points, i.e. if $(a+ib,c+id)$ is one such solution, then you know the other solution to be $(a-ib,c-id)$ (if your circles were real).

Can these complex intersection points be visualized spatially?

Connecting these points you get the radical axis of the circle, just as you get for real points of intersection. You can imagine parametrizing that radical axis with a single complex parameter. To make this more precise, you could say that the line connecting the centers will intersect the radical axis at a point $t=0$, and from there a unit distance step along the radical axis will take you to a point $t=1$. You have to arbitrarily choose a direction for the axis here. You could come up with a formula to compute the point on the axis for every $t\in\mathbb C$.

Now you could draw a 2d diagram of that radical axis alone. You'd have the real and imaginary component of $t$ as two coordinates. A complex pair of points would be mirror images with respect to the real axis. Furthermore, for reasons of symmetry, the real component of your points of intersection would have to be zero in this coordinate system. In other words, the real coordinate of the solutions in the original coordinate system is the point where the radical axis meets the line joining the circle centers.

Visualizing two points on the imaginary axis all by themselves is not that interesting. Things do become more interesting if your setup is dynamic in some way. For example in this slide of a talk on a project I'm involved with, one can see the $x$ coordinates of the points you get by intersecting a circle with a horizontal line which you can move around. As you move the line from an intersecting position to a non-intersecting one, the points will come closer together along the real axis, and after meeting at the origin they will drift apart along the imaginary axis. (The slide in question also illustrates how to avoid the singular situation, so the points there won't move through the center but curve around it instead.) The picture of two circles intersecting would look pretty much the same in the parameter space of the radical axis I described above.

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  • $\begingroup$ impressive! How about geometry meaning for Sphere-Sphere Intersection? mathworld.wolfram.com/Sphere-SphereIntersection.html $\endgroup$
    – Black Mild
    Jan 31, 2021 at 16:12
  • $\begingroup$ @BlackMild: Sphere-sphere will typically intersect in a circle which is already 2d. Considering these in complex coordinates makes things hard to imagine. You can consider its radius alone, and how that becomes imaginary when the spheres no longer touch. But since negative radius is uncommon, the number of solutions here is somewhat arguable. You might instead consider $r^2$ , the square of the radius of the circle of intersection. And see that becomes negative. Which makes the whole thing entirely real, so you can even visualize in 1d. $\endgroup$
    – MvG
    Feb 1, 2021 at 8:56

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