Geometric interpretation of a complex solution

Question

A straight line in 2-D $x+y=3$ and a circle in 2-D $x^2+y^2=4$ do not have a point of intersection in the plane containing the two. But on solving these equations analytically, on gets 2 complex solutions $x=1.5+0.5i,y=1.5-0.5i$ and $x=1.5-0.5i,y=1.5+0.5i$. I was trying to interpret this geometrically and since it did not make sense on a plane, I moved to 3-D. I defined a complex space with the $z$-axis being the "imaginary" axis and retained $x$ & $y$ as such. But though I am able to plot these 4 points and visualize them, I am not able to relate them to the plane. Any help in understanding the solution geometrically either through this approach or any other approach would be appreciated…

Answer 1

3D is only oflimited use, since both your original dimensions are complex, so you'd have to move to 4D do accurately embed this. And I for one have some trouble imagining 4D, so I'm not sure there is anything to be gained from this endeavor.

You can simplify the situation a bit by considering a line like $x=3$, i.e. one which is parallel to one of the axes of your original coordinate system. Then you can concentrate on the part of $\mathbb C^2$ where $x\in\mathbb R$, and interpret that in three dimensions. So we are essentially speaking about points of the form $(x, y + iz)$.

So what is the line in this setup? It's a plane, namely the plane $x=3$. And what is the circle? Let's compute that.

$$x^2+(y+iz)^2 = x^2 + y^2 - z^2 + 2yzi = 4 \\ yz=0 \quad\text{and}\quad x^2+y^2-z^2 = 4$$

The first of these conditions is the union of the $y=0$ and the $z=0$ planes. For $z=0$, the second equation is our well-known circle $x^2+y^2=4$ in the original $xy$ plane. For $y=0$ this is the hyperbola $x^2+z^2=4$. So you can imagine the 3d image of your circle as the union of a circle in one plane and a hyperbola in a different plane, where both of these planes are orthogonal to one another.

As you move the line from e.g. $x=1$ to $x=3$ in the 2D image, you are moving that plane in 3d from the situation where it intersects the circle to the situation where it intersects the hyperbola. In between there is the situation where there is only a single point of intersection, where your line is a tangent to the circle.

Keep in mind that this whole image depends a lot on the orientation of the line. You can try to imagine that by rotating the orientation of your line, you'd have to rotate the plane of the hyperbola as well, because hyperbola and circle still have to touch in the tangency situation. So you get a glimpse at the 4d structure after all: this rotation of the line would be your fourth dimension.