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**General Science / what are the complex solutions for x[sup]2[/sup] + y[sup]2[/sup] = 1 ?**

« **on:**11/02/2018 21:54:21 »

The familiar equation for the unit circle centered at the origin of the (x,y) coordinate system is:

x

Using only real numbers for both x and y only yields useful solutions for –1 ≤ x ≤ 1 and –1 ≤ y ≤ 1. But if we allow for either x or y to have nonzero imaginary components (x or y can be imaginary or complex), then the solution is no longer just a unit circle, but an odd 2-dimensional surface in 4-dimensional space (x, a, y, b) such that (x + a×i)

Try as I might, I am having great difficulties trying to visualize this surface. I can generate a few different 3-d slices, like a = 0 for which all of the points have only real components of the x coordinate. This slice contains a unit circle in the x,y plane, and a hyperbola in the x,b plane (x

Which other slices would be useful for me to try to graph to visualize the rest of this shape? (I am tempted to look at x=y or just something like several parallel planes using integer values of a or b.

x

^{2}+ y^{2}= 1Using only real numbers for both x and y only yields useful solutions for –1 ≤ x ≤ 1 and –1 ≤ y ≤ 1. But if we allow for either x or y to have nonzero imaginary components (x or y can be imaginary or complex), then the solution is no longer just a unit circle, but an odd 2-dimensional surface in 4-dimensional space (x, a, y, b) such that (x + a×i)

^{2}+ (y + b×i)^{2}= 1.Try as I might, I am having great difficulties trying to visualize this surface. I can generate a few different 3-d slices, like a = 0 for which all of the points have only real components of the x coordinate. This slice contains a unit circle in the x,y plane, and a hyperbola in the x,b plane (x

^{2}– b^{2}= 1). Similarly for the b = 0 slice we also see the same unit circle and a hyperbola in the y,a plane (y^{2}– a^{2}= 1).Which other slices would be useful for me to try to graph to visualize the rest of this shape? (I am tempted to look at x=y or just something like several parallel planes using integer values of a or b.