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Question from Thomas:

Why is this 3d plane flat?
if m ≠ n
arcsin(sin(m*arctan(x/y))), arcsin(cos(n*arctan(x/y))), Pi-arcsin(sin(m*arctan(x/y)))-arcsin(cos(n*arctan(x/y)))
This is using radians.
These are angles of a triangle, when m and n are the same it is a right triangle. All three inputs are not equal to each other. The last input is pi minus the first two inputs, which makes a complete triangle.
I was kind of hoping to find more lissajous curves. It doesn't make sense to me that this is a flat plane, a diagonal one at that.

I'm not a student or anything. I'm just wasting time.

Nonsense! Doing mathematics for fun is never a waste of time! That said, for the answer to your question you must know that a plane is represented using coordinates by a "linear equation." What this means is that for any plane, there exists four constants a, b, c, d for which the points of space lie on the plane exactly when its coordinates x, y, and z satisfy the equation

\[ax + by + cz = d.\]

Your points satisfy $x + y + z = \pi.$ (That is, $a = b = c = 1, d = \pi.$)

Chris

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