Imaginary solutions outside the circle
- Thread starter imaginary
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I am sorry (and am willing to be corrected), but I simply do not see that it makes any sense to try to understand the locus of points in 4 space described by an equation by looking at two very special planes and wondering why the locus in these two different planes shows different properties. Would it not make much more sense to start by considering what the locus looks like in an arbitrary plane and then seeing how that general locus translates into a specific locus in a specific plane?
Dr.Peterson
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As I said in my first post, "I understood that there wasn't much significance to it." But it can still be a little interesting to explore this space curve (not really two planes, but a 3D slice, as it turns out), if only to find it comes to a dead end. It may be that no one else has ever looked at these particular details, precisely because they are unimportant; yet one may learn something from such exploration. You might say I'm just helping a little kid play, even though no useful work is accomplished, because I remember playing in similar ways.
But, @imaginary, are you aware that we are really looking at a tiny slice of a 4-dimensional object, and not seeing the whole picture? Or that the subject of "complex variables", which I mentioned in passing, is big enough to require a whole course? There you learn about more useful ways to think about such things.
My apologies to any who find my question irksome and unclear.
And yes the motivation is just playful. Was thinking of moving at relativistic speeds in space, and the effects on observing spherical planets and elliptical orbits. Then I wanted to picture the imaginary hyperboloids around length-contracted elliptoids extended through space. For a circle, I could see the hyperbolas were oriented to the choice of x and y axes. For ellipses, I wrongly assumed they would orient with the vertex.
I have learned about and used complex numbers. But these details are so low-level I didn't think I needed to correct the picture in my head, and assumed it must be clearly laid out somewhere. Having asked others, I at best got a recitation of textbook treatments of complex numbers, but never an actual answer. @DrPeterson your willingness and patience to walk through the details has made this a delight!
I have wondered the same, but glad you said it
And yes the motivation is just playful. Was thinking of moving at relativistic speeds in space, and the effects on observing spherical planets and elliptical orbits. Then I wanted to picture the imaginary hyperboloids around length-contracted elliptoids extended through space. For a circle, I could see the hyperbolas were oriented to the choice of x and y axes. For ellipses, I wrongly assumed they would orient with the vertex.Yes it is a 4D object, which is why my intuition wanted to treat it like a rigid object that I could grab and move around with my imaginary hands. But the ellipse and hyperbolas are not a rigid object like a 4D hypercube, which does not change when you rotate it. Now I see why the imaginary hyperbolas are not rigidly fixed to the ellipse. They are artifacts of performing directed (choice of x-y axes) projections of the Cartesian form of the equation (ie they do not arise from polar form, I think). I can also picture how the hyperbolas move around in complex space when rotating the ellipse. So not a dead end for me, but very satisfying result.
I have learned about and used complex numbers. But these details are so low-level I didn't think I needed to correct the picture in my head, and assumed it must be clearly laid out somewhere. Having asked others, I at best got a recitation of textbook treatments of complex numbers, but never an actual answer. @DrPeterson your willingness and patience to walk through the details has made this a delight!
I have no objection whatever to play. I enjoy it.
It seemed to me, however, that imaginary did not view what he was doing as play. If we are thinking seriously about graphing the locus of points generated by the equation equating the sum of two squared complex numbers that equal 1, isn't it most effective to start with that general equation
[MATH](a + bi)^2 + (c + di)^2 = 1 \implies a^2 + c^2 - (b^2 + d^2) + 2i(ab + cd) = 1.[/MATH]
The first conclusion we reach is that [MATH]ab + cd = 0 \implies ab = - cd.[/MATH]
In other words, we can think about this as
[MATH]a^2 + c^2 - (b^2 + d^2) = 1.[/MATH]
We are back in the domain of real numbers, but in 4 dimensions rather than 2. Moreover, we can now see why there is a circular aspect and a hyperbolic aspect to the locus. It is not a matter of transforming a circle into a hyperbola or vice versa; they are different perspectives on an object impossible to visualize as a whole. Furthermore, we can create graphs in two dimensions from any perspective we want.
Setting |b| and |d| equal to |p|, the locus simplifies into a circle described by
a^2 + c^2 = 1 + 2p^2. From this perspective, we can extrapolate as the absolute value of p increases to an infinite number of circles with expanding radii above and below the plane described by b = 0 and d = 0.
Setting |a| and |c| equal to |p| such that |p| > 1/sqrt(2), the locus simplifies into a circle described by
b^2 + d^2 = 2p^2 - 1. If |p|= 1/sqrt(2), the locus reduces to a point. If 0 < |p| < 1/sqrt(2), the locus disappears. We can extrapolate to two sets of an infinite number of circles shrinking to a point as |p| approaches 1/sqrt{2} with a void if |p| < 1/sqrt{2}.
Setting |b| and |c| equal to |p|, the locus simplifies into a hyperbola described by a^2 - d^2 = 1. We can extrapolate to an infinite set of congruent hyperbolas. The same kind of thing happens by setting |a| and |d| equal to |p|
Now we could define other planes and look at the locus in them. Maybe they would be interesting.
It seemed to me, however, that imaginary did not view what he was doing as play. If we are thinking seriously about graphing the locus of points generated by the equation equating the sum of two squared complex numbers that equal 1, isn't it most effective to start with that general equation
[MATH](a + bi)^2 + (c + di)^2 = 1 \implies a^2 + c^2 - (b^2 + d^2) + 2i(ab + cd) = 1.[/MATH]
The first conclusion we reach is that [MATH]ab + cd = 0 \implies ab = - cd.[/MATH]
In other words, we can think about this as
[MATH]a^2 + c^2 - (b^2 + d^2) = 1.[/MATH]
We are back in the domain of real numbers, but in 4 dimensions rather than 2. Moreover, we can now see why there is a circular aspect and a hyperbolic aspect to the locus. It is not a matter of transforming a circle into a hyperbola or vice versa; they are different perspectives on an object impossible to visualize as a whole. Furthermore, we can create graphs in two dimensions from any perspective we want.
Setting |b| and |d| equal to |p|, the locus simplifies into a circle described by
a^2 + c^2 = 1 + 2p^2. From this perspective, we can extrapolate as the absolute value of p increases to an infinite number of circles with expanding radii above and below the plane described by b = 0 and d = 0.
Setting |a| and |c| equal to |p| such that |p| > 1/sqrt(2), the locus simplifies into a circle described by
b^2 + d^2 = 2p^2 - 1. If |p|= 1/sqrt(2), the locus reduces to a point. If 0 < |p| < 1/sqrt(2), the locus disappears. We can extrapolate to two sets of an infinite number of circles shrinking to a point as |p| approaches 1/sqrt{2} with a void if |p| < 1/sqrt{2}.
Setting |b| and |c| equal to |p|, the locus simplifies into a hyperbola described by a^2 - d^2 = 1. We can extrapolate to an infinite set of congruent hyperbolas. The same kind of thing happens by setting |a| and |d| equal to |p|
Now we could define other planes and look at the locus in them. Maybe they would be interesting.
@JeffM that looks interesting and powerful, thanks! I see your point about the circles and orthogonal hyperbolas.
What about going from circle to ellipse? Do I just divide a and c by some constants a' and c'? Then rotating would require some ac cross-term? I'm a little stuck on how to proceed. I want to see how the hyperbolas move with respect to the ellipse as the ellipse is rotated around (a,c)=(0,0).
What about going from circle to ellipse? Do I just divide a and c by some constants a' and c'? Then rotating would require some ac cross-term? I'm a little stuck on how to proceed. I want to see how the hyperbolas move with respect to the ellipse as the ellipse is rotated around (a,c)=(0,0).
@imaginary
Obviously the "circle-hyperbola" is a limiting case of the "ellipse-hyperbola," but I have not begun to consider what complexities are introduced.
I shall think about this, but I have a lot on my plate. At 75, my wife and I are caring for my 5 month old grandson every weekday from 9 until 6; at our age, a new baby is a huge drain on our energy. Plus I have other responsibilities and like to play too. So I shall probably respond slowly and in pieces. Perhaps Dr. Peterson will chime in too.