Hi,
My first post to these forums. I hope this is a good place for this post.
I have a problem with complex numbers, but it can be framed to avoid concerning ourselves with complex numbers, and I shall do so. Also, it's mostly an algebra problem, but also involves a bit of geometry. Let's say I've got two variables, alpha (\(\displaystyle \alpha\)) and beta (\(\displaystyle \beta\)), such that:
\(\displaystyle \alpha = A_\alpha + B_\alpha\)
and
\(\displaystyle \beta = A_\beta + B_\beta\)
Now, there are additional relationships between alpha and beta (which I will outline shortly), but it can be shown that \(\displaystyle B_\alpha\) can always be forced to zero. That's my problem. I need help working out the algebra such that \(\displaystyle B_\alpha = 0\), where all the other equalities stay unchanged.
Okay, first, no matter how we change \(\displaystyle B_\alpha\), or, for that matter, \(\displaystyle A_\alpha\), \(\displaystyle A_\beta\), or \(\displaystyle B_\beta\), the values for both \(\displaystyle \alpha
\) and \(\displaystyle \beta\) must remain unchanged.
The primary relationship between \(\displaystyle \alpha\) and \(\displaystyle \beta\) is:
\(\displaystyle A_\alpha^2 + B_\alpha^2 + A_\beta^2 + B_\beta^2 = 1\)
I believe that this relationship provides enough to get most of it done. However, let's not forget that any of \(\displaystyle A_\alpha\), \(\displaystyle B_\alpha\), \(\displaystyle A_\beta\), or \(\displaystyle B_\beta\) can be negative. Therefore, the above relationship doesn't sort out our signs for us. That's where a bit of geometry comes in. To sort out the geometry, I shall introduce theta (\(\displaystyle \theta\)) and phi (\(\displaystyle \varphi\)). These are polar coordinates (in radians) such that:
\(\displaystyle 0 \le \theta \le \pi\)
and
\(\displaystyle 0 \le \varphi < 2\pi\)
It may be useful to think of these in terms of a unit sphere (a sphere in 3D space with radius = 1, and centered on the 0,0,0 origin). Then, we can imagine a unit vector (a vector anchored on the 0,0,0 origin, that's one unit long, and pointing out in any direction, with the vector's arrow just touching the surface of our unit sphere):
\(\displaystyle \theta\) is an angle of rotation about the Y axis. It can rotate the vector from straight up (0) to straight down (\(\displaystyle \pi\)). \(\displaystyle \varphi\) is an angle of rotation about the Z axis. Looking straight down from the top of the sphere, it rotates the vector around in the sphere. \(\displaystyle \theta\) and \(\displaystyle \varphi\) provide unique positions on any/all points of the surface of the sphere (with the exceptions of when \(\displaystyle \theta = 0\) or \(\displaystyle \theta = \pi\) at which times, the value of \(\displaystyle \varphi\) doesn't matter (and is just twisting rather than rotating the vector).
Okay, here are the relationships between our \(\displaystyle \alpha\) and our \(\displaystyle \beta\) with our newly introduced \(\displaystyle \theta\) and \(\displaystyle \varphi\). But this is where my problem comes in. These relationships assume that \(\displaystyle B_\alpha\) is already zero. So, assuming that \(\displaystyle B_\alpha\) is zero, here are the relationships:
\(\displaystyle \theta = 2 \times ArcCos(A_\alpha)\)
and
\(\displaystyle \varphi = ArcCos(A_\beta / Sin(\theta / 2))\), recognizing that \(\displaystyle \theta\) must be calculated first.
... or alternatively ...
\(\displaystyle \varphi = ArcSin(B_\beta / Sin(\theta / 2))\), still recognizing that \(\displaystyle \theta\) must be calculated first.
However, there's still one more consideration for calculating \(\displaystyle \varphi\)...
If \(\displaystyle B_\beta\) is less than zero, we must subtract it from \(\displaystyle 2\pi\). So, we might add something like:
IF \(\displaystyle B_\beta < 0\) THEN \(\displaystyle \varphi = 2\pi - \varphi\)
(and this needs to be done regardless of which of the above methods is used to initially calculate \(\displaystyle \varphi\)).
Still assuming that \(\displaystyle B_\alpha\) is zero, if we have \(\displaystyle \theta\) and \(\displaystyle \varphi\), we can solve for \(\displaystyle A_\alpha\), \(\displaystyle A_\beta\), and \(\displaystyle B_\beta\) as follows:
\(\displaystyle A_\alpha = Cos(\theta / 2)\)
\(\displaystyle A_\beta = Cos(\varphi) \times Sin(\theta / 2)\)
\(\displaystyle B_\beta = Sin(\varphi) \times Sin(\theta / 2)\)
Recognizing that the first two quadrants of the XY plane (i.e., rotations around the Z axis, i.e., \(\displaystyle \varphi\)) are positive for Y, with quadrants #3 and #4 being negative for Y, this should provide the information necessary to sort out the signs of the final \(\displaystyle A_\alpha\), \(\displaystyle A_\beta\), and \(\displaystyle B_\beta\) based on the signs of the original \(\displaystyle A_\alpha\), \(\displaystyle B_\alpha\), \(\displaystyle A_\beta\), and \(\displaystyle B_\beta\). It may also be useful to envision the XZ plane (with rotations around the Y axis). In this plane, the first and second quadrants have negative Y values (in the 3D space). Given this information, I can tell you one more piece of information. If \(\displaystyle B_\beta\) starts out as zero, and \(\displaystyle B_\alpha\) starts out as positive, then, once \(\displaystyle B_\alpha\) is forced to zero, \(\displaystyle B_\beta\) will be negative (but not necessarily the same value). And if \(\displaystyle B_\beta\) starts out as zero, and \(\displaystyle B_\alpha\) starts out as negative, then, once \(\displaystyle B_\alpha\) is forced to zero, \(\displaystyle B_\beta\) will be positive (but not necessarily the same value). However, in the beginning, neither \(\displaystyle B_\alpha\) nor \(\displaystyle B_\beta\) may be zero. And again, that's the problem. If \(\displaystyle B_\alpha\) isn't zero, I need to know how to change \(\displaystyle A_\alpha\), \(\displaystyle A_\beta\), and \(\displaystyle B_\beta\) such that \(\displaystyle B_\alpha\) becomes zero.
Just to restate the problem:
Final \(\displaystyle \alpha\) must equal original \(\displaystyle \alpha\).
Final \(\displaystyle \beta\) must equal original \(\displaystyle \beta\).
In all cases, \(\displaystyle A_\alpha^2 + B_\alpha^2 + A_\beta^2 + B_\beta^2 = 1\).
The task is, if an input (original) \(\displaystyle B_\alpha\) isn't zero, how do we change \(\displaystyle A_\alpha\), \(\displaystyle A_\beta\), and \(\displaystyle B_\beta\) such that \(\displaystyle B_\alpha\) is zero, without changing our equality criteria?
Thanks in Advance for All Helping Posts,
Elroy
My first post to these forums. I hope this is a good place for this post.
I have a problem with complex numbers, but it can be framed to avoid concerning ourselves with complex numbers, and I shall do so. Also, it's mostly an algebra problem, but also involves a bit of geometry. Let's say I've got two variables, alpha (\(\displaystyle \alpha\)) and beta (\(\displaystyle \beta\)), such that:
\(\displaystyle \alpha = A_\alpha + B_\alpha\)
and
\(\displaystyle \beta = A_\beta + B_\beta\)
Now, there are additional relationships between alpha and beta (which I will outline shortly), but it can be shown that \(\displaystyle B_\alpha\) can always be forced to zero. That's my problem. I need help working out the algebra such that \(\displaystyle B_\alpha = 0\), where all the other equalities stay unchanged.
Okay, first, no matter how we change \(\displaystyle B_\alpha\), or, for that matter, \(\displaystyle A_\alpha\), \(\displaystyle A_\beta\), or \(\displaystyle B_\beta\), the values for both \(\displaystyle \alpha
\) and \(\displaystyle \beta\) must remain unchanged.
The primary relationship between \(\displaystyle \alpha\) and \(\displaystyle \beta\) is:
\(\displaystyle A_\alpha^2 + B_\alpha^2 + A_\beta^2 + B_\beta^2 = 1\)
I believe that this relationship provides enough to get most of it done. However, let's not forget that any of \(\displaystyle A_\alpha\), \(\displaystyle B_\alpha\), \(\displaystyle A_\beta\), or \(\displaystyle B_\beta\) can be negative. Therefore, the above relationship doesn't sort out our signs for us. That's where a bit of geometry comes in. To sort out the geometry, I shall introduce theta (\(\displaystyle \theta\)) and phi (\(\displaystyle \varphi\)). These are polar coordinates (in radians) such that:
\(\displaystyle 0 \le \theta \le \pi\)
and
\(\displaystyle 0 \le \varphi < 2\pi\)
It may be useful to think of these in terms of a unit sphere (a sphere in 3D space with radius = 1, and centered on the 0,0,0 origin). Then, we can imagine a unit vector (a vector anchored on the 0,0,0 origin, that's one unit long, and pointing out in any direction, with the vector's arrow just touching the surface of our unit sphere):
\(\displaystyle \theta\) is an angle of rotation about the Y axis. It can rotate the vector from straight up (0) to straight down (\(\displaystyle \pi\)). \(\displaystyle \varphi\) is an angle of rotation about the Z axis. Looking straight down from the top of the sphere, it rotates the vector around in the sphere. \(\displaystyle \theta\) and \(\displaystyle \varphi\) provide unique positions on any/all points of the surface of the sphere (with the exceptions of when \(\displaystyle \theta = 0\) or \(\displaystyle \theta = \pi\) at which times, the value of \(\displaystyle \varphi\) doesn't matter (and is just twisting rather than rotating the vector).
Okay, here are the relationships between our \(\displaystyle \alpha\) and our \(\displaystyle \beta\) with our newly introduced \(\displaystyle \theta\) and \(\displaystyle \varphi\). But this is where my problem comes in. These relationships assume that \(\displaystyle B_\alpha\) is already zero. So, assuming that \(\displaystyle B_\alpha\) is zero, here are the relationships:
\(\displaystyle \theta = 2 \times ArcCos(A_\alpha)\)
and
\(\displaystyle \varphi = ArcCos(A_\beta / Sin(\theta / 2))\), recognizing that \(\displaystyle \theta\) must be calculated first.
... or alternatively ...
\(\displaystyle \varphi = ArcSin(B_\beta / Sin(\theta / 2))\), still recognizing that \(\displaystyle \theta\) must be calculated first.
However, there's still one more consideration for calculating \(\displaystyle \varphi\)...
If \(\displaystyle B_\beta\) is less than zero, we must subtract it from \(\displaystyle 2\pi\). So, we might add something like:
IF \(\displaystyle B_\beta < 0\) THEN \(\displaystyle \varphi = 2\pi - \varphi\)
(and this needs to be done regardless of which of the above methods is used to initially calculate \(\displaystyle \varphi\)).
Still assuming that \(\displaystyle B_\alpha\) is zero, if we have \(\displaystyle \theta\) and \(\displaystyle \varphi\), we can solve for \(\displaystyle A_\alpha\), \(\displaystyle A_\beta\), and \(\displaystyle B_\beta\) as follows:
\(\displaystyle A_\alpha = Cos(\theta / 2)\)
\(\displaystyle A_\beta = Cos(\varphi) \times Sin(\theta / 2)\)
\(\displaystyle B_\beta = Sin(\varphi) \times Sin(\theta / 2)\)
Recognizing that the first two quadrants of the XY plane (i.e., rotations around the Z axis, i.e., \(\displaystyle \varphi\)) are positive for Y, with quadrants #3 and #4 being negative for Y, this should provide the information necessary to sort out the signs of the final \(\displaystyle A_\alpha\), \(\displaystyle A_\beta\), and \(\displaystyle B_\beta\) based on the signs of the original \(\displaystyle A_\alpha\), \(\displaystyle B_\alpha\), \(\displaystyle A_\beta\), and \(\displaystyle B_\beta\). It may also be useful to envision the XZ plane (with rotations around the Y axis). In this plane, the first and second quadrants have negative Y values (in the 3D space). Given this information, I can tell you one more piece of information. If \(\displaystyle B_\beta\) starts out as zero, and \(\displaystyle B_\alpha\) starts out as positive, then, once \(\displaystyle B_\alpha\) is forced to zero, \(\displaystyle B_\beta\) will be negative (but not necessarily the same value). And if \(\displaystyle B_\beta\) starts out as zero, and \(\displaystyle B_\alpha\) starts out as negative, then, once \(\displaystyle B_\alpha\) is forced to zero, \(\displaystyle B_\beta\) will be positive (but not necessarily the same value). However, in the beginning, neither \(\displaystyle B_\alpha\) nor \(\displaystyle B_\beta\) may be zero. And again, that's the problem. If \(\displaystyle B_\alpha\) isn't zero, I need to know how to change \(\displaystyle A_\alpha\), \(\displaystyle A_\beta\), and \(\displaystyle B_\beta\) such that \(\displaystyle B_\alpha\) becomes zero.
Just to restate the problem:
Final \(\displaystyle \alpha\) must equal original \(\displaystyle \alpha\).
Final \(\displaystyle \beta\) must equal original \(\displaystyle \beta\).
In all cases, \(\displaystyle A_\alpha^2 + B_\alpha^2 + A_\beta^2 + B_\beta^2 = 1\).
The task is, if an input (original) \(\displaystyle B_\alpha\) isn't zero, how do we change \(\displaystyle A_\alpha\), \(\displaystyle A_\beta\), and \(\displaystyle B_\beta\) such that \(\displaystyle B_\alpha\) is zero, without changing our equality criteria?
Thanks in Advance for All Helping Posts,
Elroy