WDC, I totally agree. That section was just a "place holder" for a better argument. I offer no excuse, the small section expresses an idea very loosely, as if they are just notes to myself.
You will probably ask, how did that get mixed with your other arguments? Again I give no excuse. I deal with many issues all on my own free time and sometimes informal notes act as "place holders" within a more formal argument.
I also generally issue a disclaimer: Check all work yourself (as you have done)
Anyway, here is a better argument:
Here we see the fixed relations which each point on the rigid body must undergo over the first 3 degrees as seen from the Sauret projection. Someone with better graphics arts skills than me (like femr) can actually show the exact relations for each point over a continuously changing angle from 0 to 3 degrees. It can be seen in a totally graphic way with no algebra.
A better argument for movement over the first 3 degrees would be based on physics, for example...
bold means vector...w means "omega" (can't do greek letters on the keyboard), the conventional term for angular velocity.
We know that the velocity of each point on a rigid body undoergoing rotation about an axis is
v = wxr
where r is the position vector from the axis of rotation to any point you wish. (w is a vector parallel with the axis of rotation)
We know the trajectory of each point for a rigid body undergoing rotation about an axis with certainty. It's most mathematically precise form (I believe) is in the form of a vector cross product.
Why didn't I express it this way before? Because, quite honestly, I didn't think the large majority of the readers would know what a freakin' cross product is.
You are a much better proof-reader and I have no doubt you know what a vector cross product is. I know it is not advanced stuff, but that is the way it is...really.
>>>>>>>>>>>>>
Anyway, we know there is a fixed relation between the movement of points in the case of rigid rotation. I am showing with mathematical certainty that that these fixed relations were not held.
Why would I care? I am not "anal" about deformity vs rigidity. I am more interested in how it deforms (and it is not what the NIST told you). It deforms concavely along the roof.
Back to your criticism, you are correct that that small part of the argument was expressed in a way that is sloppy and informal, but after I reformulate the argument, I will still be correct.
Anyone can think of what the v=wxr fixed relations means for pure rotational motion. (Think about it: I am trying to express these ideas to an audience that may not know basic vector operations. Vectors are the natural language of rigid rotation)
Anyone can see they are not obeyed. Anyone can measure the type of deformation from drop curves.
I'll replace that part with a much better argument in the next few days. Again, when I express it better, I will still win the argument. Deformity is a certainty. You can see it in the graphs and the visuals. Ain't no doubt about it.
And I also wrote "C" incorrectly in the link. It should read something like:
for b(s)-r(s)... C= sqrt[(b-r)*(b-r)]
and the same for a(s)-b(s).
You will probably ask, how did that get mixed with your other arguments? Again I give no excuse. I deal with many issues all on my own free time and sometimes informal notes act as "place holders" within a more formal argument.
I also generally issue a disclaimer: Check all work yourself (as you have done)
Anyway, here is a better argument:
Here we see the fixed relations which each point on the rigid body must undergo over the first 3 degrees as seen from the Sauret projection. Someone with better graphics arts skills than me (like femr) can actually show the exact relations for each point over a continuously changing angle from 0 to 3 degrees. It can be seen in a totally graphic way with no algebra.
A better argument for movement over the first 3 degrees would be based on physics, for example...
bold means vector...w means "omega" (can't do greek letters on the keyboard), the conventional term for angular velocity.
We know that the velocity of each point on a rigid body undoergoing rotation about an axis is
v = wxr
where r is the position vector from the axis of rotation to any point you wish. (w is a vector parallel with the axis of rotation)
We know the trajectory of each point for a rigid body undergoing rotation about an axis with certainty. It's most mathematically precise form (I believe) is in the form of a vector cross product.
Why didn't I express it this way before? Because, quite honestly, I didn't think the large majority of the readers would know what a freakin' cross product is.
You are a much better proof-reader and I have no doubt you know what a vector cross product is. I know it is not advanced stuff, but that is the way it is...really.
>>>>>>>>>>>>>
Anyway, we know there is a fixed relation between the movement of points in the case of rigid rotation. I am showing with mathematical certainty that that these fixed relations were not held.
Why would I care? I am not "anal" about deformity vs rigidity. I am more interested in how it deforms (and it is not what the NIST told you). It deforms concavely along the roof.
Back to your criticism, you are correct that that small part of the argument was expressed in a way that is sloppy and informal, but after I reformulate the argument, I will still be correct.
Anyone can think of what the v=wxr fixed relations means for pure rotational motion. (Think about it: I am trying to express these ideas to an audience that may not know basic vector operations. Vectors are the natural language of rigid rotation)
Anyone can see they are not obeyed. Anyone can measure the type of deformation from drop curves.
I'll replace that part with a much better argument in the next few days. Again, when I express it better, I will still win the argument. Deformity is a certainty. You can see it in the graphs and the visuals. Ain't no doubt about it.
And I also wrote "C" incorrectly in the link. It should read something like:
for b(s)-r(s)... C= sqrt[(b-r)*(b-r)]
and the same for a(s)-b(s).
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