technobabble vs solid geometry, part 5
I rewrote the whole deformation and roof-line concavity arguments to remove any reference to the word "linear" and present the info a bit more clearly.
An introduction to measuring deformity and viewpoint projection using vectors can be found here:
http://www.sharpprintinginc.com/911...op=view_page&PAGE_id=187&MMN_position=373:373
That page contains the following false claim, which appears to be central to
Major_Tom's "argument":
Major_Tom said:
The constraint relations for a rigid body can be rewritten as
d(a-b)/df = c*d(b-r)/df where f is frame and c is a constant. This means the slopes of the blue and yellow lines must remain proportional to one another
There are actually three errors in that claim. Video frames are discrete, so taking derivatives with respect to the "frame" f makes no sense.
Major_Tom must be using frames as a proxy for time. More fundamentally, he appears to be making a claim about how the coordinates of points on a rigid body transform as that body is rotated. If so, then he should have taken his derivatives with respect to the tilt angle θ instead of taking derivatives with respect to frames (as proxy for time).
Both of those errors are relatively minor, because it is not altogether unreasonable to assume that the tilt angle is an approximately linear function of time, which means that proportionality of the derivatives with respect to time would follow from proportionality of the derivatives with respect to the tilt angle. As shown below, however, neither proportionality holds for rigid bodies.
I will also consider the possibility that
Major_Tom meant only to claim that, if the upper section were rigid, the differences would remain proportional under rotations, and that the derivatives were just more technobabble. It's hard to tell, so I'll prove that both of the plausible interpretations of his claim are false.
For both refutations, it is enough to consider the actual vertical drops instead of the "drops" measured
twelve degrees off plumb. (In terms that
Major_Tom might understand, the actual drops are those that would be seen by viewer at infinity, level with the roof. The error can be demonstrated using viewers at other distances and angles, but the calculations would be more complicated.)
Proof that the differences aren't always proportional:
As
has been shown previously, the true vertical coordinates of the points a, b, and r on an assumed-rigid upper section, considered as functions of tilt angle θ, are
[latex]
\begin{align*}
a(\theta) &= s_2 \sin \theta + a_0 \cos \theta \\
b(\theta) &= s_2 \sin \theta + b_0 \cos \theta \\
r(\theta) &= r_0 \cos \theta
\end{align*}
[/latex]
If
Major_Tom's claim were true (when interpreted as being about the differences), then there would be some constant
c such that
[latex]
\begin{align*}
a(\theta) - b(\theta) &= c (b(\theta) - r(\theta)) \\
(a_0 - b_0) \cos \theta &= c (s_2 \sin \theta + (b_0 - r_0) \cos \theta) \\
&= c s_2 \sin \theta + c (b_0 - r_0) \cos \theta \\
(a_0 - b_0 - c b_0 + c r_0) \cos \theta &= c s_2 \sin \theta \\
\frac{(a_0 - b_0 - c b_0 + c r_0)}{c s_2} &= \frac{\sin \theta}{\cos \theta}
\end{align*}
[/latex]
where each line of that derivation follows immediately from the preceding line. The left hand side of the last equation is a constant, but the right hand side is not. Hence the equation cannot hold independent of θ, hence the claimed constant of proportionality
c cannot exist. That ends the first proof.
Proof that the derivatives aren't always proportional:
If
Major_Tom's claim were true (when interpreted as being about derivatives with respect to θ), then there would be some constant
c such that
[latex]
\begin{align*}
\frac{d (a(\theta) - b(\theta))}{d \theta} &= c \frac{d (b(\theta) - r(\theta))}{d \theta} \\
\frac{d ((a_0 - b_0) \cos \theta)}{d \theta} &= c \frac{d (s_2 \sin \theta + (b_0 - r_0) \cos \theta)}{d \theta} \\
(b_0 - a_0) \sin \theta &= c s_2 \cos \theta + c (r_0 - b_0) \sin \theta \\
(b_0 - a_0 + c (b_0 - r_0)) \sin \theta &= c s_2 \cos \theta \\
\frac{\sin \theta}{\cos \theta} &= \frac{c s_2}{b_0 - a_0 + c (b_0 - r_0)}
\end{align*}
[/latex]
where each line of that derivation follows immediately from the preceding line. The right hand side of the last equation is a constant, but the left hand side is not. Hence the equation cannot hold independent of θ, hence the claimed constant of proportionality
c cannot exist. That ends the second proof.
