[From Rupert Young (2016.09.14 1700)]
Perhaps we could reboot this discussion, and related threads, and
remember the three P’s of scientific discourse, and we might be
able to make some progress, and avoid clogging up this forum with
bad tempered and condescending posts:
Professionalism - we are all here for the same reason, to
advance a scientific understanding of the natural world. This is
best by realised by an atmosphere of mutual respect, recognising
the diversity of background and technical experience of our
colleagues and, consequently, that different people will develop
understanding at different rates.
Politeness - there is little to be achieved by insulting
and demeaning language, this just results in ad hominem attacks
and does nothing to address the issues in question. It may be
exasperating if someone appears to have got something wrong, but
there is no advantage in attacking them for it. The best thing to
do might be to walk away from the discussion, and maybe come back
to it later and try again. Though it would be worth confirming
that you are not talking at cross purposes (very often the case on
this forum) and that there is a common understanding of the
terminology involved.
Patience - clearly there are those on this forum with
superior mathematical knowledge. However, it should not be assumed
that because others may not understand some points right away that
they are stupid or unable to grasp the concepts given some time
and encouragement. There have been a number of times on this forum
when Rick has been willing to accept that he has got something
wrong. He seems willing to have another look at his reasoning so
perhaps he should be given the benefit of doubt. What might seem
basic and obvious to maths experts is probably not to the rest of
us, so it would be useful if it could be taken step by step at a
level more amenable to the majority.
Rather continually repeating concepts and formulae on a multitude
of posts what might be useful would be to have the reasoning laid
out in a separate single document, with each step agreed by all
parties, hopefully in language that we could all understand. A
challenge perhaps, but it could save a lot of time.
Rupert
···
On 14/09/2016 07:38, Alex Gomez-Marin
wrote:
martin, the csgNet is now a hybrid of a basic mathaccademy with an entertainment program for writing and joking with
those who write back. stay tuned!On Wednesday, 14 September 2016, Martin Taylor <mmt-csg@mmtaylor.net >wrote:
[Martin Taylor 2016.09.13.22.49]
[From Rick Marken (2016.09.13.1610)]
MT: but thecritical point of my suggestion that you
read my earlier message is contained in
this:
RM: Contained in what?
RM: Oh, I see. It’s contained in this:
MT: Now we have tosee how they came to equation (9). That’s a
bit more complicated, so please bear with me.MT: Theypresumably either used someone else’s
derivation or made their own, starting from
one of several equivalent measures of
curvature, one of which is C = 1/R where R is
the radius of the osculating circle at the
point of concern. Another one is developed
using vector calculus, which I have no
intention of introducing into this discussion.
It is C = dx/dsd2y/ds2 - dy/ds * d2x/ds2 , where s is distance
along the curve from some arbitrary starting
point.
For G+O this
formula was not very convenient, because they
would have had to measure these first and
second derivatives of x and y with respect to
distance along the curve fairly accurately.
But they had a trick available, in the “chain
rule” of differentiation: dx/dydy/dz = dx/dz.
The "dy"s cancel out just like ordinary
variables. Using the chain rule on the first
derivative gives you the rule for the second
derivative, and so on. For the second
derivative the rule is (d2x/dy2)(dy/dz)2 = d2x/dz2.
Using the chain
rule, G+O could multiply the formula for C by
(ds/dz}3/(ds/dz)3 = 1, for any
variable z that allowed the differentiation,
to get C = ((dx/ds)(ds/dz)(d2y/ds2)(ds/dz)2)(ds/dz)3 - (dy/ds)(ds/dz)(d2x/ds2)__(ds/dz)__3)/(ds/dz)3 . This formula is
true (allowing for typos) for variable “z”
whatever (as with the divide by zero example),
but it wouldn’t have helped G+O very much, had
it not been that for one particular variable
they already had measures they could use.
Those measures were the ds/dt velocity and the
derived d2s/dt2 values they had
obtained from their observations of movement.
Using those measures, they could set “z” = t
(time), making dx/dt = dx/ds*ds/dt. They could
then take advantage of their measured
velocities to substitute for ds/dt, and writeC = (dx/dt*d<sup>2</sup>y/dt<sup>2</sup>)/V<sup>3</sup> - (dy/dt*d<sup>2</sup>y/dt<sup>2</sup>)/V<sup>3</sup> Oh goody! Wedon’t have to measure anything new to get our
curvatures. We can use the values of dx/dt and
dy/dt that we got before! Very handy. … But
also very confusing, because it made the
published equations look as though the V3/V3 multiplier was
special to the velocities they measured,
whereas it was simply a convenient choice from
a literally infinite variety of choices they
could have made. G+O made it even more
confusing in the publication by using the
Newton dotty notation, which made it look as
though there was something necessary about the
time differentiation in the curvature
equation.When we put allthis together, we come to the way this is a
variant of the “divide by zero” error. That
error depends on the fact that you can put any
variable at all in for “x” in “x/0 =
infinity”. The – shall we call it – the
“curvature error” depends on the fact that you
can use anything at all for V (including the
measured values), provided only that V is
defined as ds/dz where z is some variable for
which ds/dz exists everywhere. You therefore
cannot use the curvature equation in any way
to determine V.Does this "Sunday" explanation help?
RM: Not really. Are you saying that G+O usedthe wrong formulas for V and R? Or that the
formulas they published are not actually the
ones they used to compute V and R? Or that there
is no way to compute R since we can’t measure
ds? Either way, you can’t say I made a math
error since I did the math correctly on the
formulas I was given. (And, as I mentioned, the
results came out exactly right).
If you actually read and thought about what you quote, Icannot read your question as anything other than a not too
subtle joke, so I have to assume that you either did not read
it or the maths, which I tried to explain at the most basic
level O could manage without seeming to insult you (which I
was afraid I was doing anyway), was a bit too deep for you.Isuggest you try again, concentrating on figuring out whythis DOES explain why you have been making the same kind of
mistake as the “divide by zero” error. Perhaps I should repeat
the last lines: " When we put
all this together, we come to the way this is a variant of
the “divide by zero” error. That error depends on the fact
that you can put any variable at all in for “x” in “x/0 =
infinity”. The – shall we call it – the “curvature error”
depends on the fact that you can use anything at all for V
(including the measured values), provided only that V is
defined as ds/dz where z is some variable for which ds/dz
exists everywhere. You therefore cannot use the curvature
equation in any way to determine V."I emphasize "IN ANY WAY". Since I may be a little too subtlewhen I say this, it simply says that your equation V =
D1/3*C1/3 means nothing at all, because it is true when V is
any variable at all that satisfies a very loose condition.
If it is a velocity it can be any velocity at all, or it can
be any value of any variable (and here I will repeat myself)
that depends on any other variable “z” whatever for which
ds/dz is everywhere calculable.
RM: Rather than saying that I was making amath error, it would have helped if you had just
said: “these are the correct formulas for
computing V and R” and showed me the formulas.
G+O showed the correct formulas, and I said so. You did thecorrect FORMAL algebra. Your math error was and apparently
continues to be the equivalent of the “divide by zero” error,
which also depends on doing the algebra correctly. The “divide
by zero” or its “curvature error” equivalent is a math error
if ever there was one. It could be and should be easily
correctable, but apparently it isn’t. I don’t know why.MartinThat would have saved a lot of trouble. Sohow about it; what are the correct formulas for
computing V and R?
Best
Rick
–
Richard
S. Marken
"Thechildhood of
the human race
is far from
over. We have
a long way to
go before most
people will
understand
that what they
do for others
is just as
important to
their
well-being as
what they do
for
themselves."
– William T.
Powers
–
Richard S.
Marken
"The childhood of thehuman race is far from
over. We have a long way
to go before most people
will understand that
what they do for others
is just as important to
their well-being as what
they do for themselves."
– William T. Powers