OK, I can see how your terminology works now, but it seems odd to me. I
never consider re-expressing the coordinates on a curve as a vector and
basing geometric properties on those constructed vectors. I either
consider the points on the curve, or its tangent or its normal - none of
which is the value you are expressing. You are, essentially, operating
on tangent vectors, but you aren't calling them that, you are calling
them something like "the vector of the derivative" which is a relative
(direction only) version of the tangent vector (which has location and
direction). When one talks about curves and being parallel, my mind
tends to think of the tangents of the curves being parallel and tangents
are directed by the first derivative.
Also, if you are going to use your definition of "vector" then parallel
is an odd term to use for values that originate from the same point
(points considered as a vector are taken to originate from 0,0) - really
you want those "vectors" to be collinear, not (just) parallel.
So, either || means "the coordinates of the curves expressed as vectors
are collinear" or it means "the curves (i.e. the tangents of the curve
at the indicated point) are parallel". Saying "vector I() is parallel
to vector B()" didn't really have meaning to me based on the above biases.
So, I get your comment now and all of the math makes sense, but the
terminology seemed foreign to me...
...jim
On 10/20/10 10:48 AM, Denis Lila wrote:
Also, how is A(t) and B(t) are parallel not the same as "the curves A
and B are parallel at t"?
Well, suppose A and B are lines with endpoints (0,0), (2,0) for A
and (0,1),(2,1) for B. Obviously, for all t, A and B are parallel at t.
However let t = 0.5. Then A(t) = (1,0) and B(t) = (1, 1). The vectors
(1,0) and (1,1) are not parallel, so saying A(t) || B(t) is the same
as saying that there exists c such that (1,0) = c*(1,1), which isn't true.
However, A'(t)=(2,0) and B'(t)=(2,0), and the vectors
(2,0) and (2,0) are parallel.
Does this make more sense?
Regards,
Denis.
----- "Jim Graham"<james.gra...@oracle.com> wrote:
On 10/20/10 7:54 AM, Denis Lila wrote:
In #2, you have a bunch of "I'() || B'()" which I read as "the
slope
of the derivative (i.e. acceleration) is equal", don't you really
mean
"I() || B()" which would mean the original curves should be
parallel?
Otherwise you could say "I'() == B'()", but I think you want to
show
parallels because that shows how you can use the dxy1,dxy4 values
as
the parallel equivalents.
Not really. I've updated the comment explaining what || does, and
it should be clearer now. Basically, A(t) || B(t) means that
vectors
A(t) and B(t) are parallel (i.e. A(t) = c*B(t), for some nonzero
t),
not that curves A and B are parallel at t.
I'm not sure we are on the same page here.
I'() is usually the symbol indicating the "derivative" of I(). My
issue
is not with the || operator, but with the fact that you are applying
it
to the I'() instead of I().
Also, A(t) = c*B(t) is always true for all A and B and all t if you
take
a sample in isolation. Parallel means something like "A(t) = c*B(t)
with the same value of c for some interval around t", not that the
values at t can be expressed as a multiple.
Again, I'() || B'() says to me that the derivative curves are
parallel,
not that the original curves are parallel...
...jim
