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From: Dan on 18 Dec 2009 11:31
On Dec 16, 9:38 am, "BGB / cr88192" <cr88...(a)hotmail.com> wrote:
> "Dan" <dant...(a)aol.com> wrote in message
>
> news:d18cd432-2d86-4091-8de2-5417fcf5f2ed(a)a10g2000pre.googlegroups.com...
>
>
>
>
>
>
>
> >> > A point *has* a location. A vector has a magnitude and direction.
> >> > Although
> >> > both points and vectors are representable in the same coordinate
> >> > spaces,
> >> > the semantics of those representations are very different.
>
> >> what is a point? (x, y)
> >> what is a vector? <x, y>
>
> >> magnitude and direction can be calculated from a vector, but they are
> >> not,
> >> in effect, the vector (as I see it).
>
> > Actually...  a vector as used in engineering and science has magnitude
> > and direction.   If it doesn't,  then...  most engineering books ever
> > written need to be thrown away.
>
> we can consider that, in these books, there is an implicit conversion going
> on.
> (here V` will mean V with the little hat)
>
> if for V` it is referenced simply as V, then it actually means V=|V`|,
> hence, the magnitude is implicitly extracted from the vector, but magnitude
> != vector.
>
> direction refers usually either to the vector itself, to a unit vector in
> the same direction, or (IMO sloppily) to an angle.
>
> regardless, the vector remains <x,y>, regardless of whether or not people
> believe it "contains" a magnitude and direction, or if we just have a very
> lightweight notation for extracting it and/or composing vectors from them (I
> will draw an analogy to the use of type-casting in many PL's...).
>
> but, this much is more philosophical than practical...
>
> > > IMO, this would be similar to claiming that <p, q, r> (in spherical
> > coords)
> >> are also a vector, since they also be said to have direction and
> >> magnitude,
> >> but spherical coords are a different entity from a vector (although, they
> >> are convertible, which was a big part of my trick of shoving a quat into
> >> the
> >> form of 3 angles...).
>
> > A line originating at the cooridinate axes and pointing towards
> > <p,q,r> can be considered a vector,  but the point in space that such
> > a vector points to is certainly not iself a vector  (in math or in
> > science).   It has no magnitude and it has no direction and thus, by
> > definition,  it is not a vector.
>
> <p,q,r> is not a point.
>
> rather, we could say
> <x,y,z> <= <p,q,r> ==> <x,y,z> = <r*cos(p)*sin(q), r*sin(p)*sin(q),
> r*cos(q)>
>
> hence, <p,q,r> would also have a magnitude and direction, but would be in a
> different coordinate space, but it is not a vector in a more strict sense of
> the word (since it is not in the form <x,y,z>...).
>
> likewise would go for <theta, rho> in 2D, even though this actually would
> "contain" magnitude and direction...
>
> > I make no comment as to how the op should write his code,  as points,
> > vectors, lines, etc..  all have a lot of overlap in how you do
> > calculations with/on them.   But....  when all is said and done,   a
> > point is still not a vector.
>
> as see it, the distinction is moot, since a point is still relative from the
> origin.
>
> (x,y,z)=<x,y,z>+(0,0,0)
> <x,y,z>=(x,y,z)-(0,0,0)
>
> all fairly moot, IMO...- Hide quoted text -

It is moot if you think of the only way of defining a point in space
as existing at the end of a vector that begins at the origin.
Suppose the point is defined as existing at the end of a vector that
starts somewhere other than the origin ( ie. relative to some other
point ). Or... if you simply think of the location of a point in
the x,y,z space as existing at a given coordinate. Put another
way.... you really don't even need to have any understanding of
vectors of any type in order to have a point. Rather, a point is
totally indepedent of the concept of vectors. I'd say, if anything,
that vectors (if you're refering to spatial concepts that is) are a
concept that is Dependent on points and not the other way around. An
algorithm that defines a vector simply by a starting and ending point
can be quite useful in many applications, both for it's simplicity
mentally and for it's use in calculations. In fact, I'm working on
such an application right now that uses just this concept because it
is extremely simple to understand and executes extremely fast compared
to other methods (which is important since the applications running
time can easily be on the order of hours).

Of course, I do agree that the whole thing of Point verses Vector is
rather moot. It ALL depends on exactly what it is that the op is
attempting to create with his application.

Oh yeah... in answer to Daniel's (the ops) original post... I find
your analogy to be reasonable... that "a point is to a vector as an
angle is to a rotation". This fits if you think of a rotation as
giving you the directions to find an angle when you start from some
given line or fixed axis. The other way of viewing it might be that
a rotation is an "operation" that you perform to transform a given
line or angle.

Dan
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