transformations2 - Graphics

dragonslayer008@hotmail.com wrote:

> Walking one unit on the x-axis in frame A corresponds to walking 2
> units in the u direction in frame B.

Confusion starts here. What is a "unit", in each system?

> So there is a difference in scale between the two frames. So this
> means I should have ||u|| = 2, right?

That would depend on which coordinate system you want to express that
measure in.

> That is, the u vector describes the x-axis of frame A relative to
> frame B,

Not really. The vector that "describes" the x basis vector of frame A
is called "x" in system B, too. What you call "u" are the numerical
coordinates of x in frame B.

> Also, in particular, if you transform a circle from frame A into frame
> B, it becomes an ellipse. Should this be interpreted as a change of
> frame or not?

That's entirely your choice. Active or passive interpretation of
transformation are equally valid.

On Jul 11, 1:00 pm, Hans-Bernhard Bröker <HBBroe...@t-online.de>
wrote:
> dragonslayer...@hotmail.com wrote:
> > Walking one unit on the x-axis in frame A corresponds to walking 2
> > units in the u direction in frame B.
>
> Confusion starts here. What is a "unit", in each system?

Thanks for your reply.

Well what I mean is, say you walk one unit (say meters) in the
positive x-axis in frame A, but in frame B, this was equivalent to
walking a distance of 2 units (say half-meters) in the u direction.

> > So there is a difference in scale between the two frames. So this
> > means I should have ||u|| = 2, right?
>
> That would depend on which coordinate system you want to express that
> measure in.

Okay, let me rephrase. ||u|| = 2 in frame B?

> > That is, the u vector describes the x-axis of frame A relative to
> > frame B,
>
> Not really. The vector that "describes" the x basis vector of frame A
> is called "x" in system B, too. What you call "u" are the numerical
> coordinates of x in frame B.

Well, I don't really have an "x" basis vector. I am using coordinate
systems as used in multivariable calculus and intro physics books,
where they don't mention bases at all, but still use vectors
effectively. So my u is the vector aimed along the positive x-axis of
frame A, which I describe numerically by coordinates relative to frame
B.

> > Also, in particular, if you transform a circle from frame A into frame
> > B, it becomes an ellipse. Should this be interpreted as a change of
> > frame or not?
>
> That's entirely your choice. Active or passive interpretation of
> transformation are equally valid.

What do you mean by "active" and "passive"? I'm not familiar with
these terms from linear algebra.

dragonslayer008@hotmail.com wrote:
> On Jul 11, 1:00 pm, Hans-Bernhard Bröker <HBBroe...@t-online.de>
> wrote:

>>> So there is a difference in scale between the two frames. So this
>>> means I should have ||u|| = 2, right?
>> That would depend on which coordinate system you want to express that
>> measure in.
>
> Okay, let me rephrase. ||u|| = 2 in frame B?

Still depends on the system the measure ||.|| belongs to. In system B,
||x|| = 2, but it may still be that ||u|| = 1 (because the way you
describe it, u would seem to be the definition of "length = 1" in your
system B).

>>> That is, the u vector describes the x-axis of frame A relative to
>>> frame B,

>> Not really. The vector that "describes" the x basis vector of frame A
>> is called "x" in system B, too. What you call "u" are the numerical
>> coordinates of x in frame B.

> Well, I don't really have an "x" basis vector. I am using coordinate
> systems as used in multivariable calculus and intro physics books,

Then I guess you have the wrong books. Most physics books I remember do
distinguish between vectors and their coordinates better than that.

> So my u is the vector aimed along the positive x-axis of
> frame A,

Now you're contradicting yourself. This "positive x-axis of frame A",
_is_ the x basis vector, but you just said you didn't have that.

> But what's its length? Is it the same length as
> which I describe numerically by coordinates relative to frame
> B.

Depends on how you compute lengths from coordinates in frame B.
Coordinate systems with different scales make that job more interesting
that it would usually be. The standard dot product, i.e.
root-of-sum-of-squares-of-coordinates, only works consistently among
frames with the same scaling factors. So you have to decide what you
want to conserve: the formula, or the length of vectors.

>>> Also, in particular, if you transform a circle from frame A into frame
>>> B, it becomes an ellipse. Should this be interpreted as a change of
>>> frame or not?
>> That's entirely your choice. Active or passive interpretation of
>> transformation are equally valid.

> What do you mean by "active" and "passive"? I'm not familiar with
> these terms from linear algebra.

Active transformation means you actually move, rotate or deform objects
(i.e. points get assigned new coordinates in the same system).

Passive transformation means objects stay the same, coordinate systems
change.

The underlying maths doesn't care one way or another. Physics and
computer graphics will.