Dear subscribers,

regarding the recent posting by Joshua Hale, I would like to point
out that there's no inconsistency in the use of the term "homogeneous" in
mathematics. On the contrary, the term is used with perfect consistency and
formally defined with no ambiguity (i.e. in formal-logic terms) in the
following expressions:

- homogeneous transformation
- homogeneous coordinates
- homogeneous function
- homogeneous equation
- homogeneous space
- homogeneous polynomial....

The expression "homogemeous coordinates" refers to scaled
coordinates, as Joshua explained. Although I have never used homogeneous
coordinates I do understand that, in 3D, the fourth homogeneous coordinate
of a point is the scaling factor used for the first three coordinates. Thus,
if [x1, x2, x3, x4] is the position of point P expressed with homogeneous
coordinates, and [x, y, z] is the same position expressed with Cartesian
coordinates:
[x1, x2, x3] / x4 = [x, y, z].
Scaling, as I wrote, is a homogeneous transformation. Therefore,
there's no inconsistency in the use of the word homogeneous in the two
expressions "homogeneous transformation" and "homogeneous coordinates".

I don't know what are the different branches of mathematics which
Joshua was referring to. Anyway, this discussion is about geometry, a single
branch of mathematics. As far as I know, geometry is a perfectly refined
tool developed much before we were born and much before its specific
applications in our fields (biomechanics and CG) were developed.

I will defend my point with two new arguments. First, if somebody is
willing to defend the current (ab)use of the expression "homogeneous
transformation" as the name of the 4x4 "roto-translation" (or "position")
matrix, he should find a decent definition of the expression "homogeneous
transformation" which can be applied to such a matrix and not for instance,
to a 3x3 pure-rotation matrix. I invite him to find a textbook or a
scientific dictionary giving such a definition in formal-logic
(mathematical) terms. It is not appropriate to oppose a vague definition to
a different definition specified in mathematical terms. I might even accept
that the definition given by some biomechanists or CG experts be conflicting
with a previously given definition which was tested, known and accepted
worldwide before they were born. But I can't accept that this new definition
be vague. Let's avoid to oppose fuzzy ideas to crystal clear concepts
defined in formal-logic terms.

Second, if many experts in biomechanics and CG use incorrect
terminology, that's not a good reason to follow suit. One of the
requirements to be a good scientist is to know suitable methods and produce
correct results (and many can do it). Another requirement is to be able to
use a correct terminology and a crystal clear structured language, for
instance to explain methods and discuss results (and only a few can do it,
most unfortunately).

Finally, in my original posting I presented two reasons against the
use of the expression "homogeneous transformation" to indicate a 4x4
roto-translation matrix. Each reason alone was strong enough to support my
conclusion. Joshua only addressed the second one.

With kind regards,

Paolo de Leva
Sport Biomechanics
University Institute of Motor Sciences
Rome, Italy

P.S. I apologize for the multiple postings of my previous message. I tried
several (about 5) times to post my message and received automatic replies
from the list-server stating that I was not authorized to post on BIOMCH-L.
Only the last time I was authorized, but two of my previous attempts were
somehow successful, notwithstanding the contrary had been stated in the
automatic replies.

-----Messaggio originale-----
Da: Joshua Hale [mailto:josh@joshhale.com]
Inviato: giovedÃ¬ 7 luglio 2005 10.35
A: Paolo de Leva; BIOMCH-L@NIC.SURFNET.NL
Oggetto: Re: [BIOMCH-L] Homogeneous transform? What one? For sure not that
one.

Dear Subcribers / Paolo,

It's true that there is an inconsistency in the usage of the term
"homogeneous transformation", and this results from multiple definitions
of the term originating in different branches of mathematics.
Unfortunately however, these are very well established and there is not
much chance of renaming either. It's probably best to be aware of both
meanings and be happy with that.

I believe when talking about 4x4 transformations and 4x1 / 1x4 position
vectors the term homogeneous refers to the fact that multiple 4
dimensional vectors can describe the same position. Such a family of
vectors is in that sense "homogeneous", and the term "homogeneous
transformation" is derived from the concept of "homogeneous coordinates".

i.e. The point (1,2,3) can be written (1,2,3,1) or (2,4,6,2), or (0.5,
1, 2, 0.5). The last element is a scaling factor, and the actual
coordinates are typically extracted by scaling the vector such that the
last element is 1. Among other things, this allows for example a
translation, or an arbitrary orthogonal coordinate transform to be
effected using a matrix multiplication operation. The concept is not
limited to 3D coordinates of course...

http://mathworld.wolfram.com/HomogeneousCoordinates.html

Best wishes,

Josh.