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View Full Version : R: Homogeneous transform? What one? For sure not that one

Paolo De Leva
07-18-2005, 12:21 AM
Dear subscribers,

There were excellent contributions on this topic. Useful references
to the concepts of "homogeneous deformation" and "homogeneous coordinates"
were suggested. It was probably not clear to someone to what extent these
contributions were needed. This problem deserved to be addressed. Here's
where I was (and many students, in the future, are likely to be) starting
from:

DEFINITION 1
[...] "describing both translation and rotation through one
mathematical operation, the matrix product" [...] "Sistems permitting such a
unified description are called HOMOGENEOUS".

DEFINITION 2
"The transformation matrix is a HOMOGENEOUS matrix: both rotation
and translation can be described by a common mathematical operation."

These definitions can be found in a successful biomechanics
textbook, a particularly exhaustive book, rich of useful information and
generally well written. I would advise everybody to buy it. However, in this
specific sentences, it clearly gives incorrect information.

Ton van den Bogert, at the beginning of his excellent contribution,
suggested that we shouldn't worry too much about incoherent terminology, and
should "allow multiple meanings (of the word "homogeneous") to co-exist
peacefully".

Perhaps, this statement was meant to be specific to this discussion.
However, it was stated in general terms. In general, I would suggest not to
be superficial about terminology. It cannot be denied that the principle of
"terminological coherence" deserves attention, although specific
circumstances may justify transgressions. There are evident practical
advantages in the use of coherent, well defined, conventional terminology,
and the clarity and refinement of Ton's language supports my thesis :-)

As for the specific case which is the subject of this discussion, it
might help to be aware that I was not dealing with two different yet
homonymous methods, developed in different fields. I was dealing with the
applications of one general method, developed within ONE field: linear
algebra. In my opinion, when we use algebraic methods, we should call them
with their correct "algebraic" name. We are supposed to study algebra before
using it. In fact, all of the contributors to this discussion appear to be
excellent mathematicians.

Indeed, it has been shown by Ton that, in a 4D space, the use of the
expression "homogeneous transformation" to indicate translation might be
even compatible with its algebraic definition: "in the 4D space of
homogeneous coordinates, pure translation is a nice linear transformation
again". That was a good point.

As another example of terminological coherence, I'll present you my
conclusion about the mechanical concept of homogeneous DEFORMATION. First
remember that, according to McGraw-Hill's scientific dictionary:

HOMOGENEITY: T(a+b) = Ta + TB and T(l*a) = l*T(a)
Or, according to some contributors
HOMOGENEITY: simply T(l*a) = l*T(a)

Whatever definition you use, it can be easily shown that, when
vectors are represented using the usual Cartesian notation, translation
meets none of the above conditions, i.e.:

TRANSLATION = NOT-HOMOGENEOUS TRANSFORMATION.

I am sure you agree. However, obviously:

TRNSLATION = RIGID TRANSFORMATION
and
RIGID TRANSFORMATION means NO DEFORMATION
(i.e. def gradient tensor F = 1)
Thus,
TRANSLATION means NO DEFORMATION

This simple Aristotelian syllogism is the reason why I am now
convinced that "homogeneous transformation" and "homogeneous deformation"
are different, but not conflicting concepts. They are compatible and make
sense when compared with each other. For different reasons and in different
ways, they don't "include" translation!

Thanks again to all the contributors to this discussion. I
appreciated both their kindness, and the valuable contents of their
contributions.

With my best regards,

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

-----Original message-----
Da: van den Bogert, Ton [mailto:BOGERTA@ccf.org]
Inviato: domenica 17 luglio 2005 19.43
A: Paolo de Leva; biomch-l@nic.surfnet.nl
Oggetto: RE: [BIOMCH-L] R: Homogeneous transform? What one? For sure not
that one

Dear subscribers,

This would not be the first instance of a scientific or mathematical term
having multiple meanings. "Homogeneous" may be such a word. So I suggest
let's not get too fussy about terminology and allow multiple meanings to
co-exist peacefully.

Similarly, the so-called direct linear transform (DLT) which is used for
camera calibration, is a linear transformation in the 4D space of
homogeneous coordinates.

--

Ton van den Bogert

--

A.J. (Ton) van den Bogert, PhD
Department of Biomedical Engineering
Cleveland Clinic Foundation
Phone: (216) 444-5566
http://www.lerner.ccf.org/bme/bogert/
http://www.isb2005.org