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



Paolo De Leva
07-14-2005, 08:39 PM
Dear subscribers,

Jeffrey A. Weiss gave a different definition of a homogeneous
transformation. Let's call it a "mechanical definition". It would be nice if
this definition were correct and applicable to the multi-purpose T.M. we
were discussing about! In this case, the arguments I presented in my
previous postings would dissolve "as tears in rain", and I would be happy to
be allowed to call that matrix an "homogeneous transformation matrix".

The definition I gave (let's call it the "mathematical definition")
can be found on the McGraw-Hill Dictionary of Scientific and Technical
Terms, and it looks coherent with the etymology of the word homogeneous
(from greek = equal race, birth, family). The McGraw-Hill scientific
dictionary is authoritative, and I found a lot of correct definitions there.
But it might be wrong or incomplete in this case. It was written by men and
no man is perfect.

However, consider that a transformation is just an ordered set of
functions (just 3 functions applied to 3 coordinates, in 3D)! Thus, it
appears quite obvious and desirable that, in a coherent scientific
terminology, the definition of "homogeneous transformation" should agree
with the definition of "homogeneous function". Unfortunately, the mechanical
definition given by Jeffrey doesn't, while the "mathematical definition"
does. See also:

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

In fact, the McGraw-Hill dictionary specifies that the homogeneous
transformation is: "also known as linear transformation, linear function,
linear operator"

Luckily, the "mechanical definition" is coherent with the etymology
of the word "homogeneous". However, unfortunately I believe that, contrary
to what Jeffrey wrote, his "mechanical definition" is not applicable to
shearing. Shearing does not seem to be an "homogeneous transformation",
neither according to the "math definition" nor according to the "mechanical
definition". In fact, with shearing, the transformation along the x axis
depends on the y and/or z position of the point, i.e. the x coordinate is
multiplied by a coeefficient s1*y+s2*z, which is clearly dependent on the
point position. Unfortunately, shearing is one of the affine transformations
typically performed by using the multi-purpose 4x4 general T.M. we have been
discussing about...

Notice also that shearing meets Jeffrey's "intuitive" version of the
"mechanical definition" of the expression "homogeneous transformation"
(which coincides with the intuitive definition of affine transformations),
but not his formal definition, as I pointed out above. Thus, the two
definitions given by Jeffrey (formal and intuitive) seem to be not
equivalent to each other.

Here's the "mathematical definition" I gave in my previous posting,
equivalent to that given by the McGraw-Hill Dictionary of Scientific and
Technical Terms:

An “homogeneous transformation” (also called “linear transformation”
or "linear function") is defined as a transformation T such that, when
applied to three different vectors, a, b, and c = a+b, it yields three other
vectors, respectively a’, b’ and c’, for which the relationship c’ = a’+b’
is still valid. Thus,
T(a+b) = T(a) + T(b)

Also, a homogeneous transformation T is such that scaling after a
homogeneous (linear) transformation is the same as scaling before it. Thus,
if “a” is a vector and “s” is a scalar, then

s*T(a) = T(s*a)

Is any subscriber willing to find and quote other authoritative
sources for the definition of the expression "homogeneous transformation"?
Please, quote always the formal definition, not only the intuitive version.

Regards,

Paolo De Leva


-----Messaggio originale-----
Da: * Biomechanics and Movement Science listserver
[mailto:BIOMCH-L@NIC.SURFNET.NL] Per conto di Jeff Weiss
Inviato: venerdì 15 luglio 2005 0.11
A: BIOMCH-L@NIC.SURFNET.NL
Oggetto: Re: homogeneous transformation

In solid mechanics, a homogeneous deformation / transformation /
deformation map is one where the components of the deformation gradient
tensor do not have any dependence on the spatial coordinates. A more
intuitive definition is that a homogeneous deformation maps straight
lines into straight lines. This admits homogeneous shear, stretching,
compression, rotation, translation, etc.

A rigid deformation / transformation / deformation map is one that obeys
the above definition but also can be decomposed into a proper orthogonal
rotation and a translation (again, both constant in space).

Any decent textbook on continuum mechanics will confirm these definitions.

The whole idea of a "matrix", whether 4x4 or 3x3 + a 3x1 translation, is
simply a computational tool.

Jeff

--

Jeffrey A. Weiss, Ph.D.
Department of Bioengineering, University of Utah
jeff.weiss@utah.edu http://hodad.bioen.utah.edu/~weiss/mrl