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Van Den Bogert, Ton
07-17-2005, 03:43 AM
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.

I think most people use the term "homogeneous transformation matrix" (HTM). This contains the word "matrix" so this is clearly within the class of linear transformations. However, I interpret the term "HTM" to mean "a linear transformation of homogeneous coordinates". Homogeneous coordinates (HC) have a very precise definition:

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

In the 3D case, a 4th coordinate is added so the transformation operates in a 4D space of homogeneous coordinates. The Mathworld page explains that with HC, you can specify points infinitely far away without requiring infinite values in the coordinate vector. This then allows pure translations to be described as rotations about a point at infinity.

Paolo gives an example of shear: "the x coordinate is multiplied by a coeefficient s1*y+s2*z". I don't think this is correct. In shear, the x coordinate would be *incremented* by an amount that is proportional to y and z. Consequently, the deformation gradient tensor would still be independent of position, and Jeff Weiss' two definitions remain consistent.

The term "linear" may also have multiple meanings. Pure translation is an operation that leaves straight lines straight, but it is not a linear transformation (in the mathematical sense, as defined at the end of Paolo's posting) of 3D spatial coordinates. But in the 4D space of homogeneous coordinates, pure translation is a nice linear transformation again.

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


> -----Original Message-----
> From: * Biomechanics and Movement Science listserver
> [mailto:BIOMCH-L@NIC.SURFNET.NL]On Behalf Of Paolo de Leva
> Sent: Friday, July 15, 2005 6:40 AM
> To: BIOMCH-L@NIC.SURFNET.NL
> Subject: [BIOMCH-L] R: Homogeneous transform? What one? For sure not
> that one
>
>
> 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: venerdý 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
>
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