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R: Homogeneous transform? What one? For sure not that one

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  • R: Homogeneous transform? What one? For sure not that one

    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

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

    "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

    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.:


    I am sure you agree. However, obviously:

    (i.e. def gradient tensor F = 1)

    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

    With my best regards,

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

    -----Original message-----
    Da: van den Bogert, Ton []
    Inviato: domenica 17 luglio 2005 19.43
    A: Paolo de Leva;
    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