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

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