Paolo De Leva

07-12-2005, 05:50 AM

Dear subscribers,

referring to the posting by Andreas Boehm on behalf of Joshua Hale,

I would like to make it clear that I did not and will not propose new

terminology. On the contrary, I believe that new terminology introduced in a

specific field was inappropriate and clearly conflicting with previously

defined terminology. This previously defined terminology belongs to the

level of abstraction of mathematics, thus its applications are not limited

to a specific field such as computer graphics (CG) or physics or mechanics

or biomechanics.

There's no need to introduce a new expression to indicate the 4x4

transformation matrix (T.M.) described in my first posting. Several other

expressions are available. These other expressions are widely known and

habitually used by computer graphics experts and biomechanists as well.

THE MATRIX ITSELF SEEMS QUITE HETEROGENEOUS...

The particular format of that 4x4 transformation matrix (T.M.) can

be and is used for several DIFFERENT kinds (species) of transformations:

translation, rotation, scaling, shearing, reflection, and any combination of

these tranformations. By the way, "different" means HETEROGENEOUS, i.e. not

homogeneous...

GENERAL TRANSFORMATION MATRIX

The 4x4 T.M. matrix format is general, i.e. not specific, i.e.

multi-purpose. Thus, someone calls it the "GENERAL T.M.". I am not saying I

like this expression. It's too generic, it might be ambiguous, but I believe

it's correct.

ROTO-TRANSLATION MATRIX

The 4x4 matrix used in biomechanics is typically a "ROTO-TRANSLATION

MATRIX", performing only rotation and translation, no shearing, no

reflection, and no scaling. In this case, "roto-translation matrix" is the

clearest name, in my opinion.

POSITION MATRIX

In biomechanics, the roto-translation matrix is also correctly

called the "POSITION MATRIX" (referring to both the angular and linear

positions of a reference system and its origin). That's an appropriate and

meaningful name.

AFFINE TRANSFORMATION MATRIX

The 4x4 T.M. will perform an "homogeneous (also called linear)

transformation" when used to perform just rotation and/or scaling and/or

reflection. Unfortunately, these transformations can be all performed also

with a 3x3 matrix!

Translation and shearing are not homogeneous (i.e. not linear)

transformations. Translation, shearing, rotation, scaling and reflection all

belong to the group of affine transformations (notice that affine

transformations include homogeneous transformations). Thus, the 4x4 T.M.

will perform an "affine transformation" when a non-null translation vector

is included in its fourth row or column or when some of its elements are

multiplied by a "shearing factor". Hence, you might want to call this matrix

an "AFFINE T.M.".

Notice that the concept "affine" means something "not perfectly

equal", but similar. On the contrary, "homo-geneous" means "of equal kind"

(homos = equal, in greek). And this stresses my point: ethimologically,

affine clearly means not-necessarily-homogeneous!

RIGID TRANSFORMATION MATRIX

The 4x4 T.M. might also be used to perform a "rigid" transformation

when scaling and shearing are not included. The roto-translation, for

instance, is a rigid transformation. Thus, the roto-translation matrix might

be called a "RIGID T.M.". However, consider that a pure rotation is a rigid

transformation as well, and pure rotations can be performed using simple 3x3

rotation matrices.

TRANSFORMATION IN HOMOGENEOUS COORDINATES

Finally, the 4x4 T.M. can be viewed as a matrix performing a

"transformation in homogeneous coordinates" (i.e. between two vector spaces

represented in homog. coordinates), as some computer graphics experts

correctly write, although it is not necessary to know the complex concept of

homogeneous coordinates to understand format and "behaviour" of a 4x4 matrix

using only three zeros and a one [0, 0, 0, 1] in the fourth column or row.

HOMOGENEOUS MATRIX

It is true that, if you like, you can call the 4x4 T.M. a

"homogeneous matrix" (not referring to the transformation, but to the

matrix). But that is, in my opinion, a poor terminological choice, because

that matrix is not really homogeneous. It looks quite "heterogeneous": it

performs several different kinds of transformations and, more importantly,

it contains at least two clearly different types of vectors:

1) a triad of vectors with null fourth element, e.g.

a = [a1, a2, a3, 0]

or its transpose a', representing several kinds of

transformations, and

2) a single translation vector with its fourth element equal to 1,

t = [t1, t2, t3, 1]

or its transpose t'.

Probably some of you won't agree, but in my opinion vector types 1

and 2 are heterogeneous (i.e. different, not homogeneous) with respect to

each other.

Actually, I don't care much about the exact name of the 4x4 T.M. I

just would like to read and hear names that have a clear meaning and help me

to understand what the writer is talking about, rather than mixing me up! I

hope that, at least on this last generic concept, everybody will be kind

enough to agree.

With kind regards,

Paolo de Leva

Sport Biomechanics

Uiversity Institute of Motor Sciences

Rome, Italy

-----Messaggio originale-----

Da: * Biomechanics and Movement Science listserver

[mailto:BIOMCH-L@NIC.SURFNET.NL] Per conto di Andreas Boehm

Inviato: venerd́ 8 luglio 2005 10.34

A: BIOMCH-L@NIC.SURFNET.NL

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

Hello,

I post this on behalf of Joshua Hale .

regards,

Andreas Boehm

-----------Posting copied below:- (included previous post)

Dear Subscribers / Paolo,

My point is simply that "homogeneous transformation matrices" are a very

well established, and using an alternative terminology is likely to

confuse more people than are intellectually liberated by inventing a new

term to describe them. -Not that I want to stand in the way of progress...

In defence of the term itself, perhaps "homogeneous" refers to the

matrices themselves, and not the transformation it encodes? i.e., if all

the elements excluding the bottom right corner element are scaled by

some factor k, and the bottom right corner element is scaled by 1/k, the

resultant matrix is homogeneous with the original matrix. (The latter is

not allowed to be zero).

Best,

Josh.

referring to the posting by Andreas Boehm on behalf of Joshua Hale,

I would like to make it clear that I did not and will not propose new

terminology. On the contrary, I believe that new terminology introduced in a

specific field was inappropriate and clearly conflicting with previously

defined terminology. This previously defined terminology belongs to the

level of abstraction of mathematics, thus its applications are not limited

to a specific field such as computer graphics (CG) or physics or mechanics

or biomechanics.

There's no need to introduce a new expression to indicate the 4x4

transformation matrix (T.M.) described in my first posting. Several other

expressions are available. These other expressions are widely known and

habitually used by computer graphics experts and biomechanists as well.

THE MATRIX ITSELF SEEMS QUITE HETEROGENEOUS...

The particular format of that 4x4 transformation matrix (T.M.) can

be and is used for several DIFFERENT kinds (species) of transformations:

translation, rotation, scaling, shearing, reflection, and any combination of

these tranformations. By the way, "different" means HETEROGENEOUS, i.e. not

homogeneous...

GENERAL TRANSFORMATION MATRIX

The 4x4 T.M. matrix format is general, i.e. not specific, i.e.

multi-purpose. Thus, someone calls it the "GENERAL T.M.". I am not saying I

like this expression. It's too generic, it might be ambiguous, but I believe

it's correct.

ROTO-TRANSLATION MATRIX

The 4x4 matrix used in biomechanics is typically a "ROTO-TRANSLATION

MATRIX", performing only rotation and translation, no shearing, no

reflection, and no scaling. In this case, "roto-translation matrix" is the

clearest name, in my opinion.

POSITION MATRIX

In biomechanics, the roto-translation matrix is also correctly

called the "POSITION MATRIX" (referring to both the angular and linear

positions of a reference system and its origin). That's an appropriate and

meaningful name.

AFFINE TRANSFORMATION MATRIX

The 4x4 T.M. will perform an "homogeneous (also called linear)

transformation" when used to perform just rotation and/or scaling and/or

reflection. Unfortunately, these transformations can be all performed also

with a 3x3 matrix!

Translation and shearing are not homogeneous (i.e. not linear)

transformations. Translation, shearing, rotation, scaling and reflection all

belong to the group of affine transformations (notice that affine

transformations include homogeneous transformations). Thus, the 4x4 T.M.

will perform an "affine transformation" when a non-null translation vector

is included in its fourth row or column or when some of its elements are

multiplied by a "shearing factor". Hence, you might want to call this matrix

an "AFFINE T.M.".

Notice that the concept "affine" means something "not perfectly

equal", but similar. On the contrary, "homo-geneous" means "of equal kind"

(homos = equal, in greek). And this stresses my point: ethimologically,

affine clearly means not-necessarily-homogeneous!

RIGID TRANSFORMATION MATRIX

The 4x4 T.M. might also be used to perform a "rigid" transformation

when scaling and shearing are not included. The roto-translation, for

instance, is a rigid transformation. Thus, the roto-translation matrix might

be called a "RIGID T.M.". However, consider that a pure rotation is a rigid

transformation as well, and pure rotations can be performed using simple 3x3

rotation matrices.

TRANSFORMATION IN HOMOGENEOUS COORDINATES

Finally, the 4x4 T.M. can be viewed as a matrix performing a

"transformation in homogeneous coordinates" (i.e. between two vector spaces

represented in homog. coordinates), as some computer graphics experts

correctly write, although it is not necessary to know the complex concept of

homogeneous coordinates to understand format and "behaviour" of a 4x4 matrix

using only three zeros and a one [0, 0, 0, 1] in the fourth column or row.

HOMOGENEOUS MATRIX

It is true that, if you like, you can call the 4x4 T.M. a

"homogeneous matrix" (not referring to the transformation, but to the

matrix). But that is, in my opinion, a poor terminological choice, because

that matrix is not really homogeneous. It looks quite "heterogeneous": it

performs several different kinds of transformations and, more importantly,

it contains at least two clearly different types of vectors:

1) a triad of vectors with null fourth element, e.g.

a = [a1, a2, a3, 0]

or its transpose a', representing several kinds of

transformations, and

2) a single translation vector with its fourth element equal to 1,

t = [t1, t2, t3, 1]

or its transpose t'.

Probably some of you won't agree, but in my opinion vector types 1

and 2 are heterogeneous (i.e. different, not homogeneous) with respect to

each other.

Actually, I don't care much about the exact name of the 4x4 T.M. I

just would like to read and hear names that have a clear meaning and help me

to understand what the writer is talking about, rather than mixing me up! I

hope that, at least on this last generic concept, everybody will be kind

enough to agree.

With kind regards,

Paolo de Leva

Sport Biomechanics

Uiversity Institute of Motor Sciences

Rome, Italy

-----Messaggio originale-----

Da: * Biomechanics and Movement Science listserver

[mailto:BIOMCH-L@NIC.SURFNET.NL] Per conto di Andreas Boehm

Inviato: venerd́ 8 luglio 2005 10.34

A: BIOMCH-L@NIC.SURFNET.NL

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

Hello,

I post this on behalf of Joshua Hale .

regards,

Andreas Boehm

-----------Posting copied below:- (included previous post)

Dear Subscribers / Paolo,

My point is simply that "homogeneous transformation matrices" are a very

well established, and using an alternative terminology is likely to

confuse more people than are intellectually liberated by inventing a new

term to describe them. -Not that I want to stand in the way of progress...

In defence of the term itself, perhaps "homogeneous" refers to the

matrices themselves, and not the transformation it encodes? i.e., if all

the elements excluding the bottom right corner element are scaled by

some factor k, and the bottom right corner element is scaled by 1/k, the

resultant matrix is homogeneous with the original matrix. (The latter is

not allowed to be zero).

Best,

Josh.