Quote:> From the context of your question, my guess is you want something

> other than a quaternion spline, probably more of a "spherical average"

> idea. The non-commutative algebra of 3D rotations manifests itself, as

> you suggest, in a non-flat space, with the consequence that many flat

> space assumptions fall apart. Although it has been nearly two decades

> since Shoemake's first paper, surprisingly little progress has been

> made on questions like yours -- which suggests it ain't easy.

> One recent paper that may be of interest appeared in TOG.

> <http://math.ucsd.edu/~sbuss/ResearchWeb/spheremean/>

I've been busy following the sugestions people kindly made to me. So,

first of all thanks to all of you. My comments follow.

Many of you think that what I really need ("o que eu preciso

verdadeiramente", as Jerzy might say) is a spline. Well I've some

doubts about it. I had already read something about splines and squads

(yes, I've read the FAQ before), but from my point of view, they look

ugly, at least compared to slerp.

In fact, although computational mechanics and computer graphics have

much in common -- specially in what concerns the description of

rotations or general movement in 3D space --, the needs are not quite

exactly the same.

I really don't know much about splines, but it seems to me that their

principal advantages are in constructing curves smooth curves through

(or controlled by) an arbitrary number of points (let us call them

nodes). For what I understand, in most applications splines are

piecewise cubic polynomials, even when the number of nodes increases.

These piecewise polynomials have all the desired transition

properties, and I think that is the main purpose behind their design.

All this applies to both splines in 3D space and in SO(3).

On the finite element method, on the other hand, you look for

interpolations based on a number of n nodes, and you generally want

that the degree of the polynomials to be equal to n-1. In this way,

you can build increasingly refined approximations. In fact, you talk

about the h refinement (in which we refine the mesh and, accordingly

the number of elements) and the p refinement (in which you keep the

same number of elements, but increase the degree of tthe polynomials,

adding internal nodes to each element). In 3D flat space, nothing

better than a (n-1) degree Lagrange polynomial to interpolate between

n nodes. Note that, if the correct solution is a polynomial of degree

p, it suffices to have n=p+1 and you don't gain anything from

increasing your n.

So, what I really want is the generalization of Lagrange polynomials

to SO(3) -- or, even better, to SE(3). I don't readily dismiss

splines, but I really don't think they are what I am after. For

instance, in flat space, you don't need splines at all in the finite

element method. So, why should they be the answer in a curved space?

Slerp can be seen as a straigth line (a polynomial of degree one) in

SO(3). So for a two node element, slerp is all I need. In particular,

it has all the uniform properties, not depending on the choice of a

starting point or "pole" (one of the suggestions I received, a paper

by Johnstone and Williams, doesn't seem to satisfy my uniform

requirement).

Let me elaborate on why uniformity is so important.

Last year I published a paper [1], in which I used the rotation vector

to parameterize the rotations, and implemented a finite element based

on the interpolation of this rotation vector. Now, I know that for

this representation I may bump into singularities, but for the

applications I have in mind, this is not a serious problem. On the

other hand, as recently shown (actually in 1999), this kind of

interpolation is not invariant under superimposed rotations, and THAT,

from the point of view of a geometrically exact formulation, is a

capital sin.

For what I understand, most of the interpolation schemes devised for

SO(3) rely on a interpolation on a flat space which is then projected

back to SO(3). That is not acceptable for my needs, since it suffers

from the exactly same problems that the vector-like parametrization

have (OK, it could avoid singularity).

Now, the paper of Buss and Fillmore, refered by you (Just), seems very

promising. In fact by working directly on SO(3), or other Sd spheres,

it seems to have all the uniformity -- and simple elegance -- I need.

Although the paper seems to concentrate in applications to splines, it

seems to me that the same spherical avarages can be used to build my

desired Lagrange-like interpolation in SO(3). So, although the details

are still beyond my understanding, I see its great potential.

I am finishing my PhD thesis and I don't have much time to develop

this idea right now. However, somehow, all the currently used

implementations of geometrical exact beam theory seemed to lack that

special final touch. My intent in coming to this newsgroup, was the

hope that some one from the computer graphic field had some insight

that was absent from the computational mechanics comunity. I am not

disappointed.

Thank you all, for your valuable help. And of course, if you have

furher comments, I will be delighted to hear them.

Manuel Correa

[1] "On the differentiation of the Rodrigues formula and its

significance for the vector-like parameterization of Reissner-Simo

beam theory" in International Journal for Numerical Methods in

Engineering, Volume 55, Issue 9, 2002, pp 1005-1032 (is someone is

interested, ask me and I will send a pdf).