> 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.
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
Let me elaborate on why uniformity is so important.
Last year I published a paper , 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
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
Thank you all, for your valuable help. And of course, if you have
furher comments, I will be delighted to hear them.
 "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).