> Then I was mislead by the conclusion of your introduction:
> | A simpler construction is provided here when the polyhedron faces
> | are triangles. A consequence of the formulas as derived in this
> | document is that they require significantly less computational time
> | than does Mirtich's formulas. I suspect that for nontriangular
> | faces, Mirtich's formulas are reducible to simpler expressions.
A paper does not consist solely of its introduction. Clearly
your background is such that you can see the idea as discussed
in the remainder of the document applies to simple polygon
faces. But my purpose for the paper is to stress the case of
You can argue that for simple polygon cases my approach
requires you to construct planar coordinates for each vertex
of the face, whereas Mirtich's construction does not. I have
not investigated if the coordinate construction plus mass/inertia
calculations are more/less than Mirtich's. Even if you had the
face triangulation available so that coordinate construction is
not necessary, the operation count is significantly large enough
in Mirtich's approach that you would want to find the
break-even point where the number of triangles finally causes
my method to use more cycles; that point is dependent on
which case you arrive at in Mirtich's formulas.
For rigid bodies, it is quite irrelevant which formula you use
since you compute the mass and inertia tensor (in body
coordinates) just once. For deformable bodies where you
have to compute the inertia tensor each time step, using
simple polygon faces is problematic. At one time step
you have a simple polygon face. At the next time step
the vertices of that face are most likely not coplanar. What
are you going to do? No problem when you use triangle
Quote:> Agreed, triangle meshes predominate. But OP did not restrict to that,
> and otherwise your code needs adaptation which, while simple for the
> experienced, may not be so for OP.
Neither did the OP restrict his problem to rigid bodies.
If his faces are not triangular, he has to modify Mirtich's
code as well. Maybe the OP has deformable bodies with
trimmed surface patches. In that case, neither Mirtich's nor
my approach will help. I am not the one who posted the
reference to Mirtich's paper :)
Quote:> I have no complaint. In fact, it bothered me that Mirtich thought it
> necessary to project the faces.
I have no complaint about Mirtich's paper. Since the "area of
a 3D polygon" problem is solved by projecting to the appropriate
coordinate plane, it is only natural you would try the same idea
for mass/inertia. Neither did I like the projection of faces. That
is why I investigated the alternative. Regardless, none of this is
rocket science. The application of Green's theorem is a topic in
an undergraduate calculus class. I suspect Green himself investigated
all these applications that folks seem to "discover" nowadays (more
appropriate might be to say "rediscover").
Quote:> ...Perhaps I triggered this by saying his code was efficient (his claim,
> with numbers). It could be ten times slower and still suffice for OP
> I wager. What really worries me here is not the relative merits of
> your code versus Mirtich,...
Yes, this is how it was triggered, but your response to Christer's
post *was* about the relative merits :)
>... but the sense that we're killing a fly with a cannon.
> To remind us all of the original request, OP was merely trying to find
> some kind of "center" for a body, specifically mentioning a cube as a
> simple example. :)
The OP did mention some faces of the cube might be subdivided to
have more triangles than other faces. If he plans on computing the
center of mass using the triangle face information, our discussion is