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#### Barycentric Coordinates on a Cuboid ?

```Hi there,

Assume one point is within a cuboid,
The point should be represented by a linear combination of all vertices
of the coboid.
Could someone tell me how to calculate the weight for those vertices?
or any reference paper?

I only know the same case within a convex polygon and a tetrahedron..
subdivide the cuboid into several tetrahedra?

Thanks very much : )

Best Regards,
Brian

```
 0
cuckoo (3)
1/21/2005 5:30:20 AM
comp.graphics.algorithms 6674 articles. 1 followers.

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```Brian <cuckoo@cs.nchu.edu.tw> wrote:

>     Assume one point is within a cuboid,
>     The point should be represented by a linear combination of all vertices
> of the coboid.

Such a linear combination would be infinitely ambiguous, which means
the term "barycentric coordinates" from your subject line wouldn't
really apply here.  Coordinates are supposed to be a one-to-one
mapping.

>     subdivide the cuboid into several tetrahedra?

Yes.  At least four of them, to be precise.  Even then, there's some
ambiguity, because different subdivisions of the cuboid would yield
different coordinates.

A subdivision exploiting symmetries of the cuboid should be less
ambiguous.  E.g. you could divide the cuboid into 6 rectangular
pyramids (one per face, with the centroid of the volume as their
apex), and each of those into 4 triangular pyramids by adding the face
center.

--
Hans-Bernhard Broeker (broeker@physik.rwth-aachen.de)
Even if all the snow were burnt, ashes would remain.
```
 0
broeker (2903)
1/21/2005 10:11:45 AM
```On 21 Jan 2005 10:11:45 GMT, Hans-Bernhard Broeker
<broeker@physik.rwth-aachen.de> wrote:
>Brian <cuckoo@cs.nchu.edu.tw> wrote:
>
>>     Assume one point is within a cuboid,
>>     The point should be represented by a linear combination of all vertices
>> of the coboid.
>
>Such a linear combination would be infinitely ambiguous, which means
>the term "barycentric coordinates" from your subject line wouldn't
>really apply here.  Coordinates are supposed to be a one-to-one
>mapping.

Scrap that graphics library; it uses homogeneous coordinates, which
are an infinity-to-one mapping!

As for whether homogeneous coordinates can be generalized beyond the
traditional simplices, some authors seem to think so; among others:

Warren et al. Barycentric Coordinates for Convex Sets
Meyer et al. Generalized Barycentric Coordinates for Irregular
N-gons
<http://www-grail.usc.edu/pubs.html>

Floater et al.  A general construction of barycentric coordinates
over convex polygons
<http://heim.ifi.uio.no/~michaelf/pubs/abstracts.html>

```
 0
nobody-here (1531)
1/21/2005 12:59:18 PM
```On Fri, 21 Jan 2005 06:59:18 -0600, Just d' FAQs <nobody-here@mac.com>
wrote:
>As for whether homogeneous coordinates can be generalized beyond the
>traditional simplices, some authors seem to think so; among others:

Doh. They're generalizing "barycentric" coordinates, of course.

We routinely *do* use homogeneous coordinates despite their oo:1 way
of representing points. And given a fixed list of point positions, any
assignment of weights (masses) to those points gives a center of mass,
though many different weight assignments can give the same center. Our
barycentric coordinates of the usual sort ("bary" meaning weight) have
happy properties like *uniquely* representing a given center, because
the fixed points are independent (none of the fixed points is a center
of mass in terms of the others) and the weights sum to 1. Similarly,
the usual coordinates for a vector refer to a basis, which is a list
of independent vectors spanning the space; thus every vector will have
unique coordinates, which we like. But to throw away all forms of
coordinates which are not 1:1 would leave us painfully hamstrung.

```
 0
nobody-here (1531)
1/22/2005 11:31:08 AM

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