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Does anyone know a free (as in free beer) C, C++ or even FORTRAN library that supports operations not only NURBS curves and surfaces but also Volumes (i.e. trivariate NURBS)?

I know about open nurbs, but as far as I know it does not support NURBS volumes (feel free to correct me on that, since my experiences with that library are somewhat limited).

Matthias
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    Maybe this answer helps: http://stackoverflow.com/a/13712396/1413374. I didn't check if these support volumes, but I guess OpeCasCADe does. – Sebastian Dec 20 '12 at 00:16

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IRIT modelling environment supports trivatiate spline volumes.

It is free, but only for non commercial uses. You can contact the developer and ask about the commercial licencing terms.

Artium
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The opennurbs ON_NurbsCage class is a trivariate NURBS "volume".

The declaration of the ON_NurbsCage class is in opennurbs_nurbssurface.h and the implementation is in opennurbs_nurbsvolume.h

ON_NurbsCage provide evaluation, bounding box, and iso surface extraction tools.

This information is for the 2013-07-11 version of opennurbs that is available from http://www.opennurbs.com

Dale Lear
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The geometry module of Gismo may satisfy your need.