# deal.II and curved faces: how can I get the curved description

I'm not a deal.II expert, and while studying step-6 I was reading the documentation of the MappingQ1 class in the deal.II documentation. At some point in the description (https://www.dealii.org/current/doxygen/deal.II/classMappingQ1.html) , there's written:

Note, however, that in 3D the faces of a general, trilinearly mapped cell may be curved, even if the edges are not

I have two main issues with this:

1. I don't see why this is mathematically true. I don't see how the image of a trilinear map $$\phi(x,y,z)$$ gives some curved boundary.

2. Considering the case of an hyper_ball, i.e. a ball in 3D. I see indeed that boundary faces are not straight quadrilaterals (see the attached image). How can I retrieve the "curved" part of the face, from a software standpoint? I mean, I can of course ask to the Mapping object the vertices of the cell or face, but this are not enough to describe the curvature... I don't see how one can query the "curved part" in terms of Points!

I understand somehow this has to do with the concept of Manifold, but I'd like to see how in the code one can actually got this "curved" information.

Any help is highly appreciated.

• This is a question for the deal.II forum. You may want to make clear what exactly you mean by "curved part" that you'd like to retrieve. State explicitly what kind of object you would like to know (e.g., the curvature, the curvature tensor, etc.) Jul 11, 2022 at 14:46
• With "the deal.II" forum, you are referring to the mailing list I found on dealii.org, right? I think I managed to answer to the first question: the fact that the map is linear only implies that the edges are straight, while the interior part is not. Indeed, if the 4 vertices lie in the same plane, then the face would be of course straight, otherwise is not, like in the picture. For the second, what I wanted to understand is if there's a way to get those 3D triangles that are appearing in each boundary face, but yes, this is a question for the forum. Jul 12, 2022 at 11:26
• Yes, the mailing list. Jul 12, 2022 at 15:40
• The two triangles are an artifact of visualization. Paraview of Visit splits each quadrilateral into two triangles for purposes of visualization, but they are not something deal.II knows about internally. Jul 12, 2022 at 15:41