# How to reconstruct a surface given points {x, y, z} and the surface normals {nx, ny, nz} at these points

I have 3D points above a 2D regular grid. At those points I know the normal vector of the surface. I am looking a command or function in Mathematica, like ListPlot3D or ListInterpolation to help to display the surface corresponding to the given points and normals at those points.

• Try ListPlot3D with the VertexNormals option.
– user484
Jul 12 '15 at 0:59

In the meantime I found one way to do it. Let's say my 3d grid points are in an array grid3d[[k,i,j]], where i and j go from 1 to 8 and every element of grid3d is a triplet of {x,y,z} data, for example grid3d[[1,1,1]] is {16,16,2.3395}. My normal vectors are similarly in a norm[[k,i,j]] array and norm[[1,1,1] is {0.351765,0.113248,3.98289}, so they are not normalized. Because it is a regular grid, from grid3D[[k,i,j]] I need only the "k" component and from the norm array I need only the tangential vectors, that is the "i" and "j" components. So, I created an inData table:

inData=ParallelTable[Table[{grid3d[[k,i,j]][],{norm[[k,i,j]][],norm[[k,i,j]][]}},{i,1,8},{j,1,8}],{k,1,Length[norm]}];


Then I created another table containing ListInterpolations:

listInt=ParallelTable[ListInterpolation[inData[[k,All,All]],{k,1,Length[inData]}];


Now I am able to see the surfaces with:

Plot3D[listInt[[k]][x,y],{x,1,8},{y,1,8}]