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I recently meet a problem of finding the points on the boundary of a 3D object, e.g., the points on a cylinder below:

a1 = Cylinder[{{0, 0, 0}, {1, 0, 0}}, 1];Graphics3D[a1]

I'm going to work on some more complex objects, which is to be obtained by ImplicitRegion, but I guess the process should be similar.

My question is, I'd like to get a list of the boundaries points similar to what we will get in a discretization method. E.g., if I use the spherical coordinates, I may discretize the two angles and calculate the corresponding radius, or if I use the Cartesian coordinates, I may discretize z and x then seeking for the corresponding y.

So how do I get such a list of data? I prefer to get the 3D list if possible. And how do I set the desired accuracy, like the increment I set in the discretization method? My wild guess is to generate a mesh on it, e.g. using the DiscretizeRegion, but I'm not sure if it is going to work or how. Thanks a lot.

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    $\begingroup$ When you say "the points"... well, of course there is an infinite number of points on the boundary of your cylinder. What do you really want? And why? $\endgroup$
    – David G. Stork
    Commented Mar 6, 2017 at 17:15
  • $\begingroup$ @DavidG.Stork, hi, what I mean is a list of points, that are similar to what I would obtain via a discretization method, as I described in the problem. and I would like to be able to control how the points are distributed, like I discrete the x and z values with fixed increment, and find the corresponding y, and I preferably I'd like to get a 3D list of dimension n_zn_x*3 instead of a 2D list of (n_zn_x)*3. Thanks. $\endgroup$
    – larry
    Commented Mar 6, 2017 at 18:12

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You can first discretize the boundary of the region with BoundaryDiscretizeRegion, and then get the points with MeshCoordinates, like so:

ClearAll[cyl, its, trans];

cyl = Cylinder[{{0, 0, 0}, {1, 0, 0}}, 1];
pts = MeshCoordinates@BoundaryDiscretizeRegion@cyl;

ListPointPlot3D[pts, BoxRatios -> {1, 1, 1}]

points

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    $\begingroup$ Hi, this is basically what I'm looking for, but if possible, I'd like to be able to control how the points are distributed, like I discrete the x and z values with fixed increment and specify their values then find the corresponding y. Do you know how to control such things? Preferably I'd like to get a 3D list of dimension n_z*n_x*3 instead of a 2D list. I guess it's probably an option of BoundaryDiscretizeRegion, e.g., the MeshCellShapeFunction or the Method? And it would be great if I could do it both in Cartesian and Spherical coordinates. Thanks. $\endgroup$
    – larry
    Commented Mar 6, 2017 at 18:18

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