Calculation of coordinates on a curved surface

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Hello dear community,
I am currently facing a problem that probably requires mathematical understanding. Since I'm looking for an algorithm in MATLAB, or I want to use an existing function, I ask my question here.
It is about finding coordinates on a curved surface. With STL-read I import a CAD file and get all curved surfaces, points and normal vectors of the corresponding model. At the same time I create a two-dimensional rectangle which is to be placed on the curved surface. First I search for the point of intersection between the centre of the two-dimensional rectangle and the curved surface using the xy-coordinates. Now I set the z-coordinate of the centre and the normal of the centre equal to the intersecting surface. Now I want to determine the coordinates of the remaining four points. I can determine the z-coordinate with the same procedure, but not the xy-coordinates. Because the xy-coordinates change due to the curvature, I am now looking for a possibility to determine them.
My first approach was to rotate the coordinates with the rotation matrix around the z-axis with the angle between the normal vectors of the centre and the other point of the rectangle. But unfortunately I got some nonsense.
Do you have an idea for an already implemented function or an algorithm? Here is another illustration that might help to show the problem:
Thanks for your help.
Best regards
  2 Comments
Nicolas Kaiser
Nicolas Kaiser on 19 Oct 2020
Edited: Nicolas Kaiser on 19 Oct 2020
Thank you very much for your quick answer, it worked well. I used the point2trimesh, which calculates exactly the closest point (point2trimesh).
Another question you might be able to answer. When I have determined the center point I get the normal vector of the surface. How can I now rotate my 4 coordinate points of the rectangular surface into the normal vector so that the 4 points taken as surface have the same normal vector as the surface of the intersection with the center of the rectangular surface. I have already tried a lot of things from rotation matrices to Euler angles. Probably I just can't get the right angle to calculate. Do you have an idea?

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