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Situation:

  • We have a bendable, non-stretchable surface, like a piece of cloth, with a regular grid on it.
  • Unknown manipulation of the surface is done while preserving it's structure
  • We recieve 3 dimensional normal vectors from each of the grid points of the surface (but not their coordinates) (V1,...VN)
  • Length of each grid unit on a flat surface is equal (L).

The question: What methods could be used to reproduce the surface from these vectors and L?

Reduced question: If we have two (2-dimensional) vectors (angles) and know the length of the curve connecting them, as well as that one of these vectors start from coordinate (0;0) in 2-dimensional plane. How can we approximate the position of the second vector and preferrably the whole curve (assuming, that the curve is quadratic)?

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If say we have two points $p_1$ and $p_2$, we know the normal vector $n_1$ and $n_2$ associated with them respectively, but we do not know $p_1$ or $p_2$'s coordinate? –  Shuhao Cao May 31 '13 at 0:35

1 Answer 1

Just to clarify; you say that you receive the normal vectors for each grid point. Does that mean that you know which vector is associated with any grid-point? If that is the case, then for any four grid-points that make up a grid-square you should have access to the associated normal vectors. You also imply that you have no access to the grid-point coordinates, since the surface manipulation is unknown. Am I right so far? The L you're referring to; is that after you stretch the "cloth" and its grid-points on a flat surface?

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Yes, I know which vector is associated with which grid point, otherwise I am sure there is no solution. Also you are reight about your other assupmtions. In a less mathematical and more engineering point of view the problem can be imagined as level sensors swen to a non-stretching fabric, and I want to determine one of many solutions, knowing the angle of each sensor and gravity, as well as their placement on the cloth. Main assumption - the surface is the least deformed one possible. –  Krišjānis Nesenbergs Jan 19 '12 at 14:01

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