i found that the coordinates of the points in the plane changes (x, y) ___ (x ', y') but the length is always constant !!!!!!????

I guess most if your question is difficult to understand because of the terms used. A grain boundary already is a plane. What do you see from a grain boundary is only the trace of this plane. Sometimes both terms are incorrectly used or simply mixed.

Then it should be clear that the grain boundary (let us assume a flat plane) is a plane regarding grain A and grain B. Both planes described as (hkl)A and (hkl)B are different. In order to find out which plane it is you need to transfer a plane into a crystal coordinate system. I guess you should find the respective formula in Article A little geometric crystallography for EBSD and related techniques

It does not become really clear whether you actually have the plane, however, since you are talking about to GB I assume you have traces of the same plane visible on two different reference planes. This plane you can derive then with respect to these reference planes so that you should have an exact description of this plane with respect to X,Y,Z of your sample. Since the orientation of both grains are hopefully also described by the same reference coordinate system you can apply the orientation matrix to derive the (hkl)A or (hkl)B.

thank you for your reply

a grain boundary is defined by his plane (hkl) and a desorientation (teta) around an axis [uvw]

to anderstand my question better, i give a simple exemple:

we consider symmetric tilt grain boundaries (STGBs) in magnesium oxide (210)[001] (desorientation alpha= 53.3° or teta= 26.6°)

how to calculate the surface of this GB

Once again thank you

This is not really true. In your example image it only looks like this since the rotation axis is perpendicular to the plane normal. But this is a very special case... A grain boundary is simply a plane which has actually nothing to do with the misorientation which is described by an (rotation) axis and the corresponding angle. There is no plane in this definition!

sigma 5 (210) [001]

plane (210) , axes [001]

Variables that define a grain boundary. xA; yA; zA and xB, yB, zB are the axes of the coordinates parallel to crystallographic directions in grains A and B, respectively. teta is the rotation axis and  is the rotation (misorientation) angle necessary to transfer both grains to an identical position. n determines the orientation of the grain boundary plane

see this article

exemple h1k1l1 = h2k2l2

These are CSLs, i.e. very special boundaries which are mainly only defined for cubic phases. However, even there certain deviations exist which are not small, e.g. Sigma 3 with a misorientation angle tolerance of about 8° (Brandon criterion). AND: the definition of Sigma 3 does not automatically describe a boundary parallel to {111}. It simply defines the geometric condition that two lattice planes will be roughly parallel to each other after rotation of one grain by the respective angle around a specific axis. However, this does not mean that the grain boundary is the plane defined by Sigma3! The so called habit plane can be and is usually different. Assume a grain inside another grain, then they have an exact orientation relationship. However, the surrounding grain boundaries can have a continuous curvature.

ello sir Gert Nolze, I thank you for your help, you had enlightened me some points. I would like to know what is a mathematical relationship of the surface of an grain boundary

There is a 5 parameter approach for a grain boundary. The respective authors are Randle and Rohrer.