If we define the slowness curve by the normal vector of the material point of the crystal surface divided by the growth (or etching) rate of that point. Frank’s theorem states that, in the course of crystal’s growing or etching, each point of the boundary surface moves along its own straight line (or the characteristic line) which is parallel to the normal of the slowness curve. Since he didn’t find out the speed of the crystal surface point advancing along this characteristic, the etched crystal shape can not be determined quantitatively. In this paper we derive the equation in explicit form expressing the surface moving speed along the characteristics in terms of the microscopic parameters (in atomic size) and variables such as step density, step flux, and step height. Not all the microscopic variables are measurable, so we have to drive certain equations relating the microscopic variables to the macroscopic ones (such as the orientations of the crystal boundary lattice plane, slopes of the characteristic lines, etching rates, and so on) through the kinematics theory of particles. One measurement of the macroscopic variables at any particular instant is enough to determine not only the etching (or growth) rates of the crystal but also the etched crystal shape of any subsequent instant.

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