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L. V. Shmeleva et al.
surface. So far as process observation passes in a laboratory system of coordinates, it
is necessary to establish a relation between these systems. The laboratory system of
coordinates is immobile relative to the surface before and after its destruction, as well
as, for example, relative to the point of center of a cross section of the stimulating
radiation flux.
3 The Relationship Between the Coordinates of the Local
and Laboratory Reference Systems
To establish a connection between the two reference systems, we consider Fig. 4,
which geometrically explains the problem.
We arrange the laboratory reference system in such a way that its origin is at
point O. Then the surface of the crater, which is formed under the influence of the
flow, will be determined by some surface that changes in time. In Fig. 4, this surface
is determined by the radius vector r. Obviously, r(t) = x(t)e x + y(t)e y + z(t)e z ,
moreover, by the general definition of a surface: z(t) = S(t, x(t), y(t)), where
S(t, x(t), y(t)) is the form function of this surface (for each moment of time, it
determines a certain dependence of z on x and y). In the future, this function will be
called the “crater form function.” In Fig. 4, the radius vector r ends at the starting
point of the vector n. With this point, we will associate some local reference frame.
In order to establish a connection between a laboratory system with a center at
point O and a local system with a center at the point, where vector n begins, it is first
necessary to determine the relationship between the unit vectors of these systems. If
the normal n to the surface is identified with the unit vector e ζ of the local reference
frame, that is, if put n ≡ e ζ , then by definition [15] this normal in the laboratory
reference frame will be determined by the relation n ≡ e ζ = N/|N|, where
Fig. 4 Illustration of the
relationship between local
and laboratory coordinate
systems
L. V. Shmeleva et al.
surface. So far as process observation passes in a laboratory system of coordinates, it
is necessary to establish a relation between these systems. The laboratory system of
coordinates is immobile relative to the surface before and after its destruction, as well
as, for example, relative to the point of center of a cross section of the stimulating
radiation flux.
3 The Relationship Between the Coordinates of the Local
and Laboratory Reference Systems
To establish a connection between the two reference systems, we consider Fig. 4,
which geometrically explains the problem.
We arrange the laboratory reference system in such a way that its origin is at
point O. Then the surface of the crater, which is formed under the influence of the
flow, will be determined by some surface that changes in time. In Fig. 4, this surface
is determined by the radius vector r. Obviously, r(t) = x(t)e x + y(t)e y + z(t)e z ,
moreover, by the general definition of a surface: z(t) = S(t, x(t), y(t)), where
S(t, x(t), y(t)) is the form function of this surface (for each moment of time, it
determines a certain dependence of z on x and y). In the future, this function will be
called the “crater form function.” In Fig. 4, the radius vector r ends at the starting
point of the vector n. With this point, we will associate some local reference frame.
In order to establish a connection between a laboratory system with a center at
point O and a local system with a center at the point, where vector n begins, it is first
necessary to determine the relationship between the unit vectors of these systems. If
the normal n to the surface is identified with the unit vector e ζ of the local reference
frame, that is, if put n ≡ e ζ , then by definition [15] this normal in the laboratory
reference frame will be determined by the relation n ≡ e ζ = N/|N|, where
Fig. 4 Illustration of the
relationship between local
and laboratory coordinate
systems