Ryaboshtan O.F., Milenin A.M., PhD

Kharkiv Petro Vasylenko National Technical University of Agriculture

 

Construction of boundary surfaces interscapulum from the set of lines

 

Let a n - parameter set of lines

 , ,                                              (1)

The surface is constructed from the set of lines (1) overlay (n -1) bonds on the option ài  with initial conditions, as which are:

1. Curve from which the line (1) intersect.

2. The surface at which the line (1) have a touch of I or II of the order of smoothness.

3. Linear strip, for which the line (1) overlap with curved line - carrier strip, and the intersection at the points concern given the tangent planes.

4. Band II order.

Consider these conditions, in order:

1. Curve m:

,   ,                                   (2)

The condition of intersection of the curves set by substituting (2) to (1) gives two equations relating the parameters

,                                              (3)

where the total number of parameters is increased by 1 (add u), and the total number of equations in 2, so that from equations (3) we can obtain the equation of connection options

,                                               (4)

providing a set of curves persechenie with the curve m. For the surface of (1) must have (n -1) such links u, hence, (n -1) initial curves in the form (2).

2. Suppose, as an initial condition given a surface

                                           (5)

The condition of perpendicularity of tangent to the lines (1) and the surface normal (5) gives the differential equation:

                       (6)

where  , , - partial derivatives .

If the system of s equations (1), (5), (6) eliminate the coordinates of the point of contact, we obtain a single equation relating the parameters:

                                                 (7)

Hence to obtain the surface of the set (1) subject to the touch I order to smooth the surfaces must have given the initial conditions (n -1) surfaces, which can be written (n -1) equations (6), which together with (6) gives the (n + 1) equations from which to exclude the n parameters. For the second-order contact a given surface (5) and must calculate the required partial derivatives  by the formulas:

,          ,

,

,                           (8)

,

substitute them in equation (3), common to many surfaces, obtained from the line (6).

The joint consideration of (1), (5) and the equation obtained after eliminating x, y, z gives the equation of connection options:

                                               (9)

Equations (7) and (9) guarantees the contact of the second order of smoothness. The above condition is equivalent to fixing the two parameters.

3. Consider the initial condition as a linear band, i.e. curve (2), in which each point given the coordinates of p and q normal to the desired surface:

,                                                            (10)

Curve (2) leads to two equations of communication (3) and equipment (10) - an equation similar to (6).

                        (11)

Consequently, each linear strip offers three constraint equations, but introduces an additional parameter u. For the solution of the set (1) must have  bands, which makes sense for odd .

4. If the initial conditions problem in the form of a strip of the second order of smoothness, i.e., curve (2), at each point where the partial derivatives are given

, , , ,                  (12)

satisfying the second-order band smoothness, then for the second order of contact of a surface and the strip (12), it is necessary, except for the two equations (3) for the curve and equation (11) for a linear band, to make one more equation, which we substitute (2) and (12). Thus, to obtain solutions of the set (1) must be set  bands of the second order of smoothness in the form of initial conditions, which makes sense when

,                                                    (13)

In solving practical problems in the set (1) can be different surfaces, changing the number and type of initial conditions. For example, when  you can take two linear bands and a single curve, or one lane, two curves and a tangent surface, etc. The total number of  parameter set (1), which can realize the given initial conditions is:

,     (14)

where , , , , - the number of vinyl strips first and second order of smoothness, with the condition of contact surfaces and the contact of the second order, respectively

, (, )

In the practical implementation of the above issues should be taken into account consistency conditions specify the initial conditions, which impose restrictions on the relative position of geometric elements.

Specific computational algorithms must be linked to a generalized algorithm for differential-parametric method and reflect the conditions of incidence and the touch of a given order.