Section of any volume geometrical figure has to be set by several parameters and so that it could be unambiguously found. The plane in space is set by three points, a straight line two. All this demonstrates that at least three parameters for this purpose are necessary. Than the secant the plane whatever these parameters were was set, they can always be counted. In the most general case – it is a corner under which the secant the plane cuts this cube and the line crossing of the plane containing the lower basis of a cube and this secant of the plane. The cube and its position are set automatically.
It is required to you
- - paper;
- - handle;
- - ruler;
- - compasses.
Instruction
1. Try to sort in more detail the general problem of creation of section of a cube. Let the secant the plane be set by a straight line of crossing of its own plane with the plane containing the lower basis of a parallelepiped l and a tilt angle to this plane f. All principle of construction illustrates the drawing.
2. Decision. Any corner in geometrical tasks on construction is set not by a corner, but its any trigonometrical function, let it will be a cotangent (ctg). It is necessary to measure in any metric system compasses solution length Nctgf = d. Transfer this size to the scale of this task and, relying on the principle of similarity of all rectangular triangles with the general acute angle, execute the following.
3. On direct l take two any points of N and F (it is desirable so that further everything proceeded in the lower basis of a cube of ABCD). From them as from the centers, carry out d radius arches to ABCD. To these arches carry out the general tangent l before its crossing with AV and CD (it is possible and further). Points of contact designate N1 and F1.
4. From N1 and F1 it is necessary to lift perpendiculars of M1 and W1 on the top basis of A1B1C1D1 which length equals N. Therefore points of crossings do not need to be looked for though it is rather simple. Now prolong M1W1 piece up to suppression from B1C1 and C1D1 in M and W respectively. Thus you found the first party of required section of MW.
5. Further it is necessary to carry out within the plane containing a side side of DCC1D1 direct WE from W point (E – its crossing from a straight line l). Crossing of WE with D1D – R point. WR piece – the second edge of required section.
6. Prolong a side edge of a cube of BB1 in the direction from In to B1. In the plane of diagonal section of a cube BB1D1D from R draw a straight line before its crossing with extension of BB1 in E2 point. From it lower a straight line before its crossing with l in E1. The straight line of E1E2 crosses side edges of a cube of A1B1 and AA1 in points of L and Q respectively. Then ML, LQ and QR are the remained required edges of section of a cube.