This task on creation of a point of intersection of a straight line with the plane is classical it is aware of engineering graphics and is carried out by methods of descriptive geometry and their graphic decision on the drawing.
Instruction
1. Let's consider definition of a point of intersection of a straight line with the plane of private situation (figure 1). Direct l crosses the frontal projecting plane Σ. The point of their crossing of K belongs also to a straight line and the plane, so the frontal projection of K2 lies on Σ2 and l2. That is, K2 = l2×Σ2, and its horizontal projection is defined by K1 on l1 by means of the line of projective communication. Thus, the required point of intersection of K(K2K1) is under construction directly without application of the auxiliary planes. Similarly straight line points of intersection decide on any planes of private situation.
2. Let's consider definition of a point of intersection of a straight line with the plane of the general provision. In figure 2 in space the located the plane Θ and direct l are set randomly. The method of auxiliary secants of the planes in the following order is applied to definition of a point of intersection of a straight line with the plane of the general provision:
3. Through direct l the auxiliary secant the plane Σ.Для simplifications of constructions it is carried out there will be a projecting plane.
4. Further the line of crossing of MN of the auxiliary plane from set is under construction: MN=Σ×Θ.
5. The point K crossings of a straight line of l and the built line of crossing of MN is noted. It is also a required point of intersection of a straight line and the plane.
6. It is applicable this rule for the solution of a specific objective on the complex drawing. Example. To define l straight line point of intersection with the plane of the general provision set by a triangle of ABC (figure 3).
7. Through direct l the auxiliary secant the plane Σ, the perpendicular planes of a projection Π2 is carried out. Its projection Σ2 coincides with l2 straight line projection.
8. The MN line is under construction. The plane Σ crosses AB in a point of M. Its frontal projection by M2 = Σ2×A2B2 and horizontal M1 on A1B1 through projective communication is noted. The plane Σ crosses the party of AC in N point. Its frontal projection of N2=Σ2×A2C2, horizontal projection of N1 to A1C1. The straight line of MN belongs at the same time to both planes, and, so is the line of their crossing.
9. The point of K1 of crossing of l1 and M1N1 is defined, then by means of the communication line K2 point is under construction. So, K1 and K2 are projections of a required point of intersection K straight lines of l and the plane ∆ ABC:K(K1K2) = l(l1l2) × ∆ ABC(A1B1C1, A2B2C2). By means of the competing points of M,1 and 2.3 the visibility of a straight line of l of rather this plane ∆ is defined by ABC.