## The reciprocal position of a straight line and a plane, two planes

## Intersection of a straight line with a projecting plane

When constructing the point of intersection of a straight line with a projecting plane, proceed from the position discussed above that a plane perpendicular to the plane of projections is projected onto it in the form of a straight line (see § 3.2). Therefore, on this line there is also a corresponding projection of the point of intersection of the given line with the projecting plane.

In Fig. 4.1 *, b * the horizontally projecting plane p is given by the traces of P and p '(the visual image is in Fig. 4.1a), the straight line * AB * is in general position. The point of their intersection coincides simultaneously with the straight line * AB * and the plane p. Consequently, its horizontal projection * K '* belongs simultaneously to the horizontal track P' and the horizontal projection of the line, that is, the point of their intersection. On the horizontal projection * K '* of the point * K * on the front projection * A B * we find the frontal projection * K' * > points of intersection. From the horizontal projection it is seen that to the right of the project -

Fig. 4.1

* K * 'the projection * K' B '* is between the axis * x * and the trace β', τ. ie the plane β is in front of the line * AB * and closes it on the frontal projection. Conditionally, the plane is considered to be opaque, so in the drawing, the front projection * K In * is shown for clarity as invisible by a dashed line. In complex drawings, dashed lines are not used.

** Some conventions of the image of invisible points, lines, planes. ** Conditionally believe that this plane is opaque. Therefore, points, lines, sections of another plane, located between the plane of projections and a given plane, are invisible to the observer, between which and the plane of projections are the objects depicted. If the lines, points, sections of the other plane are between this plane and the observer, they are visible and close the points, lines, parts of the given plane, lying on some projecting lines.

The visible line segments are represented by solid lines, invisible - by dashed lines.

Line visibility analysis is usually performed by analyzing the visibility of points, as done in the analysis of the visibility of points on crossed lines (see § 2.4, Figures 2.19, 2.20).

An example of constructing the point of intersection of a line in general position with the projections E & F; E'F ' with a horizontally projecting plane in the form of a triangle with projections * A In & quot ;, A 'B *' * C '* is shown in Fig. 4.2.

The front projection * M * of the intersection point * M * is constructed from its horizontal projection * A */', which is the intersection point of the horizontal projections * E 'F * 1 line and * A'CB' * of the triangle. Similarly visibility is noted: to the left of the point * M * the plane is a triangle -

Fig. 4.2

Fig. 4.3

Fig. 4.4

nick * ABC * when viewed from the front, closes a segment of a straight line, i.e., on the front projection to the left of the point * M * the line is invisible to the boundary of the projection of the plane of the triangle * A ; . *

Drawing in the drawing the point of intersection of the front-projecting plane, given by the traces ", a ', and the line with the projections * A" B *, * A' B * ', is shown in Fig. 4.3.

The front projection * of the * intersection point is the frontal track intersection point a and frontal projection * A B * direct. The horizontal projection * K '* is found on the horizontal projection * A' B '* of the straight line on the communication line. To the right of the point * K * the straight line * AB * (the ray * KB) * is closed from above by the plane a, therefore on the horizontal projection to the right of the point * K the projection* * K'B '* of the line is shown invisible.

A similar construction is shown in Fig. 4.4 for the point of intersection of the straight line * AB * with the horizontal plane y (yn). The frontal projection of the intersection point is the intersection point of the trace of y and the projection of i. In "." The horizontal projection * K '* is constructed on the horizontal projection * A' B 's * using the communication line. On the frontal projection, it is clear that to the left of the point * K * the projection * of K * is under the projection of & quot ;, that is, to the left of the point * K * the straight line * АВ * (the ray * КА) * is under the y plane. On the horizontal projection to the left of the sharpening * K <**, the projection* * K'A '* of the line is shown invisible.

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