## Line and point in the plane

Among the main tasks solved on the plane include: the conduct of any straight line in the plane, the construction of a point in the plane, the construction of the missing projection of the point, the verification of the point of the plane.

The solution of these problems is based on known positions of geometry: the straight line belongs to the plane if it passes through two points belonging to the plane, or through one point of this plane parallel to a straight line lying in this plane or parallel to it. In this case, the known condition is used, that if the point belongs to the plane, then its projections lie on the same projections of the line belonging to the plane.

** Carrying out any straight line in the plane. ** To do this, it is sufficient (Fig. 3.10) to take projections of the plane on the projections of two points, for example, * A & quot ;, A '* and * 1 & quot ;, G * and through them * A 1 & quot ;, AΊ '* the projections of the straight line * A - I. * Fig. 3.11 projections * In 1 & quot ;, B'G * the lines are parallel to the projections * A C & quot ;, A'C '* of the side * AC * of the triangle given by the projections * A ; "A", A 'B'C'. * The direct * B * - * 1 * belongs to the plane of the triangle * ABC. *

** Construct a point in the plane. ** To build a point in it in the plane

Fig. 3.10

Fig. 3.11

Fig. 3.12

draw an auxiliary line and mark the point on it. In the drawing (Figure 3.12), the plane given by the projections * A & quot ;, A '* of the point and * B * "," ** C' , projections A 1, "A '1' of the auxiliary line belonging to the plane. It marks the projections D & D ;, D ' of the point D, belonging to the plane.**

** Drawing the missing projection of a point. ** In Fig. 3.13 the plane is given by the projections * A C & quot ;, A 'B *' C 'of the triangle. The point * D * belonging to this plane is given by the projection * D>. * The horizontal projection of the point * D should be completed. * It is constructed using an auxiliary line belonging to the plane and passing through point * D. * To do this, for example, carry out the front projection * In I D * direct, build its horizontal projection * B and mark the horizontal projection* * D '* of the point on it.

Fig. 3.13

Fig. 3.14

** Verify that the point on the plane belongs. ** To verify the belonging of a plane point, use an auxiliary line belonging to the plane. Thus, in Fig. 3.14 the plane is given by the projections A, B, A, B and C, D, C, D, parallel lines, the point by the projections E, E, The projections of the auxiliary line are drawn so that it passes through one of the projections of the point. For example, the frontal projection/* 2 "of the auxiliary line passes through the projection* * E". * Having constructed the horizontal projection/* '2' * of the auxiliary line, we make sure , that the horizontal projection * E '* of the point of non-belonging. Consequently, the point * E * does not belong to the plane.

## Directs of special position in the plane - the main lines of the plane

Direct, occupying a special position in the plane, include the horizontals, fronts, profile straight lines and lines of the greatest inclination to the planes of projections. These lines are called ** main lines of the plane **.

** Horizontal ** is a straight line that lies in the plane and is parallel to the plane of the projections * ni. * Fig. 3.15, the projections of the horizontal are drawn through the projections * C ", C * of the point * C * and * 1 *, * 1 '* of the point * 1 * of the straight line * AB * of the plane given by the projections of the point * C * and the line * AB. * Front projection * C 1 * is parallel to the axis * x. *

** Frontal ** - a straight line that lies in the plane and is parallel to the plane of the projections * n * 2. In Fig. 3.16, the projections of the frontal are drawn through the projections * 1, * and * 2, * of the points i and * 2 * of the projections * A . A 'B *', * C & D; C'D '* of parallel lines * AB * and * CD * of the given plane. The horizontal projection * 1 '2' * of the front is parallel to the axis * x. *

From the three lines of the greatest inclination to the planes of projections, let us note the line of the greatest inclination to the plane H |. This line is called the slope line. ** The slope line ** is a straight line, lying in the plane and perpendicular to its horizontals. In Fig. 3.17 projections * A 2 & quot ;, A '2' * the slope lines * A - 2 * in the plane of the triangle with the projections * A C & quot ;, A 'B 'C' * are perpendicular to the horizontal with projections * C 1 * & quot ;, * C'1 '. *

Fig. 3.15

Fig. 3.16

Fig. 3.17

First, a perpendicular * A '2' * to the projection * Cl '* of the horizontal is drawn from the projection * A' * on the horizontal projection, the frontal projection * 2 * points * 2 * and through it the front projection * A 2 * of the slope line is carried out.

The angle between the skate line and its horizontal projection is the linear angle between the plane to which the slope line belongs and the plane of the projections π,.

## thematic pictures

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