## Construction of points of intersection of a straight line with a polyhedron surface

The construction of points of intersection of a straight line with the surface of a polyhedron reduces to constructing a line of intersection of the polyhedron with the projecting plane into which the given line is enclosed. In Fig. Figure 6.11 shows the construction of the projections * E & quot ;, E '* and * F & quot ;, F' * of the points of intersection of the line with the projections * N *, * M 'N' * with the lateral faces of the pyramid. The pyramid is given by the projections * G, G '* of the vertex and * A C & quot ;, A'B'C * of the base. The straight line * MN * is enclosed in the auxiliary frontal projecting plane y (y ). The horizontal projections * E '* and * F' * of the sought-for points are constructed at the intersection of the projection * M'N '* with the horizontal projections * 1' 3 'and 2 '3' * segments along which the y-plane intersects the lateral faces of the pyramid. Frontal projections * E and* * F * are defined by communication lines.

## Mutual intersection of polyhedra

An image of intersecting prisms * A * and the pyramid * B * is shown in Fig. 6.12. The line of their intersection passes through the points * 1,3,4, 6 * the intersections of the edges of the pyramid with the faces of the prism and the points * 2, 5 * of the intersection of the edge of the prism with the faces of the pyramid. In the general case, at the intersection of polyhedra, we obtain a three-dimensional closed polygonal line, which in some special cases can turn out to be flat.

When constructing the intersection of polyhedrons, two methods and their combinations are used:

1. The points of intersection of the edges of one polyhedron with the faces of the other and the edges of the second with the faces of the first are constructed. Through the points constructed in a certain sequence, a polygonal line of polygonal data intersects. In this case, the straight lines are drawn only through those points constructed that lie in the same face.

Fig. 6.11

Fig. 6.12

2. Straight lines are constructed along which the faces of one surface intersect the faces of the other. These segments are links of a broken line intersecting polyhedral surfaces with each other.

Thus, the construction of the line of intersection of two polytopes reduces either to the construction of the line of intersection of two planes with each other, or to the construction of the point of intersection of a straight line with a plane. Both of these problems are considered above. In practice, both methods are usually used in combination based on the condition of simplicity and ease of construction.

As an example, consider the construction of the intersection line of a truncated regular quadrangular pyramid and an obliquely arranged trihedral prism (Figure 6.13, * a). * Before proceeding to the constructions, analyze the mutual position of the polyhedra and their location relative to the projection planes . In this case it is obvious that polyhedra can intersect only along lateral faces. The edges of the prism and the side edges of the pyramid are parallel to the plane π2, the bases of the pyramid are parallel to the plane π ,. The lower face of the prism and its bases are perpendicular to the plane π2.

The specified features of the location of the prism and the pyramid determine the most rational way of constructing the line -

Fig. 6.13

sections of their surfaces to the descendants of the intersection of the edges of the prism with the faces of the pyramid and the lateral edges of the pyramid with the prism edges.

The constructions are shown in Fig. 6.13, * b. * Consider them for the left part of the drawing (from the axis of the pyramid). Projections * 1, 1, 2, 3, 4, 4, of the points of intersection of the edges of the prism with the faces of the pyramid are found by carrying through them the front planes p (p '), , a (a '), y (y'). They cross the left side faces of the pyramid along fronts - straight lines parallel to the left edge of the pyramid. The position of their frontal projections is determined by the horizontal projections* * 2/** ',* * 22' * and * 24 '* of the intersection points of the horizontal projections P', a 'and y' planes p, a, y with a horizontal projection of the base of the pyramid. In the intersection of the frontal projections of these lines with the frontal projections of the edges of the prism, frontal projections * 1 & quot ;, 2 and 4 * "points of intersection of the edges of the prism with the left sides of the pyramid. The horizontal projections/* 2 ', 4'. *

Projections * 3 & quot ;, 3 '* the points of intersection of the edge * AD * of the pyramid with the upper trailing edge of the prism are found with the help of the auxiliary frontal plane q (y), which is drawn through this edge. The plane n intersects the face of the prism along a line parallel to the edges of the prism and passing through the point * 23 * on the basis of the prism. In the intersection of the frontal projections of this line and the edge * A D *, the frontal projection * 3 * of the intersection of the specified edge with the back top face of the prism and the horizontal projection * 3 * 'are found on the communication line. With the lower edge of the prism perpendicular to the plane * n * 2, the edge * AD * intersects at the frontal projection point 5 & quot ;. In the projection relation on the projection * A 'D' * its horizontal projection 5 'is constructed.

Thus, the projections of the intersection points of all edges of the prism with the left sides of the pyramid * - 1, 12 *, * 2 ', 4 &', 4 '* and the edges of the/lZ) pyramids with two faces of the prism - * 3 ", 3 '* and 5 & quot ;, * 5' * are constructed. We connect the projections of points belonging to one face and obtain projections * 1> 2> 3 "4" 5 " 1 & quot ;, Г2 '3'4'5'1' * the broken line of intersection.

Drawing the projections * 6 "7" 8 "9" on the right side of the drawing. 10 "6", 6'7'8'9'10'6 '* the intersection lines are similar. The order of construction is illustrated by arrows.

After constructing the projections of intersection lines of polyhedra, project the remaining parts of the edges of polyhedra.

Note that the front and back edges of the pyramid do not intersect the surface of the prism.

## Deployment of face surfaces

** The surface of a polyhedron ** is called a flat figure obtained by aligning with the plane of all its faces. Deployment of the faceted surfaces is performed for cutting the sheet material in the manufacture of parts or determining the surface area of parts covered with various materials. Determination of the area is important for various coatings, both for decorative purposes and for the purpose of imparting certain properties to the surface, for example, increased electrical conductivity, and also for various chemical methods of surface treatment.

Fig. 6.14

To build a sweep of a face, you need to determine the dimensions of its faces. We note that the construction of any face of a polyhedron can be carried out by breaking it down into triangles. The length of the sides of the triangle in turn can be determined by any of the known methods.

** The unfolding of the surface of the pyramid. ** The development of the sweep of the side surface of the pyramid can be carried out in the following sequence:

Fig. 6.15

1) determine the length of the edges and sides of the base of the pyramid;

2) Perform a sweep drawing by sequential construction of triangles - faces of the pyramid.

An example of a surface sweep of a triangular pyramid * GABC * is shown in Fig. 6.14. For convenience of construction, the lateral edges of the pyramid are extended to the intersection with the plane π. This allowed us to determine on the horizontal projection the length of the segments/* '22'3', 3'1 '* of the new base of the pyramid. The length of the lateral edges * G-1, G - * 2, * G-3 * is found by rotating them around the vertical axis - the segments * G "l", G "2" G 3 . * The segments * G A & quot ;, G B & quot ;, G C are found on them. On the basis of the found segments, a lateral surface scan* * G * 0/020J0 * h * and then * G * a * A * 0 * B * a * C * 0 . On the segment * A * 0 * C * 0, the natural value of the triangle * AB * () * C * 0 on the sides * In* * and* * C * and * B * a, found by the method of a rectangular triangle (see Figure 2.8).

** Creating a prismatic surface scan ** can be done in several ways - normal cross section, triangles.

For a normal section method, it is advisable to build the sweep of the prismatic surface in the following order (Figure 6.15):

- intersect the prismatic surface by an auxiliary plane perpendicular to its edges a _L * 1-2 (ABCD * - normal section);

- Expand the constructed polyline (D0 * B , C * 0 * D * 0 *) * of the intersection of the auxiliary plane with the prismatic surface, determining the length of its segments (AD * B * U * C * 0 *, C * 0 * D * 0);

- on the perpendiculars to the unfolded intersection line * (A * 0 * D * 0 *) * to lay the length of the segments of the edges of the prismatic surface * (A * a * I * 0 *, A * a * Z * i *, B * 0 * 3 * 0 , DL, C0J0, * C * 0 * 6o, D * 0 * 7 * 0 *, D * 0 * 8 * 0 *) * and connect their ends by straight lines.

According to the method of triangles, the development of the prismatic surface is as follows: quadrangles (faces) are divided by diagonals into triangles; determine the lengths of the sides of the triangles; perform a sweep drawing by sequential construction of triangles to which the faces are broken.

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