Building polyhedron section Practical lesson / by M.Art(Arch) George V. Pereboev December 2016.

Building polyhedron section Practical lesson by M.Art(Arch) George V. Pereboev December, 2016

The intersection of the surface plane (construction section) At the intersection of the surface or a plane geometric figure plane figure is obtained, which is called a section. The cross section of the surface in a plane generally a curve (or straight line, if the plane intersected) belonging to the cutting plane. Section Definition line projections should begin with the construction of reference points - points on the essay-forming surface (point of defining the boundaries of visibility curve projections), points at distant extreme (minimum and maximum) distance from the projection planes. Thereafter, the arbitrary point of the section line. If the arbitrary points are determined using the same reception for finding the reference points it is usually necessary to use different methods. In the future, the construction of section of the surface and the line of intersection of surfaces will be shown how to find support and arbitrary cross-section points. Building polyhedron section Practical lesson / by M.Art(Arch) George V. Pereboev December 2016.

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Polyhedra sections construction. Polyhedron is called a spatial figure bounded by a closed surface consisting of the compartments of planes in the shape of polygons (triangles in the particular case). Parties polygons form the edges, and polygons plane - faces of the polyhedron. Projections of sections of polyhedra, in general, are polygons whose vertices belong to the ribs, and the parties - the faces of a polyhedron *. Therefore, the task of defining the cross section of the polyhedron can be reduced to a multiple decision problem of determining the direct (polyhedron edges) meeting point with the plane or to the task of finding a line of intersection of two planes (and the faces of the cutting plane). The first solution is called the way of the ribs, the second - in a manner of faces. Which way should give preference, it is necessary to decide in each particular case. Building polyhedron section Practical lesson / by M.Art(Arch) George V. Pereboev December 2016.

Example 1. Determine the cross-section of four-sided prism ABCDEFGH plane by α (a || b). Building polyhedron section Practical lesson / by M.Art(Arch) George V. Pereboev December 2016.

Solution: Solving this problem means the ribs. To conclude this rib in the horizontal plane projecting γ1, γ2, γ3, γ4: γ1 ⊃ (AE), γ2 ⊃ (BF), γ3 ⊃ (CG), γ4 ⊃ (DH). Find the projection of the lines of intersection of these planes with the plane α (lines 1, 2, 3, 4, 5, 6, 7, 8). We note the point of intersection obtained direct from the corresponding edges of the prism K = (1, 2) ∩ (AE), L = (3, 4) ∩ (BF), M = (5, 6) ∩ (CG), N = (7,8) ∩ (DH). Quadrilateral KLMN - the desired section. The solution is greatly simplified if the cutting plane or the plane faces (if polyhedron refers to a prism) takes projecting position. Building polyhedron section Practical lesson / by M.Art(Arch) George V. Pereboev December 2016.

Example 2. Determine the cross-section of a three-sided pyramid SABC horizontally projecting plane α. Building polyhedron section Practical lesson / by M.Art(Arch) George V. Pereboev December 2016.

Solution: To answer to the task does not require any additional constructions. On the basis of the invariant properties of the horizontal projection of the cross-section plane of the pyramid and should belong to the trail h0α plane. Therefore, it suffices to note the points M , N, L , in which the horizontal trace h0α cutting plane α intersects the horizontal projection of the pyramid edges. Front projection section of the triangle vertices are defined by their horizontal projections. It suffices dots M , N, L hold link - vertical lines and mark the point of intersection with the frontal projections of the ribs of the pyramid. * In the particular case of the polygon section can be projected into a line segment (see. Invariant property). Building polyhedron section Practical lesson / by M.Art(Arch) George V. Pereboev December 2016.

Example 3. Determine the cross-section of the prism pentagon ABCDE, whose edges perpendicularly to the horizontal plane of projection, cutting plane α - generic. Building polyhedron section Practical lesson / by M.Art(Arch) George V. Pereboev December 2016.

Solution: Since the edge of the prism perpendicular π1 plane, the horizontal projections of the points of intersection of these edges with the plane α (1 , 2, 3 , 4, 5 ) coincide with the horizontal projections of the ribs, t. E. Points A, B , C, D , E. Front projection meeting points are determined from the conditions for membership of these points α plane. The solution is reduced to finding the missing projections point owned the plane, if you know at least one of its projections. In the drawing, the front projection points 1 , 2, 3 , 4, 5 found by the frontal plane α. Building polyhedron section Practical lesson / by M.Art(Arch) George V. Pereboev December 2016.

References: Samarkin U.P. Geometry drawing in the course of Engineering graphic for architectural specialization students. = Самаркин Ю.П. Геометрические построения в курсе инженерной графики для студентов архитектурных специальностей. – Алматы: КазГАСА, 2009. Georgievskyii O.V., Kondateva T.M., Mitina T.V. Engineering graphic tasks set. = Георгиевский О.В., Кондратьева Т.М., Митина Т.В. Сборник заданий по инженерной графике. – М.: Архитектура-С, 2007. Building polyhedron section Practical lesson / by M.Art(Arch) George V. Pereboev December 2016.

Building polyhedron section: Practical lesson / by M.Art(Arch) George V. Pereboev. – Almaty. - December, 2016. – 11 p.