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Файл:Начертательная геометрия. Курс лекций. Учебное пособие
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Figure 5.4. Pyramid
5.3. Intersecting polyhedron and plane
A plane may assume two general positions to a surface:
1) a plane may touch surface – be a tangent – the surface lies wholly on one side of the
plane;
2) a plane ay cut surface – be a secant – the surface is divided by the plane.
The line in which the cutting plane intersects the surface is termed the line of intersection, and is common both to the surface cut and to the cutting plane; hence, whatever the nature
of the surface, the line of intersection must be a plane line.
It is evident that the section of a polyhedron must be wholly rectilinear and diminish in
size as the plane approaches the vertex.
The line of intersection may be found be two ways:
1) when the intersecting points of the secant plane and edges of polyhedron are deter-
mined – method of edges; the main rules of line and plane intersection are used in this method;
2) when the intersecting lines of the secant plane and faces of polyhedron are determined –
method of faces; the main rules of two planes intersection are used in this method.
In the Figure 5.5 the section of plane T and pyramid ABCS intersection is shown. It was
determined by method of edge using: the points of intersection of the plane T and each edges
of the pyramid were founded on the vertical projection, and then the horizontal projections of
these points were determined on the horizontal projections of the consequence edges.
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Figure 5.5. Intersection of plane T and pyramid ABCS
Note that a section is projected on the coordinate plane in true size when the cutting
plane is parallel to that plane. In the other case, the true size of the section may be determined
by the any methods of drawing transformation (change of the projection plane (or groundline), parallel-plane transformation, rotation and rabattement, rotation around plane traces).
5.4. Intersecting polyhedron and line
The points of intersecting polyhedron (ABCS) and line (a) are determined as the result of
line and plane intersection (Figure 5.6).
1. The line is contained in the plane (T) that is perpendicular to one of the coordinate
planes (a ϵ T and T is perpendicular to V).
2. The section (1-2-3) of plane and polyhedron intersection is determined.
3. Points of given line and plane intersection are determined as the points in which line
crosses the edges of the drawn section and line (points K and L).
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Figure 5.6. Intersection of line a and pyramid ABCS
5.5. Intersecting of two polyhedrons
The line of two polyhedrons intersection is defined by points which are results of intersection of edges of one polyhedron and faces another (intersecting line and plane). Another
method is the lines of correspondence polyhedron faces intersection definition (intersecting
two planes).
In Figure 5.7 the result of prism and pyramid intersection is shown. The points 1, 2, 3, 4,
5, 6 are determined as the result of pyramid edges (AS, BS and CS) and faces of prism intersection. The points 7 and 8 are determined by passing the additional plane Q through the edge
D of prism which is perpendicular to H and apex of the pyramid (S).
This plane intersects the pyramid in section SKL, which intersects the edge D in two
points – 7 and 8. Visibility of the intersection line is determined according to the visibility of
the crossing edges and faces of two polyhedrons. Points 5 and 8 are hidden as that are lie on the
hidden face of prism or pyramid: (·)5 ϵ DE and (·) 8 ϵ ABS.
In common case, two polyhedrons are intersected in the line, which is the space closed
polygon. When two polyhedrons have partial crossing, the intersection termed “partial penetra-
tion” or “tapping”.
Note that the projections of the line of two polyhedrons intersection always lie inside of
the crossing contour of them.
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Figure 5.7 Intersection of prism DEFG and pyramid ABCS
CONTROL TEST
1. Give the definition of the term “polyhedrons”. Term the main elements of them.
2. Give the definition of prism and prismoid. Represent any of them.
3. Which polyhedron can be termed “pyramid”? Represent it and call its main elements.
4. How can the line of polyhedron and plane intersecting be found? Give any sample.
5. How can the point of polyhedron and line intersecting be found? Give any sample.
6. Give the main principles of two polyhedrons intersection.
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6. DEVELOPMENT OF POLYHEDRONS
A development of a polyhedron is a drawing which shows the true size and relative position of each face of the solid. It represents the surface cut open along certain edges and folded
out into a single plane. When a development is cut from sheet material and properly bent, it
reproduces the surface of the solid very closely (Figure 6.1). In practical work, the development usually shows the inside of the surface, since the working dimensions of sheet-metal
structures are often the inside dimensions. Also, certain allowances must be made for seams
and "crowding" due to the thickness of the material.
Figure 6.1. Development of the prism on space
There are tree methods of development drawing:
1) method of the normal (right) section;
2) method of flattening;
3) method of triangulation.
6.1. Development by right section determination
This method is widely used for development of prism which ends and edges are in a
common position (oblique) to the coordinate planes. The edges of prism are transformed in the
position when their will be parallel to one of the coordinate planes. Then the prism is cut by the
plane perpendicular to the edges.
A right section of a solid is a section cut by a plane perpendicular to the axis or center
line of the solid. It is commonly called a cross section. The right-section plane appears edgewise and perpendicular to the axis in a view showing the axis in true length. Points in which
edges of the solid pierce the section plane are found in this view.
The true size of a right section of a solid appears in the view in which the axis of the solid projects as a point.
The solution of the problem for prism consists of 3 main steps (Figure 6.2):
1) the polyhedron is cut by a plane perpendicular to the edge of the prism;
2) the true shape of the right section is determined;
3) the development of prism is drawn using the true sizes of the edges and section.
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In Figure 6.2a the edges AD, BE and CF of the prism ABCDEF are parallel to the verti-
a)
b)
cal plane that is why they are projected on this plane in the true length.
Pass the plane Q perpendicular to the edges of prism and define the points of right section – 1, 2, and 3.
The true shape of the right section is determined by rotation of the points 1, 2, and 3
around the horizontal trace of Q-plane – Q1.
The development of the right section drawn in the true size of its sides and the true length
of the edges are determined using their vertical projections (they are represented in the true
length on the vertical plane). Then the shapes of the ends are drawn by passing the true length
of the base sides that are appeared by drawing of the faces (Figure 6.2b).
Figure 6.2. Development of prism on drawing: a) true shape of right section definition;
b) drawing of the prism development
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6.2. Method of flattening
This method is used for drawing the development of the prism which bases are parallel to
one of the coordinate planes, and its edges are parallel to other coordinate planes – they are
assumed as the principle lines. All sizes of such a prism are projected in their true length. Each
face folds out into the plane of development by rotation around the edge, which is placed
in this plane.
In Figure 6.3 the prism has the edges which are parallel to vertical plane and bases that
are parallel to the horizontal plane. For the purpose of drawing the development, the plane T
will be the plane of developing, which is parallel to V and passes through the edge 1-4.
Figure 6.3. Trapping of prism
Then, the face 1-4-5-2 is rotated around edge 1-4 (frontal line) to coincide with the plane
T: through the points 22 and 52 draw the lines perpendicular to the edge 12-42, then, through the
points 12 and 42 draw the arcs with radius that is equal the length of the base’s side 11-21. The
points 2 and 5 are assumed the new position of the face 1-4-5-2 in the T-plane. Then, the new
axis for rotation of the face 2-5-6-3 is taken the new position of the edge 2-5. Other faces are
developed by the same way.
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6.3. Development by triangulation
This method is used for developing of pyramids. The development of a pyramid is a planar figure that consists of the number of triangles – faces of the pyramid. Hence, the developing of the pyramid includes the true length of edges and sides of base determination, and drawing the triangles by their tree sizes.
In the Figure 6.4 the development of oblique pyramid is shown. The development of the
lateral faces consists of tree triangles. The surface is opened along the edge AS, and the faces
are laid out in sequence. The true length of lateral edges is determined by their rotation around
axis i, which is passed through the apex S of pyramid and perpendicular to horizontal plane.
The line S2A0, S2B0 and S2C0 is the true length of the respective lateral edges. Base of
pyramid is projected on the horizontal plane in the true shape as being parallel to that plane.
Assume the point S0 as the vertex of the development. Using the true length of SA as
a radius and point S0 as the center, strike an arc. Starting at point A0 on that arc, set off the true
lengths of the basal edge AB, and strike an arc with radius SB through the center of arc S0.
Point of intersection of two arcs gives the point B of development. The triangular base may be
attached to any basal edge of the development.
Figure 6.4. Development by triangulation
CONTROL TEST
1. Give the definition of the term “development of polyhedrons”. List the methods of it
drawing.
2. Give the main principles and steps of development drawing by right section determination.
3. Give the main principles and steps of development drawing by flattering. Which poly-
hedrons can be developed by this method?
4. Give the main principles and steps of development drawing by triangulation. Which
polyhedrons can be developed by this method?
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TASK FOR SELF-STUDY
Problem 1. To find the missing projection of point, if the other two projections are given:
Problem 2. To find the projection of the point B that lies symmetrically against point A
and respectively to the next objects: coordinate plane H, ground line OZ, and central point O.
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Problem 3. To find the traces or piercing-points of a line the projections of which are given.
Problem 4. To find the projections of any line, its piercing-points being given.
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