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Файл:Начертательная геометрия. Курс лекций. Учебное пособие
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a)
b)
Figure 3.8. Intersecting planes in space
In this case the line of intersection gives but one piercing-point N at the intersection of
the vertical traces. The horizontal projection of intersection line will be parallel to horizontal
trace of T-plane, and vertical – will coincide with vertical trace of G-plane.
3.7. The line and plane
When a line neither be parallel or lies in a given plane it intersects this plane. One of the
main problems of descriptive geometry is to find the piercing-point of a line on a given plane.
The solution consists in passing through the given line any auxiliary plane (Figure 3.9),
in determining the line in which this plane cuts the given plane, and in finding the point in
which the given line intersects the lines thus determined. Projecting planes or planes that are
parallel to coordinate planes are used as an auxiliary plane for these purposes.
Figure 3.9. Line intersects a plane
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The construction becomes extremely simple when auxiliary secant plane passing through
the line is assumed to be the projecting plane of the line itself – to H (Figure 3.10). Find the
line of intersection between the given plane G and the projecting plane Q, and the point (K), in
which the line AB in space intersects it, is its piercing-point on the given plane.
Figure 3.10. Intersecting the line and the plane given by traces
The problem when a plane given by triangular is solved by the same method. In Figure
3.11 the plane given by triangular CDE. The auxiliary plane Q has been passed through the
given line AB perpendicular to H. Secant plane cuts the triangular in a line 1-2. The point K in
which lines AB and 1-2 intersect is a point in the line of intersection sought. Then the line AB
visible is found using method of competing points. At first, the point 3 in the line AB lies higher then point 2 in the side DC of triangular, thus the sector AK of the line AB in space lies
higher then triangular, and this sector is visible and is drawn full. Therefore, sector KB of the
line AB places under plane CDE and it concealed sectors are drawn by a shot dash on each coordinate plane respectively.
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a)
b)
Figure 3.11. Intersecting the line and the plane given by triangular
3.8. The line is parallel to plane
When a given line is parallel to any line which lies in a plane, this given line is parallel
to this plane.
Let find the geometrical relation the given line AB and the given plane CDE. The auxiliary secant plane Q is passed through the line AB perpendicular to H-plane. The line 1-2 is the
line of two planes intersection. The correspondence projections of two line (AB and 1-2) are
not parallel, thus the line AB is not parallel to plane CDE.
In the case, when the line should be drawn through the point in space and be parallel to
general plane (Figure 3.13), the general line (CD) is found in the plane (∆ ABC), and then the
projections of the sought line are drawn as a parallel lines through the given point K.
Figure 3.12. The line and plane parallel definition
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Figure 3.13. The line and plane are parallel
3.9. The line and plane are perpendicular
A right line perpendicular to a plane in space gives projections which are respectively
perpendicular to the plane traces.
In Figure 3.14 let plane G be the given plane, and AB a line perpendicular to it. The
plane projecting AB upon H is not only perpendicular to that plane, but also to the given plane
G; hence, being perpendicular to two planes, it is, by Geometry, perpendicular to their line of
intersection G1, or the horizontal trace and horizontals.
Figure 3.14. The line and plane are perpendicular
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The converse of this position is likewise true: if the projections of a line are perpendicular respectively to the traces of a plane or the lines, which are parallel to the correspondence
coordinate planes (horizontals or frontals), the line is perpendicular to the plane.
When the line perpendicular to two intersecting lines of a plane, by Geometry, it is perpendicular to a plane. The lines of plane, which are parallel the coordinate plane, are used for
drawing the perpendicular line to a plane in space.
Let K be the given point, and G the given plane. The task is to pass a line perpendicular
to the plane through a given point (Figure 3.15). By the condition of the task the projections of
the required line must be respectively perpendicular to the traces of the given plane (G1 and G2)
or to two intersecting lines of that plane – its horizontal and vertical (h and f). Through K2 lead
the line perpendicular to the trace G2 or vertical projection of frontal f2, and through K1 lead the
line perpendicular to the trace G1 or horizontal projection of horizontal h1.
Figure 3.15. Drawing the line perpendicular to the plane
3.10. The planes are parallel
The given planes are parallel when they contain any two lines which are parallel to each
other. In another words, when two projections of lines which lie in one plane are parallel to the
similar projections of the lines, which lie in the second plane, these plane are parallel (Figure
3.16a). When the planes given by traces their parallel is proved when their similar traces are
parallel (Figure 3.16b).
In Figure 3.17 the drawing of parallel planes, which are passed through the point in
space, are shown. In Figure 3.17a the required plane is drawn as two intersecting lines, which
are passed through the point A and are the horizontal and frontal of the new plane.
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a)
b)
Figure 3.16. The parallel planes
a)
b)
In Figure 3.17b the plane G given by traces. At first, the horizontal of the required plane
leads through the point A, when its horizontal projection is parallel to horizontal trace G1 and is
passed through the A1.
Figure 3.17. The parallel plans drawings
Then the piercing-point (N) of the horizontal is found. Through that piercing-point the
vertical trace of required plane T2 is passed as a parallel trace to G2. The horizontal trace of the
new plane T1 is obtained by drawing the line parallel to G1 and passed through point TX.
3.11. The planes are perpendicular
When a plane in space is passed through the line, which perpendicular to a given plane,
the planes are perpendicular to each other (Figure 3.18). There are a lot of planes may be
passed through the point A, which will be perpendicular to the given plane P. All of these
planes have the similar axis as a line AB that is perpendicular to the given plane P.
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Figure 3.18. Perpendicular planes
In Figure 3.19 the task of drawing the plane, that is perpendicular to given plane BCE
and passed through point A, is drawn. Through the point A lead line perpendicular to the given
plane: through A2 lead line perpendicular A2K2 to vertical projection of frontal (B2C2), and
through A1 lead line A1K1 perpendicular to horizontal projection of horizontal (E1C1). Then,
through point A draw the general line AD. Intersecting lines AK and AD is the required plane,
which is perpendicular to the given plane BCE.
Figure 3.19. Perpendicular planes drawing
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CONTROL TEST
1. List the main methods of a plane determination.
2. Represent three traces of plane. How should they be related in projecting draw?
3. How must be related the traces of plane and line, which lies in that plane? Validate you
conclusions by representation.
4. When does a point lie in a plane? Represent your response.
5. List the main position of a plane to the coordinate planes.
6. What does the term “principle lines of a plane” mean? Represent these lines for any
given plane.
7. Give the definition to the line of the greatest declivity. What are the main steps of it
determination?
8. What is the result of two planes intersection? Represent the result of two plane inter-
section, which are assumed by traces.
9. How can the point of line and plane intersection be found? List the main stages of in-
tersecting point determination.
10. When does the line parallel to plane?
11. List the cases, when the line perpendicular to plane.
12. Represent two parallel planes.
13. List the cases, when two planes are perpendicular.
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4. DRAWINGS TRANSFORMATION
4.1. The main problems of drawings transformation
For engineering purposes the drawings of objects must be oriented in space so as to find
the true sizes, shape and position of them or their separate parts. It may be obtained when the
objects in space are oriented parallel or perpendicular to the coordinate or to auxiliary planes.
When an object is parallel to any plane, its projections on that plane will be presented in a truelength form. When an object is perpendicular to any plane, its projections on that plane will be
transformed into simple geometrical form – point, line or plane.
There are five main tasks of drawings transformation:
2.4.1. definition of the true-length of a line;
2.4.2. transformation of a line into projection position – end view of a line;
2.4.3. definition of the true-length view (shape) of a surface plane;
2.4.4. transformation of a surface plane into projection position – edge of the surface plane;
2.4.5. definition the distance between two skew lines.
The solution of these tasks may be found by the follow methods:
2. Change of the projection plane (or ground-line).
3. Parallel-plane transformation.
4. Rotation and rabattement.
5. Rotation around plane traces (or coinciding method).
Let to study each of them more thoroughly.
4.2. Change of the projection plane
This method is widely use for definition of the true size and shape of geometrical object,
and for transformation of them into projection position. In the other resource this method is
termed as Change of position method or Auxiliary view method. According to this method the
points of object do not move and change their position, but the auxiliary plane is introduced to
the coordinate system. The position of this auxiliary plane depends on the main problem of the
task. When the true-length of the objects is defined, the planes, which are parallel to the objects
and perpendicular to horizontal or vertical planes are used.
In Figure 4.1 the transformation of vertical projection of point A from system H/V to the
new system H/V1 is shown. In the new system the auxiliary plane V1, which is perpendicular to
H-plane, is introduced instead of vertical plane V. Intersecting planes V1 and H are formed new
ground-line OX1. Projections of A-point in the new system lies in a corresponding line, which
is perpendicular to the OX1 – projector of the planes H/V1. The distance of the A-point from H
remains unaltered. Thus, the distance A2 from OX and of A4 from OX1 are equal.
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a)
b)
Figure 4.1. Change the plane of projection – horizontal projecting plane
The last condition is very important for drawings. When the new auxiliary plane, which
is perpendicular to the vertical plane, is introduced, the distance between new plane and point
B is equal to the same distance between it and V (Figure 4.2). Hence, if B1 and B2 are the primitive projections of the points B, let fall upon the new ground-line OX1, the perpendicular B2B4,
and lay off a distance B2B4 equal to B0B1, above OX1. The point B4 is a new horizontal projection sought.
Figure 4.2. Change the plane of projection – vertical projecting plane
In Figure 4.3 the determination of the true-length of line AB is shown. As it was established in the Lecture 2, the true-length of a line is defined as a projection of it on the plane
which is parallel to this line. Therefore the auxiliary plane must be parallel to the line and perpendicular to the one of the coordinate plane: new plane P4 is parallel to the line AB (its horizontal projection) and perpendicular to the H-plane. Let fall upon the new ground-line OX1 the
perpendiculars from A1 and B1 projections, and lay off a correspondence distances of these
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