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Файл:Начертательная геометрия. Курс лекций. Учебное пособие
.pdf
Ministry of Education and Science of Russian Federation
Federal State Autonomous Educational Institution for Higher
Professional Education “North-Caucasus Federal University”
Institute of Civil Building, Transport and Mechanical Engineering
ANDREY BRATSIKHIN, MARIA SHPAK
А. А. Брацихин, М. А. Шпак
DESCRIPTIVE GEOMETRY
COURSE OF LECTURES
НАЧЕРТАТЕЛЬНАЯ ГЕОМЕТРИЯ
КУРС ЛЕКЦИЙ
for international students in oil and gas engineering
taught the program 131000.62 – Oil and Gas Engineering
Stavropol
2014
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УДК 514.18 (075.8)
ББК 22. 151. 3 я73
Б 87
Published according to the Resolution
of Educational Commission
of North-Caucasus Federal University
Reviewers:
Sergey Babenishev – Professor of Mechanical Engineering and Technological
Equipment Department at North-Caucasus Federal University, Dr.Eng.
Vladimir Samoylenko – Senior Lecturer of Automatic, Electronic Engineering,
and Metrology Department at Stavropol State Agrarian University, PhD.
Descriptive Geometry: course of lectures / authors: Andrey Bratsikhin, Maria Shpak. –
Stavropol: Рublisher NCFU, 2014. – 73 p.
Брацихин А.А., Шпак М.А.
Б 87 Начертательная геометрия. Курс лекций: учебное пособие. – Ставрополь: Изд-
во СКФУ, 2014. – 73 с.
Course lectures is generated according to the requirements of Federal State Standard for
Higher Education for Bachelor’s Degree students.
Established on meeting of Mechanical Engineering and Technological Equipment Department (Minute №3 at the 07th of October, 2014) for students taught the program 131000.62 – Oil
and Gas Engineering.
УДК 514.18 (075.8)
ББК 22. 151. 3 я73
Authors:
Andrey Bratsikhin – Head of Mechanical Engineering
and Technological Equipment Department NCFU, Dr.Eng.
Maria Shpak – Docent of Mechanical Engineering
and Technological Equipment Department NCFU, PhD.
© North-Caucasus Federal University, 2014
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TABLE OF CONTENTS
LECTURE 1. PRINCIPLE CONSIDERATION……………………………………………….
1.1. Descriptive geometry defined ………………………………………………………...
1.2. Methods of projection ………………………………………………………………...
1.3. Correspondence ………………………………………………………………………
1.4. The point in space …………………………………………………………………….
Control test ………………………………………………………………………………...
LECTURE 2. LINE PROJECTIONS ……………………………………………………………
2.1. Line projecting on the three projection planes ……………………………………..
2.2. Position of a line to the planes of projection ……………………………………….
2.3. Traces of a line ……………………………………………………………………….
2.4. True length of line definition …………………………………………………………
2.5. Position of lines in space …………………………………………………………….
2.6. Competing points of skew lines ……………………………………………………..
2.7. Right-angle projection ………………………………………………………………..
Control test ………………………………………………………………………………...
LECTURE 3. PLANE PROJECTIONS …………………………………………………………
3.1. Methods of a plane determination …………………………………………………..
3.2. Plane traces …………………………………………………………………………...
3.3. Points and lines are contained in plane ………………………………………………
3.4. Position of a plane to the coordinate planes ………………………………………….
3.5. The principle lines of a plane …………………………………………………………
3.6. Intersecting planes ……………………………………………………………………
3.7. The line and plane …………………………………………………………………….
3.8. The line is parallel to plane …………………………………………………………..
3.9. The line and plane are perpendicular ………………………………………………..
3.10. The planes are parallel ………………………………………………………………
3.11. The planes are perpendicular ……………………………………………………….
Control test …………………………………………………………………………….......
LECTURE 4. DRAWINGS TRANSFORMATION …………………………………………….
4.1. The main problems of drawings transformation ……………………………………
4.2. Change of the projection plane ……………………………………………………….
4.3. Methods of rotation …………………………………………………………………..
4.3.1. Rabbatement …………………………………………………………………...
4.3.2. Rotation around plane traces (or coinciding method) ………………………
Control test ………………………………………………………………………………...
LECTURE 5. PLANE SURFACES (POLYHEDRONS) ……………………………………….
5.1. Prism and prismoid …………………………………………………………………...
5.2. Pyramid ……………………………………………………………………………….
5.3. Intersecting polyhedron and plane …………………………………………………..
5.4. Intersecting polyhedron and line ……………………………………………………..
5.5. Intersecting of two polyhedrons …………………………………………………….
Control test ………………………………………………………………………………...
LECTURE 6. DEVELOPMENT OF POLYHEDRONS ……………………………………….
6.1. Development by right section determination ………………………………………..
6.2. Method of flattening ………………………………………………………………….
6.3. Development by triangulation ………………………………………………………..
Control test ………………………………………………………………………………...
TASKS FOR SELF-STUDY …………………………………………………………………….
GLOSSARY ……………………………………………………………………………………..
LIST OF RECOMMENDED TEXTBOOKS …………………………………………………
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1. PRINCIPLE CONSIDERATION
1.1. Descriptive geometry defined
Descriptive geometry is essentially the technique of accurately representing objects by
means of drawings and of solving graphically all problems related to their form and position.
Descriptive geometry provides the theoretical basis for technical drawing.
In general, an object or structure may be considered as a combination of elementary geometrical forms. These forms are commonly prisms and cylinders, but may also include pyramids, cones, surfaces of revolution, and warped or twisted surfaces. These various space figures can be analyzed into points, lines, and surfaces as the basic geometrical elements. Descriptive geometry deals specifically with the graphical representation on a plane (the drawing surface) of the basic geometrical elements and the solution of space problems connected with their
representation.
1.2. Methods of projection
There are two basic methods of the objects projection: central and parallel.
Figure 1.1. Central projection
The main goal of central projection is shown on the Fig.1.1. The natural triangular ABC
is projected on the plane H by passing the lines through the center of projection (point S) and
the main points of triangular – A,B C. The points of intersections these lines with plane H is
the ABC-central projection of triangular on the plane H – AHBHCH. This method is widely use
in architectural drawing and air-photography.
In descriptive geometry is used method of parallel projection (Fig.1.2). There is no center
of projection as in later method. For that method the direction of projection is introduced (S).
For ABC projection the parallel lines in a direction of vector S are passed through points ABC.
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The triangular AHBHCH produced by intersection of parallel lines and plane H is a parallel projection of ABC.
Figure 1.2. Parallel projection
The parallel projections can be orthogonal and oblique. If the angles of projection are not
perpendicular to plane that method is named “oblique projection”. If a straight line is passed
through any point of space and perpendicular to a plane, the point of space is said to be projected orthogonally on the plane at the point where the perpendicular intersects the plane. The
perpendicular is called the projector. The plane is the projection plane and is represented by
the drawing paper (Figure 1.3).
Figure 1.3. Orthogonal projection
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1.3. Correspondence
a)
b)
The elementary principles of orthographic projection can best be understood by considering the point as the unit of graphical representation.
It is evident that a single projection of a point on a plane does not completely represent
the position of the point with respect to the plane. Every point lying in the projector has the
same projection (Figure 1.4a). In order that a projection system be useful, it is necessary that a
given point of space have a unique graphical representation, and, conversely, that the graphical
representation of a point correspond to a single point of space.
Figure 1.4. Orthogonal projection of points:
a) on the H plane; b) on the H and V planes
One of the methods for obtaining a correspondence between a space point and its graphical
representation employs a second projection plane (V) taken perpendicular to the first plane – H
(Figure 1.4b). The orthographic projection of the point on the second plane measures the distance between the space point and the first plane. The representation of an object by means of
its projections on two mutually perpendicular planes is the system in which we are interested
and will be developed in the material immediately following.
The most suitable projection system for descriptive geometry is the Cartesian system that
is consists of three mutually perpendicular planes (Figure 1.5):
H – horizontal projection plane;
V – vertical projection plane;
P – profile projection plane;
X – axis of abscissa;
Y – axis of ordinates;
Z – axis of applicate.
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Octant
Signs of coordinates
Octant
Signs of coordinates
X Y Z X Y
Z
I
+ + + V – + +
II
+ – +
VI
– – +
III
+ – –
VII
– – –
IV
+ + –
VIII
– + –
Figure 1.5. Cartesian coordinate system
The positive directions of the axis are: to the left from the point O – for X-axis, to the
viewer side from V-plane – for Y-axis, and to the top side from H plane – for Z-axis. Other directions employ as a negative. The projection planes divide all space into eight parts that are
called “octant” (I – VIII on the Figure 1.5). The signs of coordinates for each octant are presented in the Table 1.1.
Table 1.1
The signs of coordinates according to the octants
The principal projections of an object are those that show its principal dimensions. It is
evident that two of these dimensions can appear in true size in a single orthographic projection.
Thus, the width and depth appear on a horizontal projection plane (Figure 1.6).
This projection is variously called the horizontal projection, top view, or plan. The width
and height appear on a vertical projection plane. This projection is called the front view or
front elevation. When the object is projected on a second vertical plane to show its depth and
height, the projection plane is called a profile plane. The projection is the profile or side view.
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Figure 1.6. Projection of the detail on the planes
1.4. The point in space
The projections of any point in space are determined by letting fall from that point perpendiculars to the three planes – horizontal, vertical and profile (Figure 1.7a).
Thus from point A let fall the perpendicular AA1 – termed the projecting line – to the horizontal plane, H. The foot A1 of this line is the horizontal projection of the point. In like manner, a perpendicular, AA2 drawn from the point A to the vertical plane, marks with its foot, A2,
the vertical projection of that point. The same principle is used for marking the profile projec-
tion of that point – A3.
For the descriptive geometry purpose coordinate drawing of objects is used instead of
isometric display. In order to represent two or three projections of an object on one plane some
modification must be made in the position of these planes. This is effected by revolving either
plane of projection around ground-line (OX and OY) until it coincides with the other plane.
Thus the plane H is rotated in the direction of the arrow (Figure 1.5) until it coincides with the
plane V, when the back portion of H-plane rest upon the upper portion V, and the lower portion
V covered by the front portion of H-plane (Figure 1.8).
The lines A1AX and A2AX are termed the corresponding ordinates of the point A, and AX
is its ground-point.
Ordinates A1AX and A2AX passing through the AX-point, and remaining perpendicular to
the ground-line OX, are the prolongations of each other (Figure 1.8b). Whence it follows that
two corresponding projections of any points of space will always lie in the same line perpendicular to the ground-line: projections A1 and A2 lie in the corresponding line A1A2 perpendic-
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ular to OX-line (vertical corresponding line); projections A2 and A3 lie in the corresponding
a)
b)
line A2A3 perpendicular to OZ-line (horizontal corresponding line).
Figure 1.8. Isometric (a) and projecting (b) drawings of A-points
There are some tasks to find the third projection of point when two others projections are
known.
1. Projection method. Through the A2 projection draw the horizontal corresponding line.
Through the A1 projection draw the perpendicular on the OY1 and find the point Ay1. Find the
point Ay3 using compass as shown on the Figure 8b. From Ay3-point draw perpendicular to the
intersection with horizontal corresponding line drawn through A2. Point of intersection, A3, is
the profile projection of A.
2. Coordinate method. Through the A2 projection draw the horizontal corresponding line.
The distance between A1 to OX-line is measured by compass – depth of A or YA coordinate,
and put it on the horizontal corresponding line from the AZ-point.
3. Method with constant drawing line using. Through the A2 projection draw the horizontal
corresponding line. Through the A1 draw the ordinate-line to intersect the constant drawing line k
(it is the bisector of Y1OY3 angle) at the A0-point. Through the A0-point draw the vertical line to
intersect with horizontal corresponding line drawn through A2 (Figure 1.8b).
A point may assume the following general positions:
1) In space. In this case its position would be defined by three coordinates and all projections do not lie on the each ground-line (Figure 1.9a).
2) On the projection plane (H, V, P). In this case its position would be defined by two
coordinates that do not equal zero. One projection coincides with the original point, and two
other lie in the ground-lines (Figure 1.9b).
3) On the one of the ground-lines (OX, OY, OZ). In this case its position would be defined by one coordinates that do not equal zero. Two projections coincide with the original
point and third lie in the point O (Figure 1.9c).
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a)
b)
c)
Figure 1.9. Point general positions
CONTROL TEST
1. What does the descriptive geometry define?
2. What are the methods of projection used in descriptive geometry? Describe of them.
3. List the main principles of orthogonal projection. What does the term “octant of space”
mean? How many octants are used in descriptive geometry?
4. Represent the projecting draw of any point in space. How many projections of point
can be represented? How do they connect to each other?
5. How many methods of definition of the third point projection do you know? List and
draw each of them.
6. Represent the drawings of the next point position: in space, on the projection plane and
on the ground-line.
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