Добавил:
ivanov666
Опубликованный материал нарушает ваши авторские права? Сообщите нам.
Вуз:
Предмет:
Файл:Descriptive geometry. Course of lectures for international students
.pdf
MINISTRY OF SCIENCE AND HIGHER EDUCATION
OF THE RUSSIAN FEDERATION
FEDERAL STATE BUDGETARY EDUCATIONAL
INSTITUTION OF HIGHER PROFESSIONAL EDUCATION
«DON STATE TECHNICAL UNIVERSITY»
S.A. Debeeva, E.I. Fisunova, E.V. Fominov
DESCRIPTIVE
GEOMETRY
Course of lectures
for international students
Rostov-on-Don
DSTU
2024
2
UDC 514.18(075.8)
D29
The reviewer
Candidate of technical sciences, professor K.G. Shuchev
Debeeva, Svetlana Alexandrovna.
D29
Descriptive geometry : course of lectures for international
students / S.A. Debeeva, E.I. Fisunova, E.V. Fominov ; Don State
Technical University. – Rostov-on-Don : DSTU, 2024. – 46 p.
ISBN 978-5-7890-2222-1
Course lectures is generated according to the requirements of Federal State Standard for
Higher Education for Bachelor’s Degree students.
The theoretical foundations of descriptive geometry with examples of problem solving are
presented. This publication is intended for independent work of international students.
UDC 514.18(075.8)
It is published by the decision of the Editorial and Publishing Council
Don State Technical University
ISBN 978-5-7890-2222-1
© Don State
Technical University, 2024
3
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. From this moment onwards we look at
a particular branch of geometry‒descriptive geometry‒developed by Gaspard
Monge.
Descriptive geometry deals with physical space, the kind that you have
been used since birth. Things you see around you and even things that you
cannot see have geometry.
All these things concern geometric objects almost always in relationship
to one another that sometimes requires us to make sense of it all ‒ in other
words, when we try to solve geometric problems albeit in architecture,
engineering, science. Descriptive geometry deals with manually solving
problems in three-dimensional geometry by generating two-dimensional
views.
1.2. Methods of projection
In geometry, projections are mappings of 2- or 3-dimensional figures
onto planes or 3-dimensional surfaces. For our purpose, we consider a
projection to be an association between points on an object and points on a
plane, known as the picture plane. This association ‒ between a geometric
figure and its image ‒ is established by lines from points on the Fig.to
corresponding points on the image in the picture plane. These lines are referred
to as projection lines.
There are two basic methods of the object’s projection: central and
parallel.
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
4
of triangular on the plane H – AHBHCH. This method is widely use in
architectural drawing and air-photography.
Fig. 1.1 Central Projection
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.
a) b)
Fig. 1.2. Parallel Projection:
a – oblique projection; b – orthogonal projection
5
The triangular AHBHCH produced by intersection of parallel lines and
plane H is a parallel projection of ABC.
Fig. 1.3. Orthogonal 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 (Fig. 1.3).
1.3. Correspondence
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 (Fig. 1.4). 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.
One of the methods for obtaining a correspondence between a space
point and its graphical representation employs a second projection plane (П2)
taken perpendicular to the first plane – П1. The orthographic projection of the
6
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.
Fig. 1.4. Orthogonal projection of point А on the H(П1) and V(П2) planes
The most suitable projection system for descriptive geometry
is the Cartesian system that is consists of three mutually perpendicular planes
(Fig. 1.5):
Fig. 1.5. Three mutually perpendicular planes:
П1 – horizontal projection plane; П2 – vertical projection plane;
П3 – profile projection plane; X – axis of abscissa;
Y – axis of ordinates; Z – axis of applicate
7
Fig. 1.6. Coordinate system
The positive directions of the axis are: to the left from the point O – for
X-axis, to the viewer side from П2-plane – for Y-axis, and to the top side from
П
1
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
Fig. 1.6). 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
Octant
Signs of coordinates
Octant
Signs of coordinates
X Y Z X Y Z I + + + V – +
+
II + – + VI
– – +
III + – – VII
– – –
IV + + – VIII
– + –
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 (Fig. 1.7).
8
Fig. 1.7. Projection of the detail on the planes
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.
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
(Fig. 1.8, a).
a) b)
Fig. 1.8. Axonometric (a) and projecting (b) drawings of A-points
9
Thus, from point A let fall the perpendicular AA1 – termed the projecting
line – to the horizontal plane, П1. 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 П2, marks with its foot, A2, the vertical projection
of that point. The same principle is used for marking the profile projection 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 П1 is rotated in the direction of the arrow (Fig. 1.6) until it coincides
with the plane П2, when the back portion of П1-plane rest upon the upper
portion П2, and the lower portion П2 covered by the front portion of П1-plane
(Fig. 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
(Fig. 1.8, b). Whence it follows that two corresponding projections of any
points of space will always lie in the same line perpendicular to the groundline: projections A1 and A2 lie in the corresponding line A1A2 perpendicular to
OX-line (vertical corresponding line); projections A2 and A3 lie in the
corresponding line A2A3 perpendicular to OZ-line (horizontal corresponding
line).
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 Fig. 1.8, b. 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.
10
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 (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 (Fig. 1.8, b).
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 each ground-line (Fig. 1.9, a).
2. On the projection plane (П1, П2, П3). 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 lies in the ground-lines (Fig. 1.9, b).
а) b)
c)
Fig. 1.9. Point general positions
3. On the one of the ground-lines (OX, OY, OZ). In this case its position
would be defined by one coordinate that do not equal zero. Two projections
coincide with the original point and third lie in the point O (Fig. 1.9, c).
Соседние файлы в предмете [НЕСОРТИРОВАННОЕ]