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Файл:Descriptive geometry. Course of lectures for international students
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When the faces of a solid are not congruent, the Fig.is irregular. The
prism, pyramid, and prismoid are common examples of this class.
The prism (Fig. 5.2) has two polygons called ends or bases which are
congruent and parallel, making all the other surfaces parallelograms which are
called faces. The solid is a right prism when each face is a rectangle; otherwise,
it is oblique. A line connecting the centers of the bases is an axis. The axis is
parallel to one set of edges. The description of a prism includes the shape of a
base. Thus, when the bases and faces are rectangles, the solid is a rectangular
right prism. When the bases and faces are parallelograms, the solid is a
parallelepiped.
Fig. 5.2. Pyramid
The pyramid can have any polygon for one face called the base; all other
faces are triangles (Fig. 5.2). The triangular faces meet in a common point
called the apex. A line connecting the apex and the center of the base is the
axis. The solid is a right pyramid when the axis is perpendicular to the base;
otherwise, it is oblique.
5.2. 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
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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 determined – 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 Fig. 5.3 the section of plane α and pyramid ABCS intersection is
shown. It was determined by method of edge using: the points of intersection
of the plane α 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.
Fig. 5.3. Intersection of plane α 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).
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Glossary
Auxiliary view A view on any projection plane other than a primary or principal
projection plane.
Bearing of a line The compass reading or angle of a line.
Descriptive geometry The theory of orthographic projection. It can be also defined as a graphical
method of solving solid (or space) analytic geometry problem.
Development A solid shape that is unfolded or unrolled onto a flat plane.
Edge view of a plane A view in which the given plane appears as a straight line. The edge view
of a plane may be found by viewing a true-length line that lies within the
plane as a point.
Frontal (line) A line that is parallel to the front (vertical) projection plane. Its projection
will be true-length in the front view.
Frontal plane Vertical plane – one of coordinate plane.
Horizontal (line) A line that is parallel to the horizontal projection plane. Its projection will
be true-length in the top view.
Horizontal plane Level plane – one of coordinate plane.
Intersecting lines When lines are intersecting, the point of intersecting is a point that lies on
both lines.
Isometric A pictorial view that shows three sides of an object – usually the top, front,
and right side.
Oblique line A straight line that is not parallel to any of the six principal projection
planes (coordinate planes).
Oblique plane A plane that does not appear as an edge view in a principal view.
Origin (point) A starting point that may be located anywhere on the object or near it.
Orthographic Right-angle projection. It is a method of drawing that uses
projection (drawing) parallel lines of sight at right angles (90°) to a projection plane.
Parallel lines Lines that are an equal distance from each other throughout their length.
Perpendicular lines Lines with a 90° angle between them.
Piercing point A point where a line intersects a plane.
Plane A surface that in not curved or warped. It is a surface in which any two
points may be connected by a straight line, and a straight line may be
always completely within the surface.
Profile (line) A line that is parallel to the profile projection plane. Its projection will be
true-length in the side view.
Profile plane Vertical plane ninety degrees to frontal plane – one of coordinate plane.
Projection line A line parallel to the line of sight and perpendicular to the projection
plane.
Projection plane A flat surface that the object is projected onto, such as paper, blackboard
True-length line The line that appears in its actual and true length.
True-shape of a plane The actual shape and size of a plane surface. A plane which appears as an
edge view in one view, and is perpendicular to a principal view line of
sight, such as the horizontal or frontal planes, is a true-shape plane.
Vertex The highest point of the pyramid.
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Recommended literature
1. Descriptive Geometry Worksheets with Computer Graphics /
I.L. Hill [et al.] // Series B (Paperback), 2011.
2. James Ambrose Moyer. Descriptive Geometry for Students
of Engineering. Publisher: Hard Press Editions, 2012. 252 p.
3. Лещик С.Д. Инженерная графика. В 3-х ч. Ч. 1. Начертательная
геометрия (на англ. языке) [Электронный ресурс]. Гродно: ГрГУ им. Янки
Купалы, 2019. URL: https://elib.grsu.by/doc/55727.2019-1275. 4141920673.
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Contents
1. PRINCIPLE CONSIDERATION ………………………………….
3
1.1.
Descriptive geometry defined ………………………………..
3
1.2.
Methods of projection ………………………………………..
3
1.3.
Correspondence ………………………………………………
5
1.4.
The point in space ……………………………………………
8 2. LINE PROJECTIONS ……………………………………………...
11
2.1.
Line projecting on the three projection planes ……………….
11
2.2.
Position of a line to the planes of projection …………………
12
2.3.
Traces of a line ……………………………………………….
15
2.4.
True length of line definition ………………………………...
16
2.5.
Position of lines in space ……………………………………..
17
2.6.
Competing points of skew lines ……………………………...
19
3. PLANE PROJECTIONS …………………………………………...
20
3.1.
Methods of a plane determination ……………………………
20
3.2.
Plane traces …………………………………………………..
21
3.3.
Points and lines are contained in plane ………………………
21
3.4.
Position of a plane to the coordinate planes ………………….
22
3.5.
The principle lines of a plane ………………………………...
24
3.6.
Intersecting planes ……………………………………………
26
3.7.
The line and plane ……………………………………………
27
3.8.
The line is parallel to plane …………………………………..
29
3.9.
The line and plane are perpendicular ………………………...
30
3.10. The planes are parallel ……………………………………….
32
4. DRAWINGS TRANSFORMATION ……………………………...
34
4.1.
The main problems of drawings transformation ……………..
34
4.2.
Change of the projection plane ………………………………
34
4.3.
Methods of rotation ………………………………………….
38
5. PLANE SURFACES (POLYHEDRONS) …………………………
40
5.1.
Prism and pyramid …………………………………………...
40
5.2.
Intersecting polyhedron and plane …………………………...
41
Glossary ………………………………………………………………
43
List of recommended textbooks ………………………………………
44
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Educational publication
Debeeva Svetlana Alexandrovna
Fisunova Elena Ivanovna
Fominov Evgeniy Valerievich
DESCRIPTIVE
GEOMETRY
Editor A.A. Litvinova
Computer processing: O.I. Pushkinа
_________________________________________________________________________________________
To print 27.06.2024.
The format is 60×84/16. The volume is 2,9 usl. p. l.
The circulation is 100 copies. Order No 812. The price is free
___________________________________________________________________________________________
Publishing Center of DSTU.
The address of the University and printing company:
344003, Rostov-on-Don, Gagarin Square, 1
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