Ministry of Education and Science of Ukraine

National Aerospace University n.a. N.E. Zhukovsky

“Kharkiv Aviation Institute”

JOURNAL

ON DESCRIPTIVE GEOMETRY

(Робочий зошит з нарисної геометрії)

Name ________________________

________________________

Group __________________________

Kharkiv "KhAI" 2019

3

PREFACE

In recent years among the separate groups of students the opinion appeared that the course of descriptive geometry, in connection with the appearance of computer graphics, loses its positions. Alas… we must disappoint you. Specifically, with the advent of comput-er technologies descriptive geometry, as science and as training discipline, occupied the determining place in training of mechanical engineers.

Before obtaining based on the computer graphic document, it is necessary to explain to computer what it must know how, know, draw and so on. But in order to learn computer to communicate with user on the proper level of geometric means, it is necessary to seri-ously and thoroughly study the course of descriptive geometry.

Furthermore, at present the new course of geometric simulation is formed to help de-scriptive geometry.

Descriptive geometry is the branch of geometry which allows the representation of three-dimensional objects in two dimensions, by using a specific set of procedures. The resulting techniques are important for engineering, architecture, design and in art. The theoretical basis for descriptive geometry is provided by planar geometric projections.

One of the most ancient written works, where attempt at the construction of projection images was made, is the treatise of Roman architect Mark Vitruviy “Ten Books about the Architecture” (1 century B.C.). In this work application of horizontal and frontal projections is given as something already known. Gaspard Monge is usually considered the "father of descriptive geometry". He first developed his techniques to solve geometric problems in 1765 while working as a draftsman for military fortifications, and later published his find-ings. So, drawing is the image obtained on the plane and executed about the specific rules. Drawing is the language of technology, and descriptive geometry is the grammar of that language.

After studying this grammar, student must know the rules of composition, reading and fulfilling of the drawing and the rules, methods and the ways of solution of the problems, dealing with three-dimensional forms.

Student also must know how to build the images of three-dimensional forms on the plane, i.e., how to compose drawing and how to solve on the drawing number of three-dimensional problems by graphic method.

Before approaching the study of descriptive geometry it is necessary to remind basic concepts, terms of determination from the school course of geometry. It is possible to use the glossary, located at the end of workbook for this aim.

Special attention should be given to geometric constructions. Training process on this discipline has such forms of the training: lecture, independ-

ent study of lecture material by the student, practical classes, consultation and examina-tion. At the lectures the students are acquainted with the theoretical bases of descriptive geometry, with the methods of the solution of the standard problems and with the new terminology.

The student is obliged with the preparation for the practical instruction on the as-signed theme to solve problems in workbook on the descriptive geometry. Tasks are car-ried out with the aid of the ruler, the compasses and the pencil. Sometimes it is possible to direct the line of construction and eventual result by colored ball-point pen.

In practical training the student presents the executed volume of domestic tasks for the checking to instructor. The tasks in the current theme are accomplished under the management of instructor. The test is conducted for each studied theme.

Admittance to the examination is carried out by instructor, who leads practical clas-ses on the basis of the executed works and current semester estimations.

The lecturer assumes examination at the end of the first semester of the first course. That’s all, and good luck at the examinations!

4

INTRODUCTION TO THE THEORY OF PROJECTION

Of all the techniques covered in descriptive geometry the way in which an object is

presented or viewed is the most important. If a part is to be manufactured and the angle of

view shown does not provide the necessary information, then the part could never be

manufactured to design expectations.

Monge's protocols allow an imaginary object to be drawn in such a way that it may be

3-D modeled. All geometric aspects of the imaginary object are accounted for in true

size/to-scale and shape, and can be imaged as seen from any position in space. All imag-

es are represented on a two-dimensional surface.

Descriptive geometry uses the image-creating technique of imaginary, parallel projec-

tors emanating from an imaginary object and intersecting an imaginary plane of projection

at right angles. The cumulative points of intersections create the desired image.

Protocols:

o Project two images of an object into mutually perpendicular, arbitrary directions.

Each image view accommodates three dimensions of space, two dimensions

displayed as full-scale, mutually-perpendicular axes and one as an invisible

(point view) axis receding into the image space (depth). Each of the two adjacent

image views shares a full-scale view of one of the three dimensions of space.

o Either of these images may serve as the beginning point for a third projected

view. The third view may begin a fourth projection, and on ad infinitum. These

sequential projections each represent a circuitous, 90° turn in space in order to

view the object from a different direction.

o Each new projection utilizes a dimension in full scale that appears as point-view

dimension in the previous view. To achieve the full-scale view of this dimension

and accommodate it within the new view requires one to ignore the previous view

and proceed to the second previous view where this dimension appears in full-

scale.

o Each new view may be created by projecting into any of an infinite number of di-

rections, perpendicular to the previous direction of projection. The result is one of

stepping circuitously about an object in 90° turns and viewing the object from

each step. Each new view is added as an additional view to an orthographic pro-

jection layout display.

Aside from the Orthographic, six standard principal views (Front; Left Side; Top;

Right Side; Bottom; Rear), descriptive geometry strives to yield four basic solution views:

the true length of a line (i.e., full size, not foreshortened), the point view (end view) of a

line, the true shape of a plane (i.e., full size to scale, or not foreshortened), and the edge

view of a plane (i.e., view of a plane with the line of sight perpendicular to the line of sight

associated with the line of sight for producing the true shape of a plane). These often

serve to determine the direction of projection for the subsequent view.

5

1. POINT AND ITS PROJECTIONS

In order to describe the location of а single specific point, either verbally or graphical-ly, its position must be explained by reference to some other point whose location is known. The known point then becomes а "reference point," or "origin of measurements," and all other points may be located from it by some system of three-dimensional meas-urements. The Cartesian rectangular coordinate system, shown in Fig. 1.1, is most gener-ally employed in mathematics, particularly in solid analytic geometry. Though the origin O pass three mutually perpendicular axes x, y, and z, and any point, such as А, may be lo-cated by stating the three distances х, у, and z. It supposed that the planes of projections are combined with the coordinate planes. So, determinant of the point can be written as A(x, y, z).

Model of the projection of point The Orthographic Projection of the point

(multiview drawing)

O

A1

y

xx

yz

z

y

y

A3A2

Fig. 1.1 Fig. 1.2

The point of intersection of the projecting ray with the plane of projections is called

the point projection. The plain drawing that consists of the projections of the depicted objects and con-

nected with the lines of projection communications (projectors) is called complex or multiview drawing. Projector is always perpendicular to the axes of projections.

The multiview drawing of point contains two point projections, connected together by the line of projection communications.

There are several rules for the complex drawing of point: 1. The frontal and horizontal projections of point are always arranged on the verti-

cal projector (A2A1 ox). 2. The frontal and profile views of point are always arranged on the horizontal pro-

jector (A2A3 oz). 3. Distance from the frontal point projection to the axis ox determines height and is

determined as far as coordinate z. Distance from the horizontal projection of point to the axis ox determines the depth of point and is determined as far as coordinate y.

6

4. It is always possible to construct profile projection of any given point using its horizontal and frontal projections. For this purpose it is necessary to put aside from the axis oz coordinate y on the horizontal projector, drawn through A2.

Problem 1.1. Construct Orthographic Projection of the points: A(40, 15, 20); B(30,

30, 0); C(40, 0, 20); D (0, 30, 30). Construct pictorial view.

Problem 1.2. Construct the third projections of the points A, B, C, D.

Problem 1.3. Construct the third projections of the points A, B, C, D.

7

Problem 1.4. Construct the third point projection. Measure and write down the co-

ordinates of points. Construct pictorial view of points.

Problem 1.5. Construct projec-

tions of the point B, located by 20 mm

higher than P1 and by 15 mm nearer to

plane P2 than the point A.

Problem 1.6. Construct projections

of the points symmetrical to point C(20,

15, 30) relative to the plane P1 (point A)

and origins of coordinates (point B).

Problem 1.7. Using the given point

projection A2 construct the projections

A1 and A3 so, that y = 2z.

Problem 1.8. Using the given point

projection B3 construct the projections B1

and B2 so, that y = 2x.

8

2. PROJECTION OF THE STRAIGHT LINE

A line is a non-curved entity that is void of any width and stretches out to infinity in

both directions, therefore completely absent of a starting or ending point. A line segment is

a portion of a line that is also void of any width, but is constrained by two end points.

Fig. 2.1. Orthographic Projection of a Line Fig. 2.2. Traces of the straight line

Fig. 2.3. True length of the line segment. The method of the right-angular triangular.

o The determinant of the straight line: two points (AB) or section (l).

9

o Point belongs to straight line, if the point projections belong to the similar projec-

tions of straight line.

o The projections of straight line, in the general case, are straight lines.

o Points of intersection of straight line with the planes of projections are called the

traces of straight line and are defined as the singular points of straight line, one of

coordinates of which is equal to zero.

o The full size of the oblique line segment is defined as the length of the hypotenuse

of the right triangle, built on one of the projections as on the cathetus. The second

cathetus of triangle equals to a difference in the distances of the ends of the sec-

tion based on that plane of projections, on which the first cathetus is undertaken.

Projections of the straight lines of the general and particular allocation

Fig. 2.4. Orthographic projection of oblique

line segment l (l1l2).

Fig. 2.5. Horizontal and frontal lines.

h(3-4) P1; f (5-6) P2

Fig. 2.6. Frontal and horizontal projecting lines

t(7-8) P2 ; k(7-8) P1.

10

Problem 2.1. Construct the third projection of straight line segment AB and the miss-

ing projection of the point K AB.

a)

b)

c)

d)

11

e)

f)

.

g)

h)

Problem 2.2. Construct projections of the points A, B, and C belonging to the given

line, if zA = 25 mm, zB = 0 mm, yC = 20 mm.

12

Problem 2.4. Determine the true size of section AB and angle of slope α to the plane

of projections P1.

Problem 2.5. Construct projections of the point B l, if |AB| = 30 mm.

Problem 2.7. Construct frontal projection of the point A, if |AB| = 40 mm.

Problem 2.8. Draw line l through the point M parallel to the line k.

13

Problem 2.12. Construct line k through the point M intersecting straight line a and z-

axis.

Problem 2.13. Construct projections of the straight line m parallel to the line a and

intersecting lines b and d.

Problem 2.14. Draw the straight line through the point E intersecting lines AB and

CD.

14

Problem 2.15. Construct projections of the isosceles ΔABC. CM is the altitude of

ΔABC; CM || P1; A P1; B P2.

Problem 2.16. AC is the diagonal of rhomb ABCD. B P1. Apex D is placed equi-

distantly from planes P1 and P2. Construct the projections of rhomb if AC || P2.

15

3. PROJECTION OF A PLANE

A plane can be defined as a non-curved region in space that may or may not extend

indefinitely in two directions. A plane can have length and breadth but will never contain a

thickness. Any two points on a plane may be connected with a straight line segment, and

the entire segment will be contained on that plane. Also, any two lines on a plane will ei-

ther intersect or be parallel. When a line is perpendicular to a plane, it will be perpendicular

to every line contained on that plane, as well as any line parallel to the plane. When a line

is parallel to a plane, it will be parallel to every line contained on the plane.

The very presence of a plane is established merely by defining the boundaries of that

plane. There are four main ways in which a plane's boundaries can be secured. They are:

a line and a point, three points, two intersecting lines and two parallel lines. As addition

two more ways can be considered: plane segment and traces of the plane.

Fig. 3.1. Model of a plane Fig. 3.2. Orthographic Projection

of the plane

Straight line belongs to plane, if it is carried out through two points, which deliberately lie

in this plane, or passes through one point and is parallel to the line, which lies in this plane.

Point belongs to plane, if it is constructed on the line, belonging to assigned plane.

Relatively to the principal planes of projections the planes are divided on the planes

of general allocation (oblique planes) and planes of particular position - projecting (per-

pendicular to one of the planes of projections) and the level planes (parallel to one of the

principal planes of projections).

It is possible to construct the principal lines of a plane (horizontal, frontal line, profile

line), and the line of the greatest inclination to each of the planes of projections.

Fig. 3.3. Horizontal principal line. Fig. 3.4. Frontal principal line.

16

Problem 3.1. Locate horizontal projections of line l that lies in plane ABC. Locate

frontal projections of line m that lies in the same plane.

Problem 3.2. Construct missing projection m2 of line m that lies in plane (k ǁ l).

Problem 3.3. Construct horizontal projection of ΔABC that lies in plane (k ǁ l).

17

Problem 3.4. Construct profile projection of plane ABC and the missing projection of

point M that lies in this plane.

Problem 3.5. Construct projections D2 and E1 of points D and E, which belong to

plane (a ǁ b).

Problem 3.6. Construct the horizontal projection of pentagon A1B1C1D1E1 on the

frontal projection and the horizontal projection of two adjacent sides A1B1C1.

18

Problem 3.9. Construct the missing projection of horizontal h1 in the plane (a b).

Problem 3.13. Construct the projections of horizontal line parallel to the plane (a ǁ

b) and passing through the point A.

Problem 3.14. Construct horizontal projection of ABC if ABC ǁ a.

19

Problem 3.15. Construct horizontal projection of ABC if ABC ǁ (h f); A2B2 ǁ f2.

Problem 3.16. Through the point M draw the profile line MN parallel to plane (f h).

Problem 3.17. Through the point A draw the line which would be parallel to plane .

a) b)

20

4. RELATIVE POSITION OF THE PLANES

Fig. 4.1. Intersection of two planes

Fig. 4.2. Parallelism of two planes

1. The line of intersection of two planes is determined either by two points, which

simultaneously belong to the given planes or by one common point and known direction of

this line.

2. If one of the being intersected planes is horizontal or frontal plane, then the in-

tersection of planes will be, correspondingly, horizontal or frontal principal line.

3. The points, which belong to the intersection of two planes, are determined by

the method of auxiliary cutting planes. The specified planes are intersected by auxiliary

plane (projecting or plane of level) and the common point for all three planes is deter-

mined; it belongs to the desired intersection of the assigned planes.

4. In the general case, three planes intersect at one point.

5. The sign of the parallelism of two planes is the parallelism of two being inter-

sected lines of one plane to the, correspondingly, two being intersected lines of the second

plane. Their similar traces of projection are parallel in the planes of particular allocation.

21

6. The similar principal lines (horizontal, frontal line, profile line) are mutually paral-

lel in parallel planes.

Problem 4.1. Determine the inter-

section of plane (a ǁ b) and the plane

(Γ P2).

Problem 4.2. Determine the inter-

section of plane (c ǁ d) and the plane

( P1).

Problem 4.3. Determine the inter-

section of plane Γ (ABC) and the plane

( P1).

Problem 4.4. Determine the inter-

section of plane Γ (a b) and the plane

(c ǁ d).

Problem 4.5. Determine the intersection of plane (f0 h0) and the plane (f h).

22

5. THE RELATIVE POSITION OF A STRAIGHT LINE AND A PLANE

Fig. 5.1. Intersection of the line and the

plane

Fig. 5.2. The parallelism of the line and the

plane

Rule of a line intersecting a plane:

The intersection of a line and a plane must lie on the intersection line of the given

plane and a cutting plane that contains the line. Point of intersection of the line and a plane

(piercing point) is defined as the point, which belongs simultaneously to the line and the

plane.

If the plane and the straight line are of general allocation, then piercing point can be

found by the following algorithm:

1. Construct cutting plane which would contain the given line.

2. Find line of intersection between the given plane and cutting plane.

3. Determine point of intersection between obtained line of intersection and given

line.

A line, parallel to a plane, is a line, which doesn’t belong to the plane and doesn’t in-

tersect it. If a line is parallel to a plane, then it is parallel at least to the one line, which lies

in that plane.

23

Problem 5.1. Construct point of intersection of a line AB and the plane P2.

Problem 5.2. Construct point of intersection straight CD and the plane Г (Г P1).

Problem 5.3. Determine point of intersection of the line m c and plane (ABC).

Problem 5.4. Construct point of intersection of line m and the plane (f h).

Problem 5.5. Construct point of intersection of line and the plane (ABC).

24

6. POLYHEDRONS

Any solid, bounded entirely by plane surfaces, may properly be referred to as poly-

hedrons. The plane surfaces are called the faces of the solid, and the lines of intersection

of the faces are called the edges.

If a plane intersects a polyhedron, it cuts a straight line on each of the intersected

faces. These lines form a polygonal plane that is called plane section of polyhedron. When

the intersecting plane is at right angles to the axis of polyhedron, the plane section is

called right section.

If at least two points of line belong to the face of polyhedron, then line belongs to

face. If a point belongs to line, and the line to face, then the point belongs to surface.

Problem 6.1. Construct the missing projections of the points, which belong to the sur-

faces of geometric figures.

25

Problem 6.2. Construct the missing projections of the lines, which belong to the sur-

faces of geometric figures.

Problem 6.3. Construct the intersection of pyramid and the plane.

.

Problem 6.4. Construct the intersection of a plane and geometric figures

26

Problem 6.5. Determine intersection of the line and geometric figures.

7. TRANSFORMATION OF A DRAWING

If the object has a particular location relatively to the plane of projection some of the

problems can be solved easier. For instance, if a straight line segment is parallel to the

plane of projection, it projects to a true size. Otherwise, to find its true length we need to

use method of a right-angular triangle. To achieve this we can apply methods called trans-

formation of a drawing.

There are three ways of transform of a drawing:

1. Replacement of planes of projection (projection on auxiliary planes);

2. Revolution of a model about an axes perpendicular to a plane of projection;

3. Planar parallel motion (revolution of a model about an axes without indicating an

axes in the drawing).

Basic problems which can be solved by transformation of a drawing :

27

o Oblique line segment transforms to a line segment parallel to the plane of projec-

tion;

o Oblique line segment transforms to a line segment perpendicular to the plane of

projection;

o Oblique plane segment transforms to a plane segment perpendicular to the plane of

projection;

o Oblique plane segment transforms to a plane segment parallel to the plane of pro-

jection.

Projection on auxiliary planes

In this method location of an object doesn’t change while one of planes of projection

is replaced by a new one, perpendicular to the other plane of projection.

Properties of transformation:

1. New projectors are perpendicular to a new axes;

2. Distance between a new projection and new axes is equal to a distance be-

tween replaced (old) projection and replaced axes.

Fig. 7.1. Projecting of a point on auxiliary plane.

Algorithm:

1. Draw perpendicular (new projector) from a point А1 to a new axes of projections x’.

2. On a newly obtained projector protract zA-coordinate of a point А from the point А’Х,

i.е. А′2А'Х = А2АХ.

Problem 7.1. Find the horizontal projection of the line AB, if the true length of the line

AB is 45 mm.

28

Problem 7.2. Find the frontal projection of the point K1, allocated at a distance of 20

mm from the horizontal line h.

Problem 7.3. Determine the distance from the point M to the plane ABC.

Problem 7.4. Determine angles of the slope of the line AB to the planes of projec-

tions.

29

Problem 7.5. Determine the distance between two parallel planes.

Problem 7.6. Determine the angles of the slope of plane ABC to the planes of projec-

tions.

Problem 7.7. Determine the distance between parallel lines a and b.

30

Problem 7.8. Determine distance from the point M to line a.

Problem 7.9. Construct the projections of rectangular isosceles ABC, whose

cathetus AB belongs to line a; A 90 .

Problem 7.10. Find the horizontal projection of the point D that is placed equidistantly

from the apexes of the ABC.

31

Problem 7.11. Distance between parallel lines CD and AB is 20 mm. Find horizontal

projection of a line CD.

Problem 7.12. Construct the projections of square ABCD using its side AB and direc-

tion of the frontal projection a2 of the side AD adjacent to AB.

Method of revolution

The alternate method of transforming is revolution, which requires the observer to

remain stationary and the object to be turned to obtain the various views.

32

Four fundamental principles of revolution show the concepts of what actually hap-

pens in space when revolution is used.

1. When a point M is revolved in space, it is

always revolved around a straight line

used as an axis (J-J’). It is important to

know how the axis actually lies, before

you attempt to revolve any point.

2. A point will revolve in a plane that is per-

pendicular to the axis, and its path is al-

ways a circle. The radius of the circle is

the shortest distance from the point to

the axis.

3. The circular path of the point is seen

when the axis appears as a point, i.e.,

when the axis is perpendicular to a

plane.

4. When the axis is shown true length, the

circular path of the point will always ap-

pear as an edge view at right angles to

the axis.

Algorithm of construction for the true size of a plane seg-ment (Fig. 8.3.):

1. Construct horizontal line h in the plane ABC.

2. Revolve horizontal projection of ABC-triangle to

locate h1 P2.

3. Construct frontal view of ABC-triangle. It projects to a line perpendicular to the frontal principal plane.

4. Revolve obtained edge view to locate it parallel to horizontal principal plane. A”B”C” ǁ to x-axes .

5. Obtained horizontal view is a true size of a ∆ABC.

Fig. 7.2. Revolution about horizontally

projecting line

Fig. 7.3. True size of a plane segment

J1=J’

1

R

TL of a Line segment

M1 J2 M2

90°

M'2

33

Problem 7.13. Determine the distance from the point D to the plane (ABC).

Problem 7.14. Find the center of the circle circumscribed around ΔABC.

34

Problem 7.15. Determine the slope of a line AD to the plane (ABC).

Problem 7.16. Revolve a point M about the axis i to combine it with the plane

(ABC).

35

Problem 7.17. Revolve a plane (ABC) about the axis i in such a way that the point M would prove to be at this plane.

Problem 7.18. Construct the projections of the square ABCD, if apex D belongs to the line a.

36

8. CURVES AND SURFACES

A curved line is the path of a point that moves in a constantly changing direction. If

the point moves so that it always lies in the same plane, then the resulting path is a plane

curve, or single-curved line. If the various positions of the moving point do not lie entirely in

one plane, the resulting path is called a space curve, or double-curved line. Plane curves

may be infinite in variety, but only a few are commonly used in engineering. Of primary

importance are the conic sections, which include the circle, ellipse, parabola, and hyperbo-

la. Other plane curves of practical importance are the roulettes, which include the cycloid,

epicycloid, and hypocycloid (used for the tooth profile on cycloidal gears); the spirals, of

which the involute is the most common (for tooth profiles on involute gears); and the trigo-

nometric curves, especially the sinusoid, which occurs in periodic-motion studies. Double-

curved lines may also be infinite in variety, but the helix in its several forms is by far the

most common. The line of intersection between curved surfaces is usually of double curva-

ture.

The classification of surfaces.

All regular surfaces may be divided into two major classes and further subdivided as

follows:

1. Ruled surfaces.

a. Polyhedrons.

b. Single-curved surfaces.

c. Warped surfaces.

2. Double-curved surfaces.

a. Surfaces of revolution.

b. Surfaces of evolution.

The above classification is based upon the fact that all these surfaces may be gener-

ated by the motion of a straight or curved line. In other words, a surface is the path of a

moving line just as a line is the path of a moving point. Although surfaces form the bounda-

ry of solids, we may also consider the surface itself as a tangible thing; chutes, hoppers,

and similar hollow objects whose walls are of negligible thickness are, for all practical pur-

poses, purely surfaces. Ruled surfaces may be generated by moving a straight line -

generatrix. Polyhedrons are objects which surfaces are composed entirely of plane surfac-

es.

Single-curved surfaces are those which may be generated by moving a straight line

in contact with a curved line (diretrix) so that any two consecutive position of the generat-

ing line are either parallel or intersecting. The cone, cylinder, and convolute are the only

examples of this class of surface. Warped surfaces are those which may be generated by

moving a straight line so that any two consecutive positions of the generating line are skew

lines. Double-curved surfaces are those which can be generated only by moving a curved

line. Surfaces of revolution are those which may be generated by revolving a line, straight

or curved, about an axis. This special classification may include single-curved, warped, or

double-curved surfaces. Surfaces of evolution are those which can be generated only by

moving a curved line, of constant or variable shape, along a noncircular curved path. . In

engineering construction the single-curved surfaces occurring most frequently are the cyl-

inder and the cone. We shall therefore devote considerable space to these particular sur-

faces. it should be observed that any method of solution which is valid for the cylinder may

37

also be applied to the prism and that the pyramid may be considered to be only a variant

of the cone. Indeed, with a little imagination the cylinder can be considered as a cone

whose vertex is at infinity; therefore all cone solutions are, with slight modifications, appli-

cable to cylinders also.

a) cone b) cylinder c) right helicoid

Fig 8.1. The most common ruled surfaces If a point belongs to the surface, then the point projections belong to the lines of sur-

face. On the Fig. 8.1(a) the point A on the surface of cone belongs to the generatrix S1,

which connects apex S with the point on the basis 1. On the Fig. 8.1(b) the point B on the surface of cylinder belongs to the generatrix,

passing through the point 1 and parallel to axis of circular sections i. On the Fig. 8.1(c) point M on the helical surface belongs to the generatrix, which

connects the point of 1 helix m with the axis i and passes parallel to the plane P1.

All right sections will cross surface of revolution throw the circle with the center on the axis of revolution. These circles are called parallels (Fig. 8.2). The greatest among the parallels is called equator, the smallest one is the throat.

The plane, passing through the axis of surface of revolution is called meridional, and the intersection of this plane with the surface of revolution - by meridian of surface.

Fig. 8.2

38

Problem 8.1. Construct the orthographic projections of the cylinder, represented by its base and horizontal projection of the generatrix A1B1. Point M belongs to the surface of cylinder.

Problem 8.2. Construct the orthographic projections of a sphere using the given projections of the center of sphere 0 and point M, which belongs to sphere surface.

Problem 8.3. Construct the orthographic projections of the cone with the apex S and the base that lying in the plane α. A radius of base is equal to the height of cone.

39

Problem 8.4. Construct the orthographic projections of the cone whose axis belongs to line a. The height of cone is equal l, the circle of base touches plane P1.

Problem 8.5. Find the missing projections of points A, B, C, D on the surface of sphere (a); M, N, P on the surface of cylinder (b); E, F, K, L on the surface of torus (c).

40

Problem 8.6. Construct the frontal projection of a line AB on the surface of the cone of the rotation (a); the horizontal projection of a line MN on the surface of the oblique cylin-der (b); the horizontal and profile views of a line ABC on the surface of sphere (c).

41

9. INTERSECTION OF A CURVED SURFACE AND A PLANE

The method of auxiliary secants planes adapts for constructing the intersection traces

in the general case. The points of the desired line are defined as the points of intersection

of lines, on which auxiliary secants of plane intersect the given surface and plane.

It is necessary to simplify constructions during the selection of auxiliary planes. Pref-

erence should be given to the projecting planes, which intersect the assigned surface

across the simplest possible lines.

It is important also to construct correctly the sequence of the singular points of the

section projections:

- the highest and lowest points of the section (they belong to the meridian, per-

pendicular to the secant plane);

- the points of tangency of the section projections to the outlines of a surface

curve, if such are located (boundary of visibility);

- the points on the closest and the most remote distance from the projections

planes.

On Fig. 9.1. the body of revolution catted by the plane is represented. For construct-

ing the points of the curved lines, obtained on the surface, the auxiliary horizontal planes,

which intersect surface on the parallels, and plane along horizontals, are used.

42

Fig. 9.1. Intersection of a plane and a surface.

Problem 9.1. Construct the missing projection of the cylinder of revolution and con-

struct its section by plane.

43

Problem 9.2. Construct the section of the given cylinder by the plane.

Problem 9.3. Construct the profile projection of a cylinder and its section.

Problem 9.4. Cut a cylinder by the plane, parallel to its axis and inclined down to the

plane at angle of 45o so that the section would be isometric to the given square.

44

Problem 9.5. Construct projections and full size of the section of cylinder.

Problem 9.6. Construct projections and true size of the section of a cone.

Problem 9.7. Construct the section of a cone by the plane.

45

Problem 9.8. Construct the plane, passing through the line a and cutting cone on the

parabola. Construct the projections of section.

Problem 9.9. Construct projections and true size of the section of a cone.

46

Problem 9.10. Construct the horizontal and profile views of the section of the cone by

the horizontal and frontal projecting planes.

Problem 9.11. Construct the sections of sphere by the plane, represented by the

straight line A on the surface of sphere.

Problem 9.12. Determine the shortest distance between the points A and B on the

surface of the sphere.

47

Problem 9.13. Construct the horizontal and profile views of the section of sphere by

horizontal and profile planes.

48

10. INTERSECTION OF STRAIGHT LINE WITH THE SURFACE

For determining the points of intersection of a straight line and a surface, as a rule,

auxiliary secant plane, passing through this straight line, can be used. Points of intersec-

tion of straight line with the obtained figure of section are the desired points of intersection

of a straight line and the surface. It is obvious that auxiliary secant plane must be selected

so that the projection of section would represent as possible the graphically simple lines:

straight lines or circle. For example, for the determination of points of intersection of the

line l and the surface of cylinder (Fig. 10.1.) the secant plane, assigned by line l and being

intersected it line m

parallel to the generatrix

of cylinder, should be

selected. This plane

intersects cylinder along

its generatricies. To de-

termine these

generatrixes let us find

the horizontal track MN

of plane. Let us mark

points 1 and 2 of inter-

sections of track MN

with the base (it is lo-

cated on the horizontal

plane). Let us draw the

horizontal projections of

generatrix 11' and 22'

through these points.

Intersection of this

generatrix with horizon-

tal projection of the line l

gives us points A and B.

Fig. 10.1. Intersection of the line l and cylinder surface.

For the determining of the points of intersection straight l and the surface of sphere (Fig. 10.2.) we can select the auxiliary horizontal plane, which intersects sphere through the circle of radius 0'212. Intersection of the horizon-tal projection of this circle with the horizontal projection of the line l shows location of the desired points of intersection A and B.

Fig. 10.2. Intersection of the line l and sphere.

49

Problem 10.1. Establish the intersection of the line m and the surface of sphere.

Problem 10.2. Establish the intersection of the line m and the surface of cylinder.

Problem 10.3. Establish the intersection of the line m and the surface of cone.

50

Problem 10.4. Establish the intersection of the line m and the surface of torus.

Problem 10.5. Determine the shortest distance from the point M to the surface of

cone (a) and from the point A to the surface of cylinder (b).

51

11. INTERSECTION OF THE SURFACES

Problem 11.1. Establish the intersection of the sphere and the prism.

Problem 11.2. Establish the intersection of the prism and the cylinder.

52

Problem 11.3. Establish the intersection of the cone and the prism.

Problem 11.4. Establish the intersection of the torus and the prism.

Problem 11.5. Establish the intersection of the two cylinders of revolution.

53

Problem 11.6. Establish the intersection of the cylinder and the pyramid.

Problem 11.7. Establish the intersection of the cylinder and the cone of revolution.

54

Problem 11.8. Establish the intersection of the hemisphere and the cylinder.

Problem 11.9. Establish the intersection of the cylinder and the inclined cone.

55

Problem 11.10. Establish the intersection of the cylinder and cone of revolution.

Problem 11.11. Establish the intersection of the cone of the revolution and self-

intersecting torus.

56

12. DEVELOPMENTS

Problem 12.1. Construct the development of the truncated pyramid.

57

Problem 12.2. Construct the development of the oblique cone.

7'1=3'1

7'2=3'2

1

S

21

543

67 8

2x 1 12=32

4131

11

21

S1

51

6'2=4'2

5 '1

81 71

61

72=52 6'2=4'2

82=4222 62 5 '2

S2

S1

2x 1

S2

8'1=2'1

1'1

8'2=2'2

1'2

P2

P1

58

Problem 12.3. Draw the development of the oblique truncated prism using stretch-out-line (right section) method. Draw the given line on the development.

A1

B1

C1

59

Problem 12.4. Construct the development of the oblique truncated cylinder using radial-line method. Draw the curve AB on the development.

A1

B1