descriptive geometry - part i

CHAPTER 1

PROJECTION SYSTEMS

A projection system is an ensemble of elements and laws or operations that make possible the passage from an m dimensional space plotting to an n dimensional space plotting.

Descriptive geometry uses two projection systems for the plane plotting of the objects from space (transformation of 3D into 2D):

• central projection system; • parallel projection system.

1.1. Central projection system

Elements that define this projection system are the projection plane [P] and the projection center S, external to plane [P] and situated at a finite distance in report to.

Central projection of a point A from space (fig. 1.1) is a point a, obtained by intersecting the line (SA), named projecting line, with the projection plane [P]. In the same way central projections of points B and D, are drawn as b and d. Points having projections situated ad infinitum (point C) and situated in the plane passing through S and parallel to plane [P] are excepted.

Between the points multitude from space and their central projections multitude situated in the plane [P] a univocal correspondence is settled: a single central projection corresponds to any point from space and an infinity of points from space correspond to a point situated in the projection plane.

Fig. 1.1 Fig. 1.2

A T curve from space is considered and its central projection on plane [P] is drawn from center S, obtaining curve t. The surface generated by all the projecting lines passing through projection center S and intersecting curve T is named projecting surface of curve T.

This surface, generated by a mobile line passing through a fix point and leaning on a curve, is a conical curve, which justifies the name conical projection, given to this projection system.

Figure 1.2 shows the central projection of a four sided polygon ABCD and illustrates the univocal correspondence between vertices A, B, C, D and their central projection (a≡a1, b≡b1, c≡c1, d≡d1). The size of the central projection of objects depends on their position in report to the projection center.

1.2. Parallel projection system

This projection system consists of projection plane [P] and projection direction (∆), the projection center is situated on ad infinitum; projecting lines are parallel to the line (∆) (fig. 1.3). This system is also named cylindrical projection.

As the size of the angle α between the projection direction (∆) and the projection plane is, the systems may be:

• oblique parallel projection system, having angle α ≠ 90˚ (fig. 1.4); • orthogonal projection system, having angle α ═ 90˚ (fig. 1.5). A univocal correspondence between the multitude of the points from spaces, for example the

vertices A, B, C and the multitude of their parallel projections (a≡a1, b≡b1, c≡c1) is settled.

Fig. 1.3 Fig. 1.4 Fig. 1.5

The double orthogonal projection exists inside an orthogonal projection system defined by two reciprocal perpendicular planes (fig. 1.6), named horizontal projection plane, [H], and vertical projection plane, [V]. The intersection of the two planes is (OX) axis.

Point A from space is considered and perpendicular projecting lines are respectively drawn on the two projection planes. The projecting line following direction (∆1) determines on horizontal plane [H] the horizontal projection a of the point A from space. The projecting line following direction (∆2) determines on vertical plane [V] the vertical projection a′ of the point A. Plane [Q] determined by projecting lines (∆1) and (∆2 intersects (OX) axis in point ax.

Passage from spatial plotting to the plane one is obtained by clockwise rotating plane [H], until it overlaps the fix plane [V]. Horizontal and vertical projections of point A are situated on the same line, named line of recall, perpendicular on (OX) axis in ax (fig. 1.7).

Considering projection planes infinite and representing only the (OX) axis, a plane plotting is obtained named draught (fig. 1.8). Between the multitude of points from space and the multitude of projection pairs (a, a′), a bi-univocal correspondence is settled.

Using the double orthogonal projection in plotting is due to Gaspard Monge (1746 – 1818), the founder of the Descriptive Geometry.

Fig. 1.6 Fig. 1.7 Fig. 1.8

1.3. Axonometrical projection system

Solving geometry problems with the help of graphic methods lead to the use of various projection systems. One of these, axonometrical projection system, is often used in technical drawing, as a complementary plotting that intuitively reveals general aspect and details of an object.

Axonometrical plotting is obtained by orthogonal or oblique projection of an object on a plane inclined face the orthogonal mark (fig. 1.9). The intersection of axonometrical plane, [P], with trihedron [H, V, L] is the axonometrical triangle ABC. The natural axes (OX), (OY) and (OZ) are axonometrically projected as (O1X1), (O1Y1), (O1Z1) (axonometrical axes).

Fig. 1.9

Orthogonal axonometry (OO1 perpendicular to plane P) proportionally projects segments (OA), (OB) and (OC) to segments (O1A), (O1B) and (O1C), being diminished by diminishing coefficients cos α, cos β, cos γ:

O1A = OA cos α;

O1B = OB cos β; O1C = OC cos γ.

Complementary angles of the angles α, β and γ are marked α1, β1 and γ1 and the relationship to their directing cosines is:

cos2 α1 + cos2 β1 + cos2 γ1 = 1

Replacing cos α1 = sin α and sin2 α = 1 - cos2 α, is obtained:

sin2 α + sin2 β + sin2 γ = 1;

1 - cos2 α + 1 - cos2 β + 1 - cos2 γ = 1;

cos2 α + cos2 β + cos2 γ = 2

Marking: k = cos α; m = cos β; n = cos γ, the fundamental relationship of orthogonal axonometry is obtained:

k2 + m2 + n2 = 2 As the diminishing coefficients values are, axonometry may be (fig. 1.10; 1.11;1.12):

Fig.1.10

Fig. 1.11 Fig. 1.12

• isometric axonometrical projection – axonometrical triangle is equilateral:

k = m = n;

• bi-metric axonometrical projection – axonometrical triangle is isosceles:

k = m ≠ n; k ≠ m = n;

• tri-metric axonometrical projection – axonometrical triangle is irregular:

k ≠ m ≠ n. Axonometrical plotting of a parallelepiped and of the circles inscribed in its faces (projected as ellipses) are presented in isometric (fig. 1.13), bi-metric (fig. 1.14) and tri-metric (fig. 1.15) axonometrical projection.

Fig. 1.13 Fig. 1.14

Fig. 1.15

CHAPTER 2

PLOTTING OF A POINT

2.1. Double orthogonal projection of a point

Two perpendicular projection points are considered: horizontal projection plane, [H], and vertical projection plane, [V], named fundamental planes, they intersect each other on (OX) axis, named projection axis or earth line. The projection axis shares fundamental planes into two semi-planes (fig. 2.1):

[Ha]; • anterior horizontal half-plane, • posterior horizontal half-plane, [Hp]; • superior vertical half-plane, [Vs]; • inferior vertical half-plane, [Vi]. The two fundamental planes share space into four dihedra: • dihedron I, limited by [Ha] and [Vs]; • dihedron II, limited by [Hp] and [Vs]; • dihedron III, limited by [Hp] and [Vi]; • dihedron IV, limited by [Ha] and [Vi].

Fig. 2.1 Fig. 2.2

Figure 2.1 presents points situated in the four dihedra. Considering point A, situated in the first dihedron, it will be projected on [H] in a, the horizontal

projection of the point, and on [V] in a′, the vertical projection of the point. The distance from point A to [V] is named distance, is marked y and represents the segment axa. The distance from point A to [H] is named quote, is marked z and is the length of the segment axa′. The distance and the quote represent two of the three descriptive co-ordinates of the point.

As the dihedron they are situated on is, points have distances and quotes positive or negative. Points situated above horizontal projection plane have positive quotes and points situated beneath [H] have negative quotes. Points situated in front of vertical projection plane have positive distances and points situated behind [V] have negative distances.

Table 2.1 contains the signs of y and z co-ordinates in the four dihedra:

Table 2.1

Descriptive co-ordinates

Dihedron I Dihedron II Dihedron III Dihedron IV

Distance + - - +

Quote + + - -

The draught is the plane conventional plotting of a spatial plotting, orthogonally projected on projection planes, using only the projection axes. The draught (fig. 2.2 a) is obtained by clockwise rotating plane [H] until it overlaps plane [V]. Thus, a point from space is defined in draught by its horizontal and vertical projections, situated on the same line of recall, fig. 2.2 b.

2.2. Bisector planes

Bisector planes of dihedra I – III and II – IV are respectively marked B1 and B2. Bisector planes represent the geometrical spot of points having quote equaled to distance in modulus (fig. 2.3). The draught of the points M and N contained by the bisector planes is drawn in figure 2.4.

Fig. 2.3 Fig. 2.4

The system consisting of projection planes and bisector planes shares space in eight dihedra, named octants, numbered 1 to 8. A point from space may lay in one of the 17 distinct positions, in

report to projection planes and bisector planes and these positions represent the descriptive alphabet of the point.

Figure 2.5 presents the 17 positions for points A … T drawn both spatial, seen orthogonal from x to O, and draught.

Fig. 2.5

2.3. Treble orthogonal of a point

In order to settle a bi-univocal relationship between the multitude of points from space and the multitude of points from plane, a third plane is introduced, the lateral projection plane, [L], perpendicular to [H] and [V]. The distance between the point from space and the lateral plane is named abscissa and is marked x.

Abscissa, distance and quote are the three descriptive co-ordinates that define a point. Points situated left-side plane [L] have positive abscissa and points situated right-side this plane

have negative abscissa. The projections of point M from space on the three projection planes are (fig. 2.6):

m – horizontal projection of the point;

m′ - vertical projection of the point;

m″ - lateral projection of the point.

Between the three projection planes trihedral angles are formed and they determine the position of co-ordinates axes (OX), (OY) and (OZ). In order to draw the draught the projection trihedron is opened by rotating, in indicated direction, planes [H] and [L], until they overlap plane [V] (fig. 2.7.a). The rotation of the lateral projection m″ describes an arc of circle having the radius equal to distance (fig. 2.7.b).

Fig. 2.6 Fig. 2.7 Planes [H], [V] and [L] divides space in eight trihedra, marked I … VIII (fig. 2.8). The point M

is situated in the first trihedron and the draught is represented in figure 2.9. Figure 2.10 presents the draughts of the point M successively situated in all the other trihedra.

Fig. 2.8 Fig. 2.9

Fig. 2.10

Table 2.2 presents the signs of x, y and z co-ordinates in the eight trihedra:

Table 2.2

Trihedron Descriptive co-ordinates

I II III IV V VI VII VIII

Abscissa + + + + - - - -

Distance + - - + + - - +

Quote + + - - + + - -

2.4. Points situated in projection planes

Points placed in projection planes represent a particular situation because their projection on the plane containing the points is the point itself; the other two projections are situated on the axes (fig. 2.11).

Fig. 2.11

2.5. Visibility

If two points are situated on the same line of recall, then: • the point having the longest quote is visible face the horizontal plane; • the point having the longest distance is visible face the vertical plane; • the point having the longest abscissa is visible face the lateral plane.

2.6. Solved problems

1. Draw the draught of the points A, B and C and specify the trihedron they are situated on:

a) A (20; 20; 30); B (0; 10; 10); C (30; -25; -30). b) A (-20; 30; 20); B (20; 15; 0); C (30; 20; 10). c) A (5; 18; 12); B (30; 0; 15); C (15; -20; 20).

Solution (fig. 2.12):

a) Co-ordinates axes are drawn. In the positive direction of (OX) axis 20 units are measured, marking point ax, and from this point the line of recall perpendicular to (OX) is drawn. The positive distance is measured on the line of recall, beneath (OX) axis, having 20 units length, thus the horizontal projection a is obtained. Being positive the quote is measured 30 units above (OX) axis, marking the vertical projection a′. From a a parallel to (OX) is drawn intersecting (OY) in ay; ay is rotated with an arc of circle with the radius equal to the distance, obtaining ay1 on (OY1) axis. With the help of the line of recall drawn from ay1, a″ is obtained, at the intersection with the lengthening of a′az. All co-ordinates are positive, so the point is situated on the first trihedron. Point B has abscissa equal to zero, so it is situated in the lateral plane. On (OY) axis are measured, in positive direction, 10 units, so that by is found coinciding to the horizontal projection of the point, b. In the positive direction of (OZ) axis the quote of 10 units is measured, obtaining bz, that coincides with the vertical projection b′. The arc of circle of (Ob) radius intersects (OY1) in by1. The lateral projection b″ is obtained. In order to represent point C the 30 units abscissa is measured on (OX) axis. The distance is negative, therefor 25 units are measured above (OX) axis, obtaining c, the horizontal projection of the point. The quote is negative, 30 units are measured beneath (OX) axis, obtaining the vertical projection of the point, c′. In order to draw to the lateral projection the segment (Ocy), equal to the distance, will be rotated until the intersection with the negative direction of (OY1) axis, and, with help of a line of recall, the parallel to (OX) axis, drawn through cz, is intersected. Point C is situated in trihedron

III. The solutions for questions b and c are drawn in figures 2.12 b, c.

Fig. 2.12

2. Draw the draught of the points D (20; 20; z) and E (30; -y; 30), knowing that they are situated in the bisector planes.


Points contained by bisector planes have distance and quote equal to one another. For point D the descriptive co-ordinates are positive yD = zD = 20, the point is situated in B1. For point E │yE│ = │zE│ = │30│, the point is situated in B2.

Fig. 2.13

3. The co-ordinates of point M are M (-10; 30; -15).

a) Draw the draught of point N, symmetric of point M in report to plane [H]. b) Draw the draught of point P, symmetric of point M in report to plane [B2].


a) Point N is symmetric of point M in report to [H], therefor zN = - zM, N (-10; 30; 15). b) Point P is symmetric of point M in report to plane [B2], therefor zP = - yM and yP = - zM, so P (-

10; 15; -30).

Fig. 2.14

2.7. Proposed problems

1. Draw the draught of the points A (20; 0; 30); B (20; -35; 10); C (60; 45; 15); M (45; -25; -40); N (30; 0; 0); P (60; -30; 35). Find the dihedra and the octants the points are situated in.

2. Consider the point A (25; 30; 30). Find the projections of the points A1 and A2, symmetric of point A in report to planes [H] and [V]. Find the projections of point A3, symmetric of point A in report to (OX) axis.

3. Consider in draught the point A situated in dihedron II, octant 3. Find the draught of symmetric points of point A in report to projection planes.

4. Draw the draught of the projection of triangle MNP having the vertex M in the lateral projection plane and the vertices N and P contained by the bisector planes of dihedron II, respectively III.

5. Consider the points A (15; 35; 40) and B (35; -15; -25). Draw the draught of the symmetric points of points A and B in report to bisector planes and precise the dihedra and the octants they are situated in.

6. Consider the point M (15; 20; -10). Draw the draught of the triangle ABC having the vertices symmetric points of point M in report to axis of co-ordinates.

CAPTER 3

PLOTTING OF A STRAIGHT LINE

3.1 Double orthogonal projection of the line

A line (D) from space is projected as (d), the horizontal projection of the line on [H] plane, and as (d′), the vertical projection of the line on [V] plane (fig. 3.1). Projections (d) and (d′) determine the position of a line, marked D (d, d′), in space. Both spatial plotting and the draught of two points M and N situated on the line (D) have projections situated on the like projections of the line. Point A is external to the line because the horizontal projection a is situated on the vertical projection (d′) and the vertical projection a′ is situated on the horizontal projection (d).

Fig. 3.1

3.2. Traces of the line

Traces of the line are the points of intersection between the line and the projection planes (fig. 3.2). In the double orthogonal projection system, a line D (d, d′) intersects the horizontal plane [H] in H, the horizontal trace of the line, and the vertical plane in V, the vertical trace of the line.

Fig. 3.2

The horizontal trace H (h, h′) can be found on draught by lengthening the vertical projection (d′) of the line until it intersects (OX) axis, h′ is situated on (the vertical projection of the horizontal trace). Hence, with a line of recall, h is found on the horizontal projection of the line (d). The vertical trace V (v, v′) can be found lengthening (d) until it intersects (OX), v (the horizontal projection of the vertical trace) is situated on; and, with a line of recall, v′ is found on the vertical projection (d′) of the line.

3.2. Regions of the line

A certain line from space generally traverses three dihedra. The parts of a line belonging to a trihedron are named regions of the line. Limiting the parts of the line situated on each dihedron is known as division of the line in regions. The line in figure 3.3 traverses dihedra I, IV and III. In order to find out in draught the regions of the line, the traces of the line are drawn and its projection are lengthened outside these traces. Thus the line was divided in three regions: two regions situated outside the traces and one between the traces. In each region a point (Q, M, P) is considered and the signs of its descriptive co-ordinates are analyzed, finding out the dihedra the line traverses.

Fig. 3.3

3.4. Treble orthogonal projection of the line

Three projections correspond to a line (D) from space in tri-orthogonal system: (d) – horizontal projection, (d′) - vertical projection, (d″) – lateral projection; the line is marked D (d, d′, d″).

Fig. 3.4

The lateral trace is the intersection point of the line with the plane [L] and it is marked L (l, l′, l″) (fig. 3.4.a). In draught (fig. 3.4.b) traces H (h, h′, h″), V (v, v′, v″) and L (l, l′, l″) have projections of a kind situated on the like projections of the line.

3.5. Characteristic positions of a line in report to projection planes

In report to projection planes a line may be in one of these situations:

Line of particular position: • line parallel to a projection plane; • line perpendicular to a projection plane; • line situated in a projection plane.

Line of common position, having a certain inclination face the projection planes.

3.5.1. Lines parallel to the projection planes

a) The level line (horizontal line) is parallel to the horizontal projection plane, [H]. All horizontal line’s points have the same quote (fig. 3.5).

The horizontal line has no horizontal trace, only vertical V (v, v′, v″) and lateral trace L (l, l′, l″). The draught shows that the horizontal projection (d) of the level line (D) is inclined with angles β and θ face axes (OX) respectively (OY). These represent the real sizes of the angles made by the vertical and lateral projection planes with the line from space.

Line segments situated on a horizontal line are projected in true size on [H] plane.

Fig. 3.5

b) The front line (frontal line) is parallel to the vertical projection plane, [V]. All frontal line’s points have the same distance (fig. 3.6).

The frontal line has no vertical trace only horizontal H (h, h′, h″) and lateral trace L (l, l′, l″). The draught shows that the vertical projection (d′) of the frontal line (D) is inclined with angles α and θ face axes (OX) respectively (OZ). These represent the real sizes of the angles between planes [H], [L] and the frontal line from space.

Segments situated on a frontal line are vertical projected in true size.

Fig. 3.6

c) The profile line is parallel to the lateral projection plane, [L]. All profile line’s points have the same abscissa (fig. 3.7).

Fig. 3.7

The profile line has no lateral trace, only horizontal H (h, h′, h″) and vertical V (v, v′, v″) ones. In draught the lateral projection (d″) of the profile line (D) is inclined to angles α and β to the (OX), respectively (OZ) axis, that represent the real sizes of the angles the line from space is inclined to [H] and [V] plane.

The line – segments contained by a profile line are laterally projected in true length.

3.5.2. Lines perpendicular to the projection planes a) The vertical line is the line perpendicular to plane [H] (fig. 3.8). This line has only horizontal

trace H (h, h′, h″). The horizontal projection (d) of the line coincides with the horizontal trace. Projections (d′) and (d″) are parallel to (OZ) axis.

b) The ending line is the line perpendicular to [V] plane (fig. 3.9). This line has only vertical trace V (v, v′, v″). The horizontal projection (d) is parallel to (OY) and the lateral projection (d″) is parallel to (OY1). The vertical projection is a point that coincides to the vertical trace v′.

c) The frontal – horizontal line is perpendicular to plane [L] (fig. 3.10). The single trace of the line is the lateral one, L (l, l′, l″). Projections (d) and (d′) are parallel to (OX) axis and the lateral projection is a point that coincides to l″.

Fig. 3.8 Fig. 3.9 Fig. 3.10

3.5.3. Lines laying in a bisector plane

a) Line lying in the first bisector plane has projections symmetrical to (OX) axis (fig. 3.11). The traces have merged projections on (OX) axis (h ≡ h′ ≡ v ≡ v′).

Fig. 3.11 Fig. 3.12

b) Line lying in the second bisector plane has projections (d) and (d′) merged and its traces are merged on (OX) axis (fig. 3.12).

3.5.4. Lines lying in a projection plane a) Line lying in the horizontal plane [H] has the horizontal projection (d) inclined face (OX)

axis, the vertical projection (d′) coinciding to (OX) axis and the lateral projection (d″) coinciding to (OY) axis (fig. 3.13).

The horizontal trace of the line (D) is undetermined, because an infinity of points belongs both to the given line and to the horizontal line. The vertical trace v′ is situated on (OX) axis and the lateral trace l″ is situated on (OY1) axis.

Fig. 3.13

b) Line lying in the vertical plane [V] has the horizontal projection coinciding to (OX), and the vertical projection inclined to this axis (fig. 3.14). The horizontal trace of the line lies in the (OX) axis, the lateral trace lies in the (OZ) axis, and the vertical trace is undetermined.

Fig. 3.14

3.6. The relative positions of two straight lines

Two lines may be in one of the following relative positions: lines parallel to each other, lines intersect, skew lines (not parallel and not intersecting) and lies that coincide.

a) Lines parallel to each other have the like projections parallel to each other. Because two projection planes drawn through parallel lines are parallel to each other, and the intersection lines of these planes with the projection planes are parallel to each other; these lines of intersection are the projection of the given lines.

Figure 3.15 presents the draught of the parallel lines (D) and (D1): (d) ║ (d1); (d′) ║ (d1′); (d″) ║ (d1″). In order to settle the parallelism of two lines checking the parallelism of their horizontal, vertical and lateral projections is compulsory.

Fig. 3.15 Fig. 3.16 Fig. 3.17

b) Lines that intersect have the like projections intersecting each other and the intersection point I (i, i′) has the projections situated on the same line of recall (fig. 3.16).

c) Common lines (skew lines) are neither parallel nor intersecting each other. In draught their like projections meet in points that are not situated on the same line of recall. Skew lines are situated in different planes.


1. Given: the line (D) (d, d′, d″), defined by the points A (35; 20; 25) and B (10; -10; -5). Is required:

a) Draw the draught of the line; b) Find the traces of the line; c) Find the dihedra the line passes through.

Solution (fig. 3.18)

a) Draw the projections of the points A and B. Binding the like projections the horizontal, vertical and lateral projections of line (D) are obtained.

b) Line projections are extended to the intersection with co-ordinates axes. The following projections of line’s traces are obtained: h′ = (d′) ∩ (OX); v = (d) ∩ (OX); l = (d) ∩ (OZ). Drawing up the adequate lines of recall helps finding the other projections, thus all the traces of the line are known.

c) The dihedra the line passes through are found by analyzing the signs of the descriptive co-ordinates of some points situated in the three regions of the line:

Points situated left to the vertical trace, for example point A, have all co-ordinates positive, therefor they are situated in the first dihedron.

Points situated between traces have positive quote and negative distance, therefore they belong to the IInd dihedron.

Points situated right to the vertical trace, for example point B, have negative distance and quote, therefor they belong to the IIIrd dihedron.

Fig. 3.18

2. For the line (D), defined by the points A (10; -20; -30) and B (30; 20; -10), is required: a) Find in draught the traces of the line; b) Draw the points of intersection I and J with bisector planes; c) Find the trihedra the line passes through.


a) In draught the construction of the projections and traces of the line is alike previous problem. b) Through h′ (d1′) is drawn, the symmetric of (d′), that intersects (d) in i; i′ and i″ are obtained on

(d′), respectively (d″), drawing the lines of recall. Thus point I (i, i′) is obtained at the intersection of the line (D) with the first bisector. Projections (d) and (d′) intersect in j ≡ j′. Point J (j, j′) represents the intersection of the line (D) with the second bisector.

c) The signs of the descriptive co-ordinates of the points situated in the three regions of the line are analyzed: Points situated left to the horizontal trace have all co-ordinates positive, therefor they

belong to the first trihedron. Points situated between traces, for example point B have positive abscissa and negative

quote, therefor they belong to the IInd trihedron. Points situated right to the vertical trace, for example point A, have positive abscissa,

negative distance and quote, therefor they belong to the IIIrd trihedron.

Fig. 3.19

3. a) Through point A (30; -10; 20) draw the lines:

• the profile line, (D), having an inclination of 60˚ to plane [H]. • the ending line, (∆).

b) Find the traces of the two lines.


a) As the properties of the line are known, through point A projections (d) – parallel to (OY) and (d″) – parallel to (OZ) are drawn. The 60˚ angle is projected in true size on lateral projection, so that through a″ (d″) is drawn, 60˚ inclined in report to (OX). The ending line (∆) is perpendicular to plane [V] and has the projections (δ) ┴ (OX), (δ′) ≡ a′, (δ″) ┴ (OZ).

b) The profile line has two traces: • the horizontal trace, (h1, h1′, h1″) • the vertical trace, (v1, v1′, v1″).

The ending line has only vertical trace (v2, v2′, v2″).

Fig. 3.20

4. Given the line (D) (d, d′) of common position and A (a, a′) a point external to it. Drop a perpendicular from A onto the line (D).


Through point A (a, a′) the horizontal line (G) (g, g′) and the frontal line (F) (f, f′), intersecting line (D) (d, d′) in the points I (i, i′) and J (j, j′), are drawn. Assume AIJ triangle. The desired line is an

altitude of the triangle, drawn from point A, perpendicular to the line (D). In order to draw the perpendicular, the altitudes i′n′ ┴ (f′) and jm ┴ (g) are first drawn, at there intersection the orthocentre C (c, c′) of the triangle is situated. Join the othocentre with the vertex A of the triangle and the desired perpendicular is obtained.

Fig. 3.21

5. In draught draw the perpendicular common both to the frontal – horizontal line (∆) (δ, δ′) and to the ordinary line (D) (d, d′).


Lateral projections (d″) and (δ″) of the given lines are drawn. Because, according to definition, the frontal – horizontal line is perpendicular to [L] plane, the desired perpendicular is a line parallel to [L] plane, meaning a profile line, denoted (E) (e, e′, e″). First (e″), the perpendicular in δ″ to (d″) is drawn. With the help of lines of recall the point A (a, a′, a″), the leg of the perpendicular on line (D), is obtained. By extension and perpendicular to (OX), (e) and (e′) are drawn. The point B, the leg of the perpendicular on the line (∆) is drawn. (AB) is the common perpendicular line to the two given lines.

Fig. 3.22


1. The line (D) (d, d′, d″) is specified by the points A (50; -18; -5) and B (25; -5; 15). Find:

a) The traces of the line; b) The intersections I (i, i′) and J (j, j′) with the bisector planes; c) Trihedra the line passes through.

2. Through point A draw:

a) The horizontal line making a 30° angle with the [V] plane; b) The frontal line 60° inclined to [H] plane;

c) The profile line making a 45° angle with the [H] plane.

3. Given triangle ABC specified by its vertices A (20; 30; 20), B (20; -30; 20) and C (10; 0; -20). Find, in draught, the co-ordinates of its orthocentre.

4. Points A (8; -10; -32) and B (35; yB; 20) specify the frontal line (F) (f, f′). Drop from point C (25; 5; 40) a perpendicular to the frontal given line.

5. Given the line (D) (d, d) specified by the points A (10; 20; 15) and B (50; 0; 45). Given the point C (25; -7; 25) external to the line.

a) Find the horizontal line (G) (g, g′) passing through point C and intersecting line D. b) Find the frontal line (F) (f, f′) passing through point C and intersecting line D. c) Draw the line (∆) (δ, δ′) containing the vertical trace V of the horizontal line (G) and the

horizontal trace H of the frontal line (F).

6. Draw the draught of the traces of the lines passing through point A (35; 45; 40) and respectively perpendicular to the three projection planes.

CHAPTER 4

PLOTTING OF A PLANE

An ordinary plane, [P], can be specified by the following geometric elements: • the projections of two intersecting lines; • the projections of two parallel lines; • the projections of three not collinear points; • the projections of a line and of a point external to it.

4.1. Plotting of a plane

In descriptive geometry the plane is specified either by the elements already enumerated or by its traces.

Traces of a plane are the intersection lines of the plane with the projection planes (fig. 4.1.a). The intersection line between [P] and [H] is the horizontal trace of the plane, denoted Ph; the intersection line between [P] and [V] is the vertical trace, denoted Pv; the intersection line between [P] and [L] is the lateral trace, denoted Pl.

The traces of the plane intersect by twos, in points situated on the projection axes, denoted Px, Py, Pz (fig. 4.1.b).

Fig. 4.1

4.1.1. Line and point lying in a plane For a line to be situated in a plane is compulsory that its traces to be situated on the alike traces

of the plane. H ≡ h ∈ Ph, V ≡ v′ ∈ Pv and in draught h ∈ Ph, v′ ∈ Pv, fig. 4.2.a. In space,

A point lies on a plane if it lies on a line situated in the plane. The projections of the point are situated on the alike projections of the line: m ∈ d, m′ ∈ d′, m″ ∈ d″ (fig. 4.2.b).

Fig. 4.2

4.1.2. Specifying a plane

a) Finding the traces of a plane specified by the projections of two intersecting lines

Given the lines (D1) (d1, d1′) and (D2) (d2, d2′) intersecting in point M (m, m′), fig. 4.3. The traces of lines are drawn, denoted H1, H2, V1 and V2. The traces of the plane are obtained joining the alike traces of the two lines: the horizontal trace, Ph, is obtained joining H1 and H2 and the vertical trace, Pv, joining V1 and V2. The traces of the plane intersect in Px on (OX) axis.

Fig. 4.3

b) Finding the traces of a plane specified by the projections of two parallel lines

For the parallel lines (D1) (d1, d1′) and (D2) (d2, d2′) the horizontal and vertical traces are drawn, joining together the alike traces of the lines the traces of the plane are obtained (fig. 4.4).

Fig. 4.4

c) Finding the traces of a plane specified by the projections of three non collinear points

Given the non collinear points A (a, a′), B (b, b′) and C (c, c′), fig. 4.5. In order to find the traces of the plane [R], the traces of the intersecting lies (AB) (ab, a′b′) and (BC) (bc, b′c′) are drawn. The solution of the problem is alike the one described in 4.1.2.a.

Fig. 4.5 Fig. 4.6

d) Finding the traces of a plane specified by the projections of a line and of a point external to it

Given the line (D) (d, d′) and the point A (a, a′) external to it, figure 4.6. Through point A a line (D1) (d1, d1′) parallel to the given line is drawn. The solution of the problem is alike the one described in 4.1.2.b.

4.2. The characteristic lines of a plane

The characteristic lines lying in a plane are: 1. Lines parallel to a projection plane: the horizontal line, the frontal line, the profile line. 2. The steepest line in report to a projection plane.

4.2.1. Lines situated in a plane and parallel to a projection plane a) The horizontal line (level line) of a plane [P] is parallel to the horizontal projection plane,

figure 4.7. The horizontal projection (d) is parallel to the horizontal trace Ph of the plane [P]. The vertical projection (d′) is parallel to (OX) axis. The level line has no horizontal trace; the vertical and the lateral traces of the line are situated on the alike traces of the plane. The angle between (D) and [V], denoted α, is horizontally projected in true size.

Fig. 4.7

b) The front line (frontal line) of a plane [Q] is parallel to the vertical projection plane, figure 4.8. The horizontal projection of this line is parallel to (OX), and the vertical projection is parallel to the trace Qv of the plane. The angle β between (∆) and [V] is vertically projected in true size.

Fig. 4.8

c) The profile line of a plane [R] is parallel to the lateral projection plane, figure 4.9. The horizontal (d) and vertical (d′) projections are one in prolongation to the other and also perpendicular to (OX). The lateral projection (d″) is parallel to the lateral trace of the plane, Rl, showing the angles between the line from space and the vertical and horizontal planes.

Fig. 4.9

4.2.2. The steepest lines of a plane

a) The steepest line in report to the horizontal projection plane

The steepest line of a plane in report to [H] is a line contained by the plane and perpendicular to all the horizontal lines of the plane, therefor to the horizontal trace of the plane, figure 4.10.

Fig. 4.10

The use of this characteristic line of the plane [P] consists in the fact that the dihedral angle between [P] and [H], denoted α, equals the angle between the steepest line, (∆), and its horizontal projection, (δ), figure 4.11. In order to find the true size of this angle the triangle h1 v1 v1′ is rotated around (h1 v1) cathetus.

The horizontal projection of the steepest line in report to [H] is perpendicular to the trace Ph.

Fig. 4.11

Another useful characteristic of the steepest line is that the line completely specifies the plane is situated on (fig. 4.12). Known (D) (d, d′) the steepest line in report to [H] of the plane to be found. The traces of the line are drawn. The perpendicular drop in h to (hv) is the horizontal trace, Ph, of the plane. Px and v′ are joined, obtaining the vertical trace, Pv, of the plane, that is completely specified.

Fig. 4.12

b) The steepest line in report to the vertical projection plane

The steepest line in report to [V] is a line contained by the plane and perpendicular to all the frontal lines of the plane, therefor to the vertical trace Qv, figure 4.13.

Fig. 4.13

The vertical projection of this line is perpendicular to the Qv trace, figure 4.14.

Fig. 4.14

In order to find the true size of the dihedral angle between [Q] and [V], denoted β, the procedure is alike the described before, but the folding is in vertical plane.

4.3. Characteristic positions of a plane

4.3.1. Planes parallel to a projection plane. Double projecting planes. a) The level plane, [N], is a plane parallel to the horizontal projection plane, [H], and is also

perpendicular to the other two projection planes, figure 4.15. In draught, the vertical and lateral traces of the level plane are one in prolongation to the other and also parallel to (OX). The level plane has no horizontal trace.

A plane geometrical figure situated in a level plane is horizontally projected in true size; the vertical and lateral projections of the figure are completely distorted (line-segments).

Fig. 4.15

b) The frontal plane, [F], is parallel to the vertical projection plane, [V], figure 4.16. In draught, traces are Fh ║ (OX), Fl ║ (OZ); the plane has no vertical trace.

Fig. 4.16

A plane geometrical figure situated in a frontal plane is projected on plane [V] in true size and on the other projection planes is projected distorted, as line-segments.

c) The profile plane, [Q], is parallel to the lateral projection plane, fig. 4.17. In draught the horizontal and vertical traces are one in prolongation to the other and also parallel to (OZ).

Fig. 4.17

A plane geometrical figure contained by a profile plane is laterally projected in true size and on the other projection planes is completely distorted (line-segments).

4.3.2. Planes perpendicular to a projection plane. Projecting planes. a) The vertical plane, [R], is perpendicular to the horizontal projection plane (fig. 4.18). In

draught, the vertical and lateral traces are perpendicular to (OX), respectively (OY1). The angles between the horizontal trace and (OX) and (OY) axes are the true sizes of the angles between plane [R] and plane [V], respectively [L].

Fig. 4.18

b) The ending plane, [S], is perpendicular to the vertical projection plane (fig. 4.19). In draught, the horizontal and lateral traces are perpendicular to (OX), respectively (OZ). The angles between the vertical trace and (OX) and (OZ) axes are the true sizes of the angles between plane [S] and plane [H] and [L].

Fig. 4.19

c) The plane parallel to (OX) axis, [T], is perpendicular to the lateral plane, (fig. 4.20). The horizontal and vertical traces are parallel to (OX) axis, and the angles between the lateral trace and (OY) and (OZ) are the true sizes of the angles between plane [T] and planes [H], respectively [L].

Fig. 4.20

d) The plane containing (OX) axis, [W], is perpendicular to the lateral plane and has the horizontal and vertical traces coinciding to (OX) (fig. 4.21).

Fig. 4.21

4.4 The relative positions of two planes

Two planes may be in one of the following relative positions:

parallel planes; intersecting planes: - under an arbitrary angle;

- perpendicular planes.

4.4.1. Parallel planes Two parallel planes have the alike traces parallel to each other: Ph ║ Qh, Pv ║ Qv, Pl ║ Ql (fig.

4.22).

Fig. 4.22

All the projecting planes in report to plane [L], parallel to (OX) axis, are mutually parallel (fig. 4.23).

Fig. 4.23

4.4.2. Intersecting planes Given two planes, [P] and [Q], intersecting along a line (D) (fig. 4.24). Finding the intersection

line of the two planes consists in finding the traces of the line, lying on the alike traces of the planes. The horizontal trace of the line, h, is the intersection point between Ph and Qh; with the help of a line of recall h′ is obtained. The vertical trace, v′, is the intersection point between Pv and Qv; with the help of a line of recall v is obtained. The horizontal projection (d) of the intersection line is drawn through h and v and the vertical projection (d′) is drawn through h′ and v′.

Fig. 4.24

When is not enough room for the traces of the plane to intersect (fig. 4.25), two auxiliary planes are drawn, a level plane [N] and a frontal plane [F]. Plane [N] intersects planes [P] and [Q] along two horizontal lines, intersecting each other in point K (k, k′). Plane [F] intersects planes [P] and [Q] along two frontal lines, intersecting each other in point M (m, m′). The desired intersection line, denoted (D) (d, d′), joins together points K and M.

Fig. 4.25

4.5. Relative positions of a line in report to a plane

In report to a plane, a line may be in one of the following positions: line lying in the plane; line parallel to the plane; line intersecting the plane: - under an arbitrary angle;

- perpendicular to the plane.

4.5.1. Line lying in a plane A line (D) (d, d′) lies in a plane [P] (Ph, Pv) if its traces lye on the alike traces of the plane: H (h,

h′) ∈ Ph, V (v, v′) ∈ Pv, figure 4.26.

Fig. 4.26 Fig. 4.27

4.5.2. Line parallel to the plane A line is parallel to a plane if it is parallel to a line situated in the plane and reverse. In figure 4.27 is shown the way to draw a line, (D1), passing through point M and parallel to

plane [P]. Given the line (D) (d, d′) contained by plane [P]. The projections (d1) ║ (d), (d1′) ║ (d′), of the line (D1), parallel to [P] are drawn through point M.

4.5.3. Line intersecting a plane In order to find out the intersection point between a line (D) and a plane [P] an auxiliary plane,

[R], containing line (D) is drawn (fig. 4.28). Usually, the auxiliary plane is a projection plane, in our case [R] is a vertical plane. The intersection line (∆), between planes [P] and [R], is drawn. The intersection point between lines (D) and (∆), denoted N (n, n′), is obtained. The intersection point between line (D) and plane [P] is N.

Fig. 4.28

4.5.4. Line perpendicular to a plane A line is perpendicular to a plane if it is perpendicular to any line contained by the plane. Given the plane [Q], specified by the intersecting lines (∆1) (δ1, δ1′) – level line and (∆2) (δ2, δ2′)

– frontal line (fig. 4.29). The intersection point of the lines is N (n, n′). A line (D) is perpendicular to plane [Q] if it is perpendicular to any line contained by the plane; therefor to the lines (∆1) and (∆2). The projections (d) and (d′) of the line are perpendicular to the projections (δ1) and (δ2′), therefor to the traces Qh and Qv. A line is perpendicular to a plane when its projections are respectively perpendicular to the alike traces of the plane.

Fig. 4.29

4.6. Visibility in draught

When settle visibility in draught points, lines and planes are thought that opaque. If the projections of the geometrical elements coincide, their visibility in report to each projection plane is analyzed. The hidden elements are conventionally drawn with thin dashed line.

4.6.1. Intersection between a geometrical figure and a line The solution of this problem consists in finding the intersection point and analyzing the visibility

in report to projection planes. Given the triangle ABC (abc, a′b′c′) intersected with an arbitrary line (D) (d, d′), figure 4.30.

Fig. 4.30

In order to find the intersection point N (n, n′) a vertical plane, [P], is drawn through line (D). This plane intersects triangle ABC along the line (12, 1′2′), that intersects line (D) in the desired point N (n, n′). According to the reasons presented in chapter 2.5 the visibility in report to [H] and [V] can be settled.

4.6.2. The intersection of two plane geometrical figures Given the triangular plates ABC (abc, a′b′c′) and MNP (mnp, m′n′p′). Through (AB) and (AC) sides the projecting planes [S] and [Q] are drawn, perpendicular to plane

[H]. The intersection line is denoted (12, 1′2′). Visibility, in draught, of the two surfaces is settled analyzing the visibility of the sides (fig. 4.31).

Fig. 4.31


1. Given the points A (45; 10; 17), B (17; -7; 32), C (60; -35; 28) and M (35; 14; 17). Is required: a) Draw the traces of the plane [P] specified by the points A, B, C; b) Draw the traces of the plane [Q], that contains point M and is parallel to plane [P].


a) Through points A and B the line (D1) (d1, d1′) is drawn and its horizontal H1 (h1, h1′) and vertical V1 (v1, v1′) traces are found.

Through points A and C the line (D2) (d2, d2′) is drawn, its vertical trace is V2 (v2, v2′). The vertical trace Pv of the desired plane intersects (OX) in Px and is obtained joining together v1′ and v2′. The horizontal trace Ph is obtained joining together Px and h1.

b) Through point M a line (D3) (d3, d3′), parallel to (D1) (d1, d1′), is drawn. Through the traces of the line (D3), Qh, Qv, respectively parallel to Ph, Pv, are drawn.

Fig. 4.32

2. Given the line (D) (d, d′) specified by the points A (25; 36; -36) and B (50; -12; 48). Is required: a) Find plane [P] having line (D) as the steepest line in report to [H]; b) Draw plane [Q], parallel to (OX) having the traces at y = 30 and z = 24; c) Draw the intersection line (∆) of the planes [P] and [Q].


a) The projections of the line (D) are drawn and its horizontal and vertical traces are found. Drop the perpendicular from h1 to (d), which is Ph. The intersection between Ph and (OX) axis is Px. Through Px and v1′ is drawn Pv.

b) Traces Qh, and Qv,, parallel to (OX), are drawn at the indicated distances. c) The traces of the intersection line (∆) are found at the intersection of the alike traces of the

plane, respectively h2 and v2′. With the help of a line of recall h2′ and v2 are found. The projections of the line (∆) (δ, δ′) are obtained joining together h2 and v2, h2′ and v2′.

Fig. 4.33

3. Given the points: A (10; 45; -20), B (80; -10; 60), M (95; 0; 45), N (55; 40; 0), S (35; 45; 50), T (60; 35; 15), Px (10; 0; 0) and Qx (105; 0; 0;). Find:

a) Plane [P] specified by the points Px, M and N; b) Plane [Q] specified by the points Qx, A and B; c) The intersecting line (∆) of the planes [P] and [Q]; d) The intersection point I between plane [P] and line (D) specified by the points S and T.


a) Points M and N are situated both on the vertical, respectively horizontal projection plane, and on the desired plane [P], therefor these points are situated on the traces Pv, respectively Ph. Binding these points to Px plane [P] is obtained.

b) Traces H (h, h′) and V (v, v′) of the line (AB) are found and the traces of the plane [Q] are obtained joining together the traces of the line and Qx.

c) The traces of the line (∆) (δ, δ′) are situated at the intersection of the alike traces of the planes. d) Line (D) is drawn and, through its vertical projection, the ending plane [R] is constructed. The

intersection line between planes [P] and [R] is found and is denoted (D1) (d1, d1′). The intersection between the lines (D) and (D1) is the desired point I.

Fig. 4.34

4. Given the plane [P] specified by the point A (24; 30; 18) and the front line (F) (f, f′) having the horizontal trace H (48; 12; 10) and being 60˚ inclined in report to horizontal projection plane. a) Draw the traces of the plane [P]; b) Draw the steepest line (∆) (δ, δ′), of the plane [P] in report to [V], knowing that it passes

through B (30; yB; 15). Solution (fig. 4.35)

a) The frontal line (F) (f, f′) is plotted and another frontal line (F1) (f1, f1′) is drawn through A (a, a′), finding its horizontal trace H1 (h1, h1′). The horizontal trace Ph is obtained joining together h and h1. The vertical trace is parallel to (f′) and (f1′).

b) The vertical projection (δ′) is drawn through b′, perpendicular to Pv. With the help of lines of recall h2 and v2 are obtained, which, joined together, give the horizontal projection (δ) of the desired line. This horizontal projection, (δ), also contains the horizontal projection b; thus it is graphically found the distance of the point B.

Fig. 4.35


1. Given the points Px (50; 0; 0), A (5; 30; 0), B (5; 0; 35), Qx (55; 0; 0), C (60; 0; 10) and M (20; 0; zM). Through M draw the line (D) contained by the plane [P], specified by Px, A and B, knowing that the line is parallel to the ending plane [Q], specified by Qx and C.

2. Given the points Px (5; 0; 0), H (25; 25; 0), V (90; 0; 35), Qx (85; 0; 0), A (40; 5; 35) and B (5; 15; zB). Find: a) Plane [P], specified by Px, H and V; b) Plane [Q], specified by Qx and the horizontal line (N), specified by A and B; c) Plane [R] parallel to (OX) having the traces y = z = 20; d) The intersection point I of the planes [P], [Q] and [R].

3. Given the points A (30; -20; 50) and B (90; 40; -10) specifying line (D). Find: a) The traces of the plane [P] having (D) the steepest line in report to [H]; b) The intersection line (D1) between plane [P] and the level plane having the quote z = 40; c) The intersection line between plane [P] and the frontal plane having the distance y = 50.

4. Given the points A (145; 50; 70), B (160; 105; 30), H (70; 50; 0), M (110; 0; 110) and N (50; 80; 0). Find:

[P] specified by A, B and Ha) The traces of the plane ; b) The projections of the horizontal line (E) passing through B and contained by plane [P]; c) The traces of the plane [Q], specified by the points M, N and Qx (170; 0; 0); d) The intersection line between planes [P] and [Q]; e) The intersection point I of the line (D), specified by points A and B, with plane [Q].

5. Given the points A (50; -40; 0), B (30; 0; 90), C (85; 25; 25), E (70; 25; 70), S (60; 50; 45), Rx (130; 0; 0) and Px (70; 0; 0). Find: a) Plane [R] specified by Rx, A and B; b) Plane [P] specified by Px, C and E; c) The intersection line of the planes [P] and [R]; d) Plane [Q], parallel to plane [P] and passing through point S;

e) Through C draw the horizontal line (E) and check-up if it is situated in the plane [P].

6. Find the intersection and analyze the visibility of the triangular plates ABC and DEF, thought as opaque, knowing that A (60; 20; 5), B (35; 5; 35), C (5; 30; 10), D (55; 25; 15), E (40; 40; 35) and F (10; 10; 0).

descriptive geometry - part i

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