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Файл:Начертательная геометрия. Курс лекций. Учебное пособие
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2. LINE PROJECTIONS
Figure 2.1. Planes intersection
Figure 2.2. Line projections
2.1. Line projecting on the three projection planes
Line in the space is a result of two planes intersection (Figure 2.1). Line in space is not
limited. The limited part of line is called “line segment”.
As a line is a succession of points, its projection on any plane will be determined by projecting each point of the line on that plane. Thus, if from the different points of the line AB
perpendicular be drawn to plane H, their feet will indicate the horizontal projections of those
points, and the line A1B1, which passes through them, the horizontal projection of the line itself. In like manner the vertical projection A2B2 will be found by drawing projecting lines to
the vertical plane V (Figure 2.1).
This regular succession of parallel projecting lines (AA1 and BB1, AA2 and BB2, AA3
and BB3) will form the surface, which in the case of the right line is projecting plane (Figure
2.2). In either case the projection of the line AB lies in the line of intersection between the projection surfaces and the planes of projection (AA2B2B∩V in A2B2).
In general, a right line will be fully determined by its two projections. The projecting
drawing of line AB is shown on the Figure 2.3. The common rules of point projection (A and B)
are used for drawing AB-line projections.
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a) line parallel to H and V planes of projection, and perpendicular to P plane
Figure 2.3. Line projections
2.2. Position of a line to the planes of projection
A right line in space may have one of three positions to the planes of projection (Figure
2.4):
1. Parallel to both. In this case its two projections on the plane, which it is parallel, are
parallel to ground-line between these planes, and the third projection is a point because the line
perpendicular to the third plane (Figure 2.4a and Figure 2.4b)
2. Parallel to one. When line inclining to the other plane (Figure 2.4c) or line lies in a
plane perpendicular to ground-line (Figure 2.4d);
3. Parallel to neither. In this case, line inclining at any angel (Figure 2.2 and 2.3).
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b) line parallel to V and P planes of projection, and perpendicular to H plane
c) line parallel to one (V) and inclining to other (H and P) planes of projection
d) line parallel to one (P) and inclining to other (H and V) planes of projection
Figure 2.4. Line position in space
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If a line is perpendicular to a plane (Figure 2.4a and Figure 2.4b), its projection on the
a) Horizontal
b) Frontal
plane is a point.
A line segment parallel to a plane projects in its true length (TL) on the plane: AB||V,
A2B2=TLAB on Figure 2.4c, and AB||P, A3B3=TLAB on Figure 2.4d. A vertical line projects as a
point on the H-plane (Figure 2.4b).
Horizontal line segment appears in true length in the top view (Figure 2.5a). The angle
between the horizontal projection, C1D1, and OX is the true angle between line CD and the vertical plane – β; angle γ is a true angle between line CD and profile plane.
A frontal line is defined as a line parallel to the F-plane (Figure 2.5b). The front view E2F2 is
a true-length projection and also shows the true angle between line EF and the H-plane – α.
A profile line is a parallel line to the profile plane (Figure 2.5c). The side view G3K3 is a
true-length view. The angle between G3K3 and OZ is the true angle between line GK and the
V-plane.
It should be noted that when a line segment is parallel to the reference line in one view,
the adjacent view is a true-length view.
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c) Profile
Figure 2.5. Straight lines
Note: If a point lies on a line, a pair of projections of the point will lie on a line which is perpendicular to the reference line (Figure 2.6). Point C lies on the line AB due to its horizontal and
vertical projections lie on the correspondent projections of the line: C1 ϵ A1B1; and C2 ϵ A2B2.
The points M and N do not lie on the line AB.
Figure 2.6. Relation between points and line
Note: If a space line is divided in a given ratio, its projection is divided into the same ratio. For example, the point K divides the EF-line in ratio 3:5. For this purpose, the other line
F1E0 draws on the horizontal projection of the line EF and it is divided into 3+5=8 equal segments. Then the point E0 connects with E1. The line K0K1, that draws parallel to line E0E1, divides the horizontal projection E1F1 into the necessary ratio – 3:5.
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a)
Figure 2.7. Method of dividing the line into required ratio
2.3. Traces of a line
A general line in space intersects the projection planes in the specific points that are
called “trace” or “piercing-point” (Figure 2.8).
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b)
Figure 2.8. Traces of line AB: a) in space; b) on the projection planes
When the vertical projection A2B2 of line is prolonged until it intersects OX, the point of
intersection M2 is the vertical projection of the required point M from which the horizontal projection M1 may be determined by means of the ordinate. The point M whose distance from H is
a minimum is that which the line AB in space pierces plane H, and is termed the horizontal
trace of that line.
When the projection A1B1 of line is prolonged until it intersects OX, the point of intersection
N1 is the horizontal projection of the required point N from which the vertical projection N2 may be
determined by means of the ordinate. The point N whose distance from V is a minimum is that
which the line AB in space pierces plane V, and is termed the vertical trace of that line.
2.4. True length of line definition
There are no projections of general line define its true length because gives projections
are shorter then itself (Figure 2.9a). In this case that line (AB) is the hypothenuse of the rightangled triangles ABB1, hence is greater then the bases AB1 or this equivalent, the projection
A1B1. Thus, another base of the triangular is the difference between height of points A and B
(ΔZAB=ZA – ZB) that may be defined on the vertical projection A2B2. If this section is drawn on
the horizontal projection at the right angle to the A1B1 the hypothenuse of that triangle may be
found as a line A1B*. This line is the true length of AB and the angle between A1B* and A1B
is the true angle of line intersection with H plane (Figure 2.9b).
The similar way of draw the true length of CD on the vertical projection (Figure 2.9c).
But in this case the difference between depth of points C and D should be drawn on the vertical
projection of CD. This difference may be found on the horizontal projection of the line as it
shown on the Figure 2.9c.The line C*D2 is the true length of CD and the angle between C*D2
and C2D2 is the true angle of line intersection with V plane.
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1
a)
b)
c)
Figure 2.9. Method of triangular drawing on the projection planes:
a) in space; b) on the H plane; c) on the V plane
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2.5. Position of lines in space
a)
b)
c)
Two lines in space may assume two general positions to each other:
1) they may either lie in the same plane, and intersect or be parallel;
2) they may have such a position that no plane can be passed through them – they can
affect neither intersect and be parallel.
Parallel lines. Two parallel lines in space give projections that are parallel, since their
projecting planes are parallel, and hence cut the coordinate planes in parallel lines (Figure
2.10a). Conversely, if the projections at the same name (horizontal or vertical) are parallel, the
lines in space are parallel (Figure 2.10b and Figure 2.10c).
Figure 2.10. Parallel lines: a) perpendicular planes through parallel lines;
b) general parallel lines; c) parallel lines that lie in one plane
The parallel position for profile lines is defined on the profile projection (Figure 2.11).
Intersecting lines. Two lines in space which intersect give projections which intersect in
points which lie in the same perpendicular to ground lines (Figure 2.12).
Figure 2.11. Parallel profile lines
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For the point of intersection being common to both lines, its projections must likewise be
common to the two projections and follow the law which governs the projections of any point.
Conversely, when the like projections of the lines intersect in points lying in a common perpendicular to ground-line, the lines in space intersect.
Figure 2.12. Intersecting lines
When two lines in space neither parallel nor intersection, there are termed skew lines.
Projections of the point of the projection lines intersection does not lie in the common perpendicular to the ground lines (Figure 2.13).
Figure 2.13. Skew lines
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