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Файл:Descriptive geometry. Course of lectures for international students
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2. LINE PROJECTIONS
2.1. Line projecting on the three projection planes
Line in the space is a result of two planes intersection. 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 П1, 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 П2
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 (Fig. 2.1).
Fig. 2.1. Line projections
Fig. 2.2. Line projections
In either case the projection of the line AB lies in the line of intersection
between the projection surfaces and the planes of projection.
In general, a right line will be fully determined by its two projections.
The projecting drawing of line AB is shown on the Fig. 2.2. The common rules
of point projection (A and B) are used for drawing AB-line projections.
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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:
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 (Fig. 2.3).
2. Parallel to one. When line inclining to the other plane or line lies in a
plane perpendicular to ground-line (Fig. 2.4).
3. Parallel to neither. In this case, line inclining at any angel (Fig. 2.2).
Fig. 2.3. Line position in space. Parallel to both:
a – line parallel to П1 and П2 planes of projection, and perpendicular to П3 plane;
b – line parallel to П2 and П3 planes of projection, and perpendicular to П1 plane;
c – line parallel to П1 and П3 planes of projection, and perpendicular to П2 plane
a)
b)
c)
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a)
b)
c)
Fig. 2.4. Line position in space. Parallel to one:
a – line parallel to one П2 and inclining to other П1 and П3 planes of projection;
b – line parallel to one П1 and inclining to other П2 and П3 planes of projection;
с – line parallel to one П3 and inclining to other П1 and П2 planes of projection
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If a line is perpendicular to a plane (Fig. 2.3), its projection on the plane
is a point.
A line segment parallel to a plane projects in its true length (TL) on the
plane: AB||П2, A2B2=TLAB on Fig. 2.4, a, and AB||П3, A3B3=TL
AB
on
Fig. 2.4, c. A vertical line projects as a point on the П1-plane (Fig. 2.3, b).
A frontal line is defined as a line parallel to the П2-plane (Fig. 2.4, c).
The front view A2B2 is a true-length projection and also shows the true angle
between line AB and the П1-plane ‒ φ1.
Horizontal line segment appears in true length in the top view (Fig. 2.4, b).
The angle between the horizontal projection, A1B1, and OX φ2 is the true angle
between line AB and the vertical plane – П2; angle φ3 is a true angle between
line AB and profile plane.
A profile line is a parallel line to the profile plane (Fig. 2.4, c). The side
view A3B3 is a true-length view. The angle between A3B3 and OZ is the true
angle between line AB and the П2-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.
Fig. 2.5. Line position in space
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 (Fig. 2.5). Point К lies
on the line AB due to its horizontal and vertical projections lie on the
correspondent projections of the line: К1 Є A1B1; and К2 Є A2B2.
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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” (Fig. 2.6, 2,7).
Fig. 2.6. Traces of line AB in space
Fig. 2.7. Traces of line AB 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
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M from which the horizontal projection Mi may be determined by means of
the ordinate. The point M whose distance from П1 is a minimum is that which
the line AB in space pierces plane П1, 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 П2 is a minimum is that which the
line AB in space pierces plane П2, 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 than itself (Fig. 2.8). In this case that line (AB) is
the hypothenuse of the right-angled triangles ABB1, hence is greater than 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 (ΔZAВ=Z
A
– Z
B
)
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*.
Fig. 2.8 True length of AB
This line is the true length of AB and the angle between A1B* and A1B
1
is the true angle of line intersection with П1 plane (Fig. 2.9).
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Fig. 2.9. True length of AB on the vertical
and horizontal projection
The similar way of draw the true length of AB on the vertical projection.
But in this case the difference between depth of points A and B should be
drawn on the vertical projection of AB. This difference may be found on the
horizontal projection of the line as it shown on the Fig. 2.9. The line A*B2 is
the true length of AB and the angle between A*B2 and A2B2 is the true angle
of line intersection with П2 plane.
2.5. Position of lines in space
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 (Fig. 2.10, 2.11). Conversely, if the
projections at the same name (horizontal or vertical) are parallel, the lines in
space are parallel.
Intersecting lines. Two lines in space which intersect give projections
which intersect in points which lie in the same perpendicular to ground lines
(Fig. 2.12).
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Fig. 2.10. Parallel lines in space
Fig. 2.11. Parallel lines:
general parallel lines
Fig. 2.12. Intersecting lines
Fig. 2.13. Skew lines
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.
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 (Fig.2.13).
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2.6. Competing points of skew lines
What does it mean – the point of projection lines intersection? It is the
projection of two points (1 and 2), one of them lies on the one line (1ЄCD),
and another lies on the second line – 2ЄAB (Fig. 2.14).
Fig. 2.14 Competing points of skew lines
It is necessary to note that these points have the same distance to the П1
plane, but the distance to the П2 planes is different: point 2 lies nearer that
point 1 to the П1 plane.
The competing points are necessary to find the points or lines visibility
at the different view direction. The point 1 that lies on the line CD block the
view of the point 2 that lies on the line AB at the П1-plane in vision direction
shown as view arrow nearby to the 12 projection (Fig. 2.14).
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3. PLANE PROJECTIONS
3.1. Methods of a plane determination
The position of a plane in space may be determined by different methods
presented in the Table 3.1
Table 3.1
Methods of plane determination on a drawing
Method
View in space
Projection drawing
1. Three points that
do not lie in the
same right line
2. By a line and
a point that does
not lie in that line
3. By two
intersecting lines
4. By two parallel
lines
5. By planar figure
(triangular)
6. By plane traces
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