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Файл:Начертательная геометрия. Курс лекций. Учебное пособие
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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 (K and L), one of them lies on the one line (LϵCD), and another lies on the second line –
KϵAB (Figure 2.14).
It is necessary to note that these points have the same distance to the V plane, but the distance to the H planes is different: point K lies nearer that point L to the H plane.
The competing points are necessary to find the points or lines visibility at the different
view direction. The point L that lies on the line CD block the view of the point N that lies on
the line AB at the H-plane in vision direction shown as view arrow nearby to the L2 projection
(Figure 2.14). The point N that lies on the line AB block the view of the point L that lies on the
line CD at the V-plane in vision direction shown as view arrow nearby to the N1 projection.
Figure 2.14. Competing points of skew lines
2.7. Right-angle projection
There are two rules for projecting of right-angle:
1) When the plane that passes through the general angle perpendicular to the projection
plane, the projection of this angle on that plane is a line.
2) When the plane that passes through the right angle do not perpendicular to any projection planes, and one of its side is parallel to projection plane, this right angle projecting to that
plane as a right angle (Figure 2.15).
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Figure 2.15. Right angle projections
In the Figure 2.15 the lines BC and AC perpendicular to each other due to BC is parallel
to horizontal projection plane (A2B2ǀǀOX) and line A1C1intersects line C1B1 at the right angle.
The lines ED and FE perpendicular to each other due to ED is parallel to vertical projection
plane (E1D1ǀǀOX) and line F2E2 intersects line D2E2 at the right angle.
CONTROL TEST
1. List the main properties of a line in space. Which principles are used for line projecting?
2. Describe three position of line to the planes of projections. Represent the samples of them.
3. Represent the horizontal, frontal and profile lines. Which properties of them can you list?
4. When does a point lie on a line? Represent the method of dividing the line into re-
quired ratio.
5. What does the term “line traces” mean? Represent the traces of oblique line.
6. Describe the right-angle triangles method for true-length of line definition.
7. Which lines in space can be termed as parallel? Represent any sample.
8. Which lines in space can be termed as intersecting? Represent the sample.
9. Which lines in space can be termed as skew lines? Give the definition of competing points
of skew lines.
10. List the main rules of right-angle projecting.
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3. PLANE PROJECTIONS
№
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
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
Methods of plane determination on a drawing
Table 3.1
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4
By two parallel
lines
5
By planar figure (triangular)
6
By plane traces
plane traces, and in ordinary practice are the means whereby the plane is determined in position (Figure 3.1).
Method 6).
the trace G1. The horizontal trace of line that lies in the plane will lie on the horizontal trace of
this plane.
plane traces may be defined by drawing the traces of line that are determined this plane.
coincide with GL (ground lines), since they represent the projections of two right lines of that
plane which must either intersect or be parallel to each other.
3.2. Plane traces
The lines in which a general plane intersects the planes of projection are termed the
In the ordinary cases the plane has three traces: horizontal, frontal and profile (Table 3.1,
When line AB lies in the plane G, it intersects the H-plane in the point M that will lie on
Note: The traces of a line that lie in a plane will lie on the related plane traces. The
The traces do not necessarily measure the angles of inclination.
The traces of a plane serve to indicate its position in all cases except that in which they
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When the traces of a plane are not parallel they must intersect each other in a point in ground
line, since the coordinate planes and the given plane form a solid angle whose vertex is the point of
intersection or the piercing-point of ground line on that plane – points Gx, Gy and Gz.
a)
b)
Figure 3.1. Plane traces
3.3. Points and lines are contained in plane
When two points of a line lie in a plane that line lies in that plane or is contained in that
plane. Line MN is contained in the plane G, which is given by traces, due to points M and N lie in
the plane traces – they lie in the plane G (Figure 3.2a). Line 1-2 has the common points with the
lines AB and CD, hence line 1-2 lies in the plane, that is given by that parallel lines (Figure 3.2b).
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When point lies on a line that is contained in a plane this point lies in that plane. Thus, it
a)
b)
is necessary to draw the line (NM or 1-2), that will be contained in the plane, and find the point
on this line. This point (E) will lie in the plane (Figure 3.2).
Figure 3.2. Point in the planes
3.4. Position of a plane to the coordinate planes
A plane in space may assume the following general positions to the coordinate planes
(Table 3.2): it may
3) passes through the ground-line, giving no traces in two coordinate planes, and hence
indeterminable except by the use of a new vertical plane;
4) be parallel to one, given but two traces parallel to GL on that plane to which it is not
parallel (table 3.2, r.4-6);
5) be parallel to GL and perpendicular to one plane and inclines to two planes, when
both traces will be parallel to that line;
6) be perpendicular to one of plane and inclines to two other planes (Table 3.2, r.1-3),
when first trace will be inclined to GL and defined the true angles of it inclining to other
planes, and two other traces be perpendicular to GL. These planes are termed projecting plane.
When this plane perpendicular to the horizontal plane it is termed horizontal projecting plane,
when it perpendicular to the vertical plane – frontal projecting plane, to the profile plane – pro-
file projecting plane.
7) Inclines to all planes, other than in the first and third cases, when all traces will in-
cline to GL.
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Some position of a plane to the coordinate planes
№
Position
in space
View in space
Projection drawings
Position
of plane traces
1
Perpendicular
to the H-plane –
horizontal
projecting plane
G1 – inclines to OX
G2 – is perpendicular
to OX
G3 – is perpendicular
to OY
2
Perpendicular
to the V-plane –
vertical projecting
plane
G1 – is perpendicular
to OX
G2 – inclines to OX
G3 – is perpendicular
to OZ
3
Perpendicular
to the P-plane –
profile projecting
plane
G1 и G2 – are parallel
to OX
G3 – inclines to OZ
4
Parallel
to H-plane – hori-
zontal plane
G1 – is absent
G2 – is parallel
to OX
G3 – is parallel
to OY
5
Parallel
to V-plane –
frontal plane
G1 – is parallel to OX
G2 – is absent
G3 – is parallel to OZ
6
Parallel
to P-plane –
profile plane
G1 и G2 – are perpendicular to OX
G3 – is absent
Table 3.2
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3.5. The principle lines of a plane
a)
b)
c)
a)
b)
c)
There are a lot of lines lie in a plane in space, some of which are parallel to the coordinate planes, and some lines perpendicular to them. Lines which are parallel to coordinate
planes are termed principle lines, and ones perpendicular to them are termed lines of greatest
declivity, which are measure the angle of the plane in space with either plane of projection.
Lines which are parallel to horizontal plane H are termed horizontals, ones are parallel to vertical plane V – frontals, and lines which are parallel to profile plane P – profiles.
Horizontals may be represented as lines of intersection of the general plane and any horizontal plane. That is why the horizontal trace can be associated as a horizontal at the zero-level.
All horizontal projections of horizontals are parallel to the horizontal trace of a plane, and
their vertical projections are parallel to OX.
When the plane given by traces (Figure 3.3b), the vertical projection of horizontal (h2) is
drawn as a line parallel to OX. Then the point of intersection that line and vertical trace of
plane is found. This point is a piercing-trace of horizontal h – N≡N2. In the next step the horizontal projection of the N-point is found as a point on the OX by fall perpendicular to the
ground-line. Then horizontal projection of horizontal h is drawn as a line that is parallel to horizontal trace of the plane and passed through the N1-point.
When the plane given by triangular (Figure 3.3c), the vertical projection of horizontal (h2) is
drawn as a line parallel to OX and passed through any two points of the triangular. Then the horizontal projections of those points are obtained and the horizontal projection of horizontal is found.
Frontals may be assumed as lines of intersection of the general plane and any vertical plane.
Hence vertical trace can be found as a vertical at the zero-level. All vertical projections of frontals are
parallel to the vertical trace of a plane, and their horizontal projections are parallel to OX.
Figure 3.3. Horizontals of planes
Figure 3.4. Frontals of planes
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The lines of greatest declivity are perpendicular to the respective traces. Thus (Figure
a)
b)
c)
а)
b)
3.5), the lines drawn perpendicular to the horizontal trace or horizontals of the planes measure
the greatest declivity of the given plane with horizontal coordinate plane H. That is why,
at first, horizontal of the plane is drawn, and then the line of greatest declivity is found as
a perpendicular to the horizontal projection of horizontal.
Figure 3.5. The lines of greatest declivity
3.6. Intersecting planes
The intersection of any two surfaces is determined, in general, by the aid of auxiliary secant planes, which pass through the surfaces and cut lines upon them. The point common to the
lines thus cut are common to both surfaces and, hence, to their line of intersection.
When the intersecting surfaces are planes, the intersection is a right line, for determination of
which two auxiliary secant planes will ordinarily prove sufficient. As we studied early, a line may
be drawn when two its points, which are common to the intersecting planes, are found.
Let find the line of intersection between two planes (G and Q), given by their traces
(Figure 3.6).
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c)
Figure 3.6. Intersecting planes given by traces
a)
b)
By the application of the preceding principles, the two coordinate planes may be considered as the auxiliary secant planes. Thus, V cuts the two given planes in the vertical traces,
which intersect each other in 1≡12, while H cuts them in the traces G1 and Q1, which intersect
in 2≡21. But the points thus determined are the piercing-points of the line of intersection sought
1-2, the projections of which may be found in ordinary way.
In general cases and in the case when traces of planes do not intersect within the limits of
the drawing, the additional secant planes, that are usually horizontal or frontal, may be using
for finding two points of the line of given planes intersection. In Figure 3.7 two planes assume
parallel lines and triangular in space. Two points of the line of intersection must be determined
by means of two horizontals as a secant planes – R and T. They cut given planes in the two
lines-horizontals, which horizontal projections intersection give two points – 1 and 2, through
which the line of intersection passes.
Figure 3.7. Additional secant planes using for intersecting planes
In Figure 3.8 plane T is intersected by horizontal plane G.
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