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Файл:Descriptive geometry. Course of lectures for international students
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21
3.2. Plane traces
The lines in which a general plane intersects the planes of projection
are termed the plane traces, and in ordinary practice are the means whereby
the plane is determined in position (Fig. 3.1).
Fig. 3.1. Plane traces
In the ordinary cases the plane has three traces: horizontal, frontal and
profile (Table 3.1, Method 6).
When line МN lies in the plane α, it intersects the П1-plane in the point
M that will lie on the trace h
1
α. The horizontal trace of line that lies in the
plane will lie on the horizontal trace of this plane.
When line МN lies in the plane α, it intersects the П2-plane in the point
N that will lie on the trace f
2
α. The frontal trace of line that lies in the plane
will lie on the frontal trace of this plane.
Note: The traces of a line that lie in a plane will lie on the related plane
traces. The plane traces may be defined by drawing the traces of line that are
determined this plane.
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
α (Fig. 3.2, a). Line 1-2 has the common points with the triangle ABC, hence
line 1-2 lies in the plane, that is given by that triangle (Fig. 3.2, b).
22
а) b)
Fig. 3.2. Point in the planes
When point lies on a line that is contained in a plane this point lies in
that plane. Thus, it 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 (K) will lie
in the plane (Fig. 3.2).
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:
1) passes through the ground-line, giving no traces in two coordinate
planes, and hence indeterminable except by the use of a new vertical plane;
2) 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);
3) be parallel to GL and perpendicular to one plane and inclines to two
planes, when both traces will be parallel to that line;
4) 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 – profile projecting plane;
5) inclines to all planes, other than in the first and third cases, when all
traces will incline to GL.
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Table 3.2
Some position of a plane to the coordinate planes
Position in space
View in space
Projection drawing
Position of plane traces
1
2
3
4
Perpendicular
to the П1-plane –
horizontal
projecting plane
G1 – inclines to OX;
G2 – is perpendicular to OX;
G3 – is perpendicular to OY
Perpendicular
to the П2-plane –
vertical projecting
plane
G1 – is perpendicular to OX;
G2 – inclines to OX;
G3 - is perpendicular to OZ
Perpendicular
to the П3-plane –
profile projecting
plane
G1 и G2 – are parallel to OX;
G3 - inclines to OZ
Parallel
to П1-plane –
horizontal plane
G1 – is absent;
G2 – is parallel to OX;
G3 – is parallel to OY
Parallel
to П2-plane –
frontal plane
G1 – is parallel to OX;
G2 – is absent;
G3 – is parallel to OZ
End of table 3.2
1
2
3
4
Parallel
to П3-plane –
profile plane
G1 и G2 – are perpendicular
to OX;
G3 – is absent
3.5. The principle lines of a plane
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 (Fig. 3.3). 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.
Fig. 3.3. Horizontals of planes
Lines which are parallel to horizontal plane П1 are termed horizontals,
ones are parallel to vertical plane П2 ‒ frontals, and lines which are parallel to
profile plane П3 ‒ 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.
24
25
When the plane given by traces (Fig. 3.4, a), the vertical projection of
horizontal (f2) 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 Ni-point.
а) b)
Fig. 3.4. Horizontals of planes
When the plane given by triangular (Fig. 3.4, b), the vertical projection
of horizontal (f2) 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.
LGO-line of greatest declivity
Fig. 3.5. The lines of greatest declivity
26
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.
The lines of greatest declivity are perpendicular to the respective traces.
Thus (Fig. 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 П1.
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.
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 is 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 (α and β), given by
their traces (Fig. 3.6).
Fig. 3.6. Intersecting planes in space
27
By the application of the preceding principles, the two coordinate planes
may be considered as the auxiliary secant planes. Thus, П2 cuts the two given
planes in the vertical traces, which intersect each other in N, while П1 cuts
them in the traces αi and βi, which intersect in M (Fig. 3.7).
Fig. 3.7. Intersecting planes given by traces
But the points thus determined are the piercing-points of the line of
intersection sought MN, the projections of which may be found in ordinary
way.
3.7. The line and plane
When a line neither be parallel or lies in a given plane it intersects this
plane. One of the main problems of descriptive geometry is to find the
piercing-point of a line on a given plane.
The solution consists in passing through the given line any auxiliary
plane (Fig. 3.8), in determining the line in which this plane cuts the given
plane, and in finding the point in which the given line intersects the lines thus
determined. Projecting planes or planes that are parallel to coordinate planes
are used as an auxiliary plane for these purposes.
28
Fig. 3.8. Line intersects a plane
The construction becomes extremely simple when auxiliary secant plane
passing through the line is assumed to be the projecting plane of the line
itself – to П1 (Fig. 3.9).
m ∩ = ?
Solution
1) m Є β ,
β – horizontal projecting plane;
2) ∩ = MN ;
3) MN ∩ m = K
Fig. 3.9. Intersecting the line and the plane given by traces
Find the line of intersection between the given plane α and the projecting
plane β, and the point (K), in which the line m in space intersects it, is its
piercing-point on the given plane.
29
The problem when a plane given by triangular is solved by the same
method. In Fig. 3.10 the plane given by triangular ABC. The auxiliary plane
α has been passed through the given line m perpendicular to П1. Secant plane
cuts the triangular in a line 1-2.
Fig. 3.10. Intersecting the line
and the plane given by triangular
The point K in which lines m and 1-2 intersect is a point in the line of
intersection sought. Then the line m visible is found using method of
competing points. At first, the point 1 in the side AB of triangular lies higher
then point 5 in the line m, thus the sector 5K of the line m in space lies lower
then triangular, and this line m is invisible and its concealed sectors are drawn
by a shot dash on each coordinate plane. Therefore, sector K4 of the line m
places above the plane ABC and it visible sectors are drawn full.
3.8. The line is parallel to plane
When a given line is parallel to any line which lies in a plane, this given
line is parallel to this plane (Fig. 3.11).
Let find the geometrical relation the given line AB and the given plane
CDE (Fig. 3.12). The auxiliary secant plane Q is passed through the line AB
perpendicular to H-plane.
30
Fig. 3.11. The line is parallel to plane
Fig. 3.12. The line is parallel to plane CDE
The line 1-2 is the line of two planes intersection. The correspondence
projections of two line (AB and 1-2) are not parallel, thus the line AB is not
parallel to plane CDE.
3.9. The line and plane are perpendicular
A right line perpendicular to a plane in space gives projections which
are respectively perpendicular to the plane traces.
In Fig. 3.13 let plane α be the given plane, and AB a line perpendicular
to it. The plane projecting AB upon H is not only perpendicular to that plane,
but also to the given plane α; hence, being perpendicular to two planes, it is,
by Geometry, perpendicular to their line of intersection П1, or the horizontal
trace and horizontals.
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