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Файл:Descriptive geometry. Course of lectures for international students
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Fig. 3.13. The line and plane are perpendicular
The converse of this position is likewise true: if the projections of a line
are perpendicular respectively to the traces of a plane or the lines, which are
parallel to the correspondence coordinate planes (horizontals or frontals), the
line is perpendicular to the plane.
When the line perpendicular to two intersecting lines of a plane, by
Geometry, it is perpendicular to a plane. The lines of plane, which are parallel
the coordinate plane, are used for drawing the perpendicular line to a plane
in space.
Fig. 3.14. Drawing the line perpendicular to the plane
Let K be the given point, and α the given plane. The task is to pass a line
perpendicular to the plane through a given point (Fig. 3.14). By the condition
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of the task the projections of the required line must be respectively
perpendicular to the traces of the given plane (α
1
and α2) or to two intersecting
lines of that plane – its horizontal and vertical (h and f). Through K2 lead the
line perpendicular to the trace α
2
or vertical projection of frontal f2, and
through K1 lead the line perpendicular to the trace α1 or horizontal projection
of horizontal h1.
3.10. The planes are parallel
The given planes are parallel when they contain any two lines which are
parallel to each other. In another words, when two projections of lines which
lie in one plane are parallel to the similar projections of the lines, which lie in
the second plane, these planes are parallel (Fig. 3.15, a). When the planes
given by traces their parallel is proved when their similar traces are parallel
(Fig. 3.15, b).
In Fig. 3.16 the drawing of parallel planes, which are passed through the
point in space, are shown.
In Fig. 3.16 the plane G given by traces. At first, the horizontal of the
required plane leads through the point A, when its horizontal projection is
parallel to horizontal trace h1 and is passed through the A1.
а) b)
Fig. 3.15. The parallel planes
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Fig. 3.16. The parallel plans drawings
Then the piercing-point (N) of the horizontal is found. Through that
piercing-point the vertical trace of required plane T2 is passed as a parallel
trace to G2. The horizontal trace of the new plane T1 is obtained by drawing
the line parallel to G1 and passed through point TX.
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4. DRAWINGS TRANSFORMATION
4.1. The main problems of drawings transformation
For engineering purposes, the drawings of objects must be oriented in
space so as to find the true sizes, shape and position of them or their separate
parts. It may be obtained when the objects in space are oriented parallel or
perpendicular to the coordinate or to auxiliary planes. When an object is
parallel to any plane, its projections on that plane will be presented in a truelength form. When an object is perpendicular to any plane, its projections on
that plane will be transformed into simple geometrical form – point, line or
plane.
There are five main tasks of drawings transformation:
– definition of the true-length of a line;
– transformation of a line into projection position – end view of a line;
– definition of the true-length view (shape) of a surface plane;
– transformation of a surface plane into projection position – edge of the
surface plane;
– definition the distance between two skew lines.
The solution of these tasks may be found by the follow methods:
1. Change of the projection plane (or ground-line).
2. Parallel-plane transformation.
3. Rotation.
4. Rotation around plane traces (or coinciding method).
Let to study each of them more thoroughly.
4.2. Change of the projection plane
This method is widely used for definition of the true size and shape of
geometrical object, and for transformation of them into projection position. In
the other resource this method is termed as Change of position method or
Auxiliary view method. According to this method the points of object do not
move and change their position, but the auxiliary plane is introduced to the
coordinate system. The position of this auxiliary plane depends on the main
problem of the task. When the true-length of the objects is defined, the planes,
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which are parallel to the objects and perpendicular to horizontal or vertical
planes are used.
In Fig. 4.1 the transformation of vertical projection of point A from
system П1/П2 to the new system П1/ П4 is shown. In the new system the
auxiliary plane П4, which is perpendicular to П1-plane, is introduced instead
of vertical plane П2. Intersecting planes П4 and П1 are formed new groundline OX1. Projections of A-point in the new system lies in a corresponding
line, which is perpendicular to the OX1 – projector of the planes П1/ П4. The
distance of the A-point from П1 remains unaltered. Thus, the distance A2 from
OX and of A4 from OX1 are equal.
a)
b)
Fig. 4.1. Change the plane of projection – horizontal projecting plane
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In Fig. 4.2 the determination of the true-length of line AB is shown. As
it was established in the Lecture 2, the true-length of a line is defined as a
projection of it on the plane which is parallel to this line. Therefore, the
auxiliary plane must be parallel to the line and perpendicular to the one of the
coordinate planes: new plane П4 is parallel to the line AB (its horizontal
projection) and perpendicular to the П1-plane. Let fall upon the new groundline OX1 the perpendiculars from A1 and B1 projections, and lay off a
correspondence distance of these points from the vertical projections. New
line A4B4 is a true-length of the line AB, and the angle between that line and
OX1 is the true-angle of it inclining to П1 plane. During next step the line AB
is transformed in its end view.
Fig. 4.2. Determination of the true-length view and the end view of AB
Determination of the true-shape and edge view of a plane.
A plane is in its simplest position with respect to the П1- and П2-planes
when it is parallel to one plane and therefore perpendicular to the second. In
this position, one projection is a normal view and the other an edge view. If
the plane is represented as a polygon, the normal view gives the true shape of
the figure. When a given plane is inclined to the H-and V-planes, the edge and
normal views can be obtained by selecting appropriate auxiliary planes.
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The plane ABC is represented as a triangular (Fig. 4.3). First of all, the
edge view of ABC is determined.
Note that a plane projects as a straight line on a projection plane taken
perpendicular to any line in the plane.
Fig. 4.3. Determination of the true-size view
and the edge view of ABC
For convenience, the projection plane is taken perpendicular to a
principal line of the plane.
Draw the auxiliary line h parallel to the horizontal plane – horizontal of
the plane ABC. Then h1 is the true-length view of h. Take plane П4
perpendicular to line h by placing X14 at right angle to h1. Project A4B4C4 on
plane П4, determining the edge view ABC. The angle a between A4B4C4 and
X14 is equal to the angle between plane ABC and the horizontal plane П1.
Note that the edge view of ABC can also be drawn on a third plane taken
perpendicular to a frontal line of ABC.
The normal view of a plane will appear on a projection plane taken
parallel to the given plane. Take plane П5 parallel to ABC by drawing X45
parallel to A4B4C4. Project ABC on plane П5 (A5B5C5) is the normal view of
the plane and the true-shape view of triangular ABC.
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4.3. Methods of rotation
Instead of employing method of changing the plane of projection, when
position of the object does not change – we add the new planes in the given
system, the same end may be attained be changing the position of the object
itself, thus affording new views and consequently new projections.
The operations by which this is accomplished are either movement
parallel to a rectilinear or plane direction, or movements of rotation.
In these changes it is to be remembered that the principal objects is to
facilitate the work of solution; hence, whatever the alteration in position, the
simplification of the constructions must be kept constantly in view.
With the movement of rotation there are necessarily implied (Fig.4.4):
1) an axis around which the object revolves – the axis of rotation i; when
a point A is revolved in space, it is always revolved around a straight line used
as an axis (i). It is important to know how the axis actually lies, before you
attempt to revolve any point.
2) the fixed distances of each point of the object from the axis during
the entire rotation – the radius of rotation R;
3) the foot of the radius, always marking a point in the axis – the center
of rotation i;
4) the locus of each point – the circle of rotation AA’;
5) the plane of that circle, always at right angle to the axis – the plane
of rotation T. A point will revolve in a plane that is perpendicular to the axis,
and its path is always a circle. The radius of the circle is the shortest distance
from the point to the axis.
6) the arc AA’ through which any point A revolves, giving the measure
of rotation AA’ (angle φ) for every other point of the object.
Taken together, these constitute a system of rotation the position of
which must evidently depend upon that of the axis.
There are two general positions which such an axis may assume to the
moving object:
1) it may have one or more points in common with it;
2) it may lie wholly outside it.
While the axis of rotation may be made to assume any position to the
planes of projection, still for practical purposes it ought to be so placed as to
render the constructions as simple as possible. Such a position is one in which
the axis is assumed to be perpendicular to either plane of projection.
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Fig. 4.4. Method of point A rotation
In order to effect the rotation of a point around an axis, a perpendicular
must always be passed through the point to the axis, given the center of
rotation. This perpendicular, which is the line of the radius, must be turned
into the required position, and the original distance of the point from the axis
set off upon it, the measurement being made from the foot of the perpendicular
or center.
Fig. 4.5. Rotation of the line
40
Method of rotation around the line, which is perpendicular to the plane
of projection, is widely used for true-length of line definition. Axis of rotation
(i) is drawn through a point in common with the line (point B) – Fig. 4.5. In
this case point A must be rotated around i to the position when AB will be
parallel to the vertical plane and AB will be projected on this plane as a natural
size, and the angel α will be projected on П2 in its true measure.
Note that object’s projections on the plane, which perpendicular to the
axis of rotation do not change their size and shape. But the other projections
are moved in directions that are parallel to the consequence axis of coordinate
planes (A
2A’2
|| OX) and changed their size and shape.
5. PLANE SURFACES (POLYHEDRONS)
5.1. Prism and pyramid
Solids which are bounded by planes are called polyhedrons. Precisely,
it is the surface of the solid that is the polyhedron. The term is commonly used,
however, to describe either the surface or the solid. The boundary planes
intersect to form the edges of the solid. A plane polygon formed by a set of
edges is a face. The point in which a set of faces intersects is a vertex
(Fig. 5.1). When all the faces of a solid are congruent, the figure is a regular
polyhedron. There are five regular polyhedrons: the tetrahedron, cube,
octahedron, dodecahedron, and icosahedron.
Fig. 5.1. Polyhedrons
Apex
Edges
Ends of bases
Faces
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