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Файл:Начертательная геометрия. Курс лекций. Учебное пособие
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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 H plane. During next step the
line AB is transformed in its end view.
Determination of the end view of AB. A line projects as a point on a plane perpendicular
to the line. The line and plane will appear perpendicular in the view in which the line projects
in true length.
Take new auxiliary plane P5 perpendicular to AB (on the projection where it determines
the true-angle to H-plane) by drawing OX2 at right angle to A4B4. Project AB on the plane P5.
Projections A5 and B5 coincide, giving the end view of AB.
Figure 4.3. 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 H- and V-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.
The plane ABC is represented as a triangular (Figure 4.4). 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.
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Figure 4.4. 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 AF parallel to the horizontal plane – horizontal of the plane ABC.
Then A1F1 is the true-length view of AF. Take plane P4 perpendicular to line AF by placing
OX1 at right angle to A1F1. Project A4B4C4 on plane P4, determining the edge view ABC. The
angle α between A4B4C4 and OX1 is equal to the angle between plane ABC and the horizontal
plane H.
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 P5 parallel to ABC by drawing OX2 parallel to A4B4C4. Project ABC on
plane P5 (A5B5C5) is the normal view of the plane and the true-shape view of triangular ABC.
Determination of the shortest distance between two skew lines
Using the principle of the method, the common perpendicular can be drawn in the view
in which one of the given lines projects as a point (end view of line). The distance between the
lines is equal to the true-length of MN (Figure 4.5).
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Figure 4.5. Determination of the shortest distance between two skew lines
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 (Figure 4.6):
1) an axis around which the object revolves – the axis of rotation MN; when a point A is
revolved in space, it is always revolved around a straight line used as an axis (MN). 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 AO;
3) the foot of the radius, always marking a point in the axis – the centre of rotation O;
4) the locus of each point – the circle of rotation AA’;
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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
A1O
(angle φ) for every other point of the object.
1A’2
Taken together, these constitute a system of rotation the position of which must evidently
depend upon that of the axis.
Figure 4.6. Method of point A rotation
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.
In connection with the first case the more ordinary positions of the axis are:
1) with a line, it has a point in common;
2) with a plane, a point or line in common;
3) With a plane figure, it coincides with an axis, diameter or side, or with any tangent to it;
4) with a surface, it coincides with an axis, element or tangent.
In the second case the ordinary positions of the axis are:
1) with a line, it is either parallel to it or lies in another plane;
2) with a plane, it is parallel to it;
3) with a plane figure, it lies in the plane of the figure or is parallel to it;
4) with a surface, it is parallel to some line or element.
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.
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.
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 (MN ┴ H) is drawn through a
point in common with the line (point B) – Figure 4.7. In this case point A must be rotated
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around MN 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 V in it true measure.
Figure 4.7. Rotation of the line
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
|| OX) and changed their size and
2A’2
shape.
These rules are used in the method of parallel-plane transformation, when the axis is not
assumed and the radius of rotation is not defined. It is necessary to change the position of one
of projection without any changes in the size of that projection, and moves it in position of
parallelism to the chosen plane projection (Figure 4.8). Then the other projections are drawn
in the lines that must be parallel to the OX-axis.
Figure 4.8. True-length of line definition by parallel-plane method
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4.3.1. Rabbatement
As it is desirable to avoid constructions, which, by the position of the object in space,
may appear confuse, and as it also frequently necessary in practice to determine the true size,
shape and position of such an object, or of its separate parts, this end is attained by the method
of rotation termed rabbatement.
By way of illustration, assume the revolving object to be a plane; then should this plane
be revolved around any line which is parallel to either coordinate plane until it is parallel also
to that plane, or should it be revolved around any line lying in either coordinate plane until it
coincides with that plane, the rotation is by rabbatement. In this method the rotation is produced around horizontals or verticals of plane until that plane become parallel to horizontal or
vertical plane respectively.
Note that it is necessary to determine the true-length of the radius of rotation by any
known methods (right-angle triangular, rotation, parallel-plane transformation).
In Figure 4.9 the true size of triangular is drawn by method of rabbatement, when it is revolved around horizontal AD. Through the point B1 and C1 fall the perpendiculars to the horizontal A1D1 as a path of their rotation around horizontal.
Figure 4.9 – True-shape of plane definition by rabbatement method
Then the true-length of radius B1O0 is determined by method of right-angle triangular.
Draw the true-length of OB through the center of rotation of point B – point O1 and determine
the new projection of B-point – B0. The horizontal projection of C0 is determined as point of
intersection line B0D1 and perpendicular passed through the C1. Triangular A0B0C0 is the trueshape of the plane that is parallel to horizontal coordinate plane.
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4.3.2. Rotation around plane traces (or coinciding method)
This method is widely used in the cases when the plane is assumed by the traces. Given
plane is revolved around one of its trace until plane is coincided with one of the coordinate
plane. In Figure 4.10 the true-length of line AB, which lies in the plane T, is determined by coinciding method.
Figure 4.10. True-length of line AB definition by coinciding method
The axis of rotation is the trace T1 of the plane. Each point of the plane and line AB are
revolved in the planes, which are perpendicular to the horizontal trace T1. New position of the
vertical trace T2 is determined by rotation of the point that lies on the trace – point C. The radius of rotation TXC2 has the true length as a line lies in the frontal plane. The centre of rotation
is the point TX. Draw the arc with radius TXC2 through the centre TX until it intersects the perpendicular passed through the C1 point – path the C point in rotation procedure. Draw the new
coincided trace T’2. Then pass the horizontals through the points A and B, and determine the
coincided position of them on the trace T’2, and draw the coincided horizontals through these
points being parallel to the trace T1 until there intersect the path of A and B points. A’2B’2 – is
the true-length of AB.
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CONTROL TEST
1. List the main tasks of drawing transformation, and methods of their solution.
2. Describe the method of changing position or auxiliary view method. Represent it in
any sample.
3. How can be used the auxiliary view method for true-length of line definition?
4. How can be used the auxiliary view method for end view of line definition?
5. List the main steps of determination of the true-shape and edge view of a plane by aux-
iliary view method.
6. List the main steps of determination of the shortest distance between two skew lines by
auxiliary view method.
7. Represent the rotation of a line and give the principles of parallel-plane method.
8. What does the term “rabbatement” mean. Give the principles of rotation by rabbate-
ment for any line.
9. Describe the method of rotation around plane traces (or coinciding method).
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5. PLANE SURFACES (POLYHEDRONS)
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 (Figures 5.1). When
all the faces of a solid are congruent, the figure is a regular polyhedron. There are five regular pol-
yhedrons: the tetrahedron, cube, octahedron, dodecahedron, and icosahedron.
Figure 5.1. Polyhedrons
When the faces of a solid are not congruent, the figure is irregular. The prism, pyramid,
and prismoid are common examples of this class.
5.1. Prism and prismoid
The prism (Figure 5.2) has two polygons called ends or bases which are congruent and
parallel, making all the other surfaces parallelograms which are called faces. The solid is
a right prism when each face is a rectangle; otherwise, it is oblique. A line connecting the centers of
the bases is an axis. The axis is parallel to one set of edges. The description of a prism includes the
shape of a base. Thus, when the bases and faces are rectangles, the solid is a rectangular right
prism. When the bases and faces are parallelograms, the solid is a parallelepiped.
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Figure 5.2. Prism
a)
b)
The prismoid has two parallel ends or bases which are dissimilar polygons having the
same number of sides (Figure 5.3a). The faces are plane quadrilaterals. The frustum of a pyramid is a limiting case of a prismoid (Figure 5.3b).
Figure 5.3. Prismoid: a) prismoid; b) frustum pyramid
5.2. Pyramid
The pyramid can have any polygon for one face called the base; all other faces are triangles (Figure 5.4). The triangular faces meet in a common point called the apex. A line connecting the apex and the center of the base is the axis. The solid is a right pyramid when the axis is
perpendicular to the base; otherwise, it is oblique.
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