Descriptive geometry
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tersection of some surfaces w hen only one surface is proje cting (fig. 6.4, а). |
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tersectio n of some surfaces of revolution with the same axis of ro tation for |
them (fig. 6.4, b). Coaxial surfaces intersect in a circle, the plane of which is perpen-
dicular to |
the axis of rotatio n surfaces. At that, if the axis of rotatio n surfaces is paral- |
lel to th |
projecti on plane, the intersection line projec s onto th is plane as a line- |
segment. |
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Fig. 6.4. In tersection of surfaces:
a – Intersection of a straight circular cone with a triangular pris m; b – Intersection of a straight circul ar cone, a s phere and a straight circular cylinder with a sphere
4. I tersectio n of some surfaces of revolution of th |
second order whi ch are de- |
scribed around sphere. (The Monge’s theorem, see fig. 6.8). |
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All these task s from p ivate cas es which were con sidered be fore can be solved |
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without any additional acti on only by surface property |
All other tasks a e general |
cases. In order to solve these tasks it is necessary to use |
some auxiliary mediators – |
planes or spheres. This depends on the type of surfaces a nd their mutual lo ation and location according to the planes of pr jection.
There are thre e methods of solvi ng task fr om general cases.
1.M ethod of auxiliary cutting planes
2.M ethod of auxiliary concentri c spheres
3.M ethod of auxiliary eccentric spheres
6.2. Method of au xiliary c utting level planes
The followin g exampl illustrate s how to apply the |
method of auxiliary cutting |
planes for construc tion of the intersection line o f a spher |
with a c one (fig. 6.5). |
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Fi g. 6.5. Determining of an intersec tion line of a straight circular cone with a sphere by the Method of auxiliary cutting level planes
In order to co nstruct the intersec tion line of the given surfaces it is advisable to
introduce the fron al plane and a nu ber of h orizontal |
planes (α V1, αV2, α 3, αV4) as |
the auxiliary plane s. Charac teristic p oints of projections |
should b e identified firstly. |
Then fro ntal proje ctions of the highest and low est points (1’’ and 5 ’’) are found as the
points of |
intersection of outlines. Horizontal p rojection s |
of the points 1’ a nd 5’ are |
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determined due to the connection lines and surface properties. |
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Aux iliary horizontal planes cut the sphere |
and the |
one in circles. Find the pro- |
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jection 4 ’’ of the p oints lying on the |
sphere eq uator with |
the help of horizontal plane |
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αV3, whi h passes through he center |
of the s |
here. The plane in tersects the sphere |
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along the equator nd the cone in the circle of |
adius r. T he horizo ntal proje ctions in- |
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tersectio |
of the la test yields the horizontal pr ojection 4’. The horizontal projections |
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of points 4’ are th e points of visibility bounds of the intersection line on this projection. The passing p oints (2, 3) are det ermined by means f auxiliar y horizontal planes
αV1 and |
V2. After all points are connected in a ccordance with the given or taken or- |
der and obtain a s ooth cur ve line in accordanc e with vi ibility. |
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6.3. Me hod of a uxiliary concentric spheres |
This |
method is widely used for solution o problems on const ruction of intersec- |
tion lines of rotatio n surfaces with int ersecting axes.
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Bef ore studying this method we consider a particular case fo r of intersection of
rotation |
urfaces, the axes of which c oincide, . e. the c ase when coaxial surfaces of |
rotation intersect. |
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This |
property is used for constru cting a line of mutual intersection of two revo- |
lution su rfaces by means of auxiliary spheres. In this cas we can u se both, oncentric |
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(constructed from one cent r) and ec centric (d rawn fro m differe nt centers) spheres.
We are going to c |
nsider application of auxiliary concentric spheres, those with have |
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a constan t center. |
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No te: if a p lane of revolution surface axes is not parallel to the |
projection |
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plane, the circles of their intersection are projected as ellipses and this |
makes the |
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problem solution |
more complicated. For this reason the method o f auxiliary spheres |
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should b used un |
er the following co nditions: |
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a) in tersectin |
surfaces are the su rfaces of revolutio n; |
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b) s urfaces ax es intersect and an intersection point is taken for the center of auxiliary sph eres;
c) th e plane produced b y the surf aces axes (plane of symmetry) is parallel to one of the pr jection planes.
Usi ng this m ethod, it is possible to constr uct the line of intersection o f the surfaces in o ne projection. The applicatio n of this method is described below (fig. 6.6).
Fig. 6. . Determining of an in tersection l ine of a straight circular cone wit h a straight circular cylinder by the Meth od of auxiliary concentric spheres
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The points 1’, 3’, 4’, 10’ are determined as the points of level generatrixes of the surfaces belonging to the plane of axes intersection (the plane of symmetry). Find the other points by the Method of auxiliary spheres.
It is necessary to draw an auxiliary sphere of an arbitrary radius from the intersection point of the axes for the given surfaces (point O’’). This sphere is simultaneously coaxial to the cone and the cylinder and cuts them along the circles the planes which are perpendicular to the corresponding rotation axes. The frontal projections of those circles are line-segments. The constructed sphere intersects the cone along the circle of diameter, and the cylinder – along the circles (represented by color in fig. 6.6) in the intersection points of the horizontal line-circle with the vertical lines-circles obtain points 2’’ and 5’’–9’’ respectively, which belong to the intersection line.
In such a way it is possible to construct a certain amount of points of the desired intersection line. Students should note that not all the spheres may be used for the problem solution. So the limits of the auxiliary spheres usage should be considered.
The maximum radius of a cutting sphere is equal to the distance from the centre O to the farthest intersection point of the level generatrixes (from O’’ to 4’’ and 10’’). The minimum cutting sphere is a sphere, which contacts one surface (the larger one) and cuts another (the smaller one).
If an auxiliary sphere cuts only one given surface, this sphere is not proper for the problem solution.
Rmax > Raux > Rmin.
In order to construct the second projection of the intersection line one may use the circles obtained while cutting the cone by auxiliary spheres or drawing additional sections of the surface. Points 13’’≡14’’ and 15’’≡16’’ lying on the level generatrixes of the cylinder are the points of visibility bounds of the intersection line on the horizontal projection.
6.4. Method of auxiliary eccentric spheres
All auxiliary eccentric spheres, which are used in this method, have different centers. Therefore, a center for all spheres should be specifying before their drawing. It is necessary to set any circular cross-section for the one surface from two given in order to find a center for an auxiliary sphere. The next part of the solving algorithm is the same as for the Method of auxiliary concentric spheres
There are three graphical conditions for applying the method of auxiliary eccentric spheres:
a)when revolution surfaces of fourth order (open or closed torus) or surfaces of elliptical cylinder intersect;
b)there is a common plane of symmetry for both surfaces and it is a level plane;
c)axes of surfaces intersect or cross.
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Fig. 6.7. Determining f an inters ection line wi h half of torus by the Method of
f a straight truncated circular cone uxiliary eccentric sph eres
6.5. Monge’s T heorem
If tw o surfaces of the s cond ord er are des cribed around a sphere, the li ne of their mutual in tersection decompo ses into t wo plane curves. Th e planes of these c urves pass through a straight line connecting the intersection points of the tang ent lines.
Fig. 6.8 presents an example with a const ruction of surface i ntersection lines on the basis of Monge’s theor m, where two cylinders, a ylinder and a con e and two cones are describe d around sphere.
Fi . 6.8. Priva te intersection case. Determining of an intersection line of two straight circular co nes which are circums cribed arou nd a spher by Monge ’s theorem
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7. UNFOLDING (DEVELOPMENT) OF SURFACES
Unfolding is a flat plane as a result of combining surface of a body with the plane. This surface is a flexible inextensible film which is not torn by this kind of transformation (involute). These kinds of surfaces are called unfoldable. They can be polygons, ruler surfaces (cylindrical, conical) and surfaces with edge of regression (torso).
There are three types of unfolding.
1.Accurate unfoldings can be constructed for faced surfaces of prism and pyramid, circle cylinders and cones.
2.Approximate unfoldings can be constructed by replacing (approximation) given surfaces with segments of developable prismatic and pyramidal surfaces. This is true for circle inclined cones, elliptical cylinders with circle sections and complex surfaces.
3.Conditional unfoldings are used for undeveloped surfaces of a sphere, torus surfaces, toroid etc.
There is only one point on the unfolding for each point of surfaces. There are several properties of inter-relation between a surface and an unfolding:
1.Lengths of the relevant lines of surfaces and unfolding are equal.
2.Lines, which are parallel on the surface, are parallel on the unfolding.
3.Angels between relevant intersecting lines on the surface and on the unfolding are equal.
4.Areas of the relevant shapes on the surface and on the unfolding, which are border by closed lines, are equal.
7.1.Unfolding of a polyhedron.
1.Determination of a true size for side faces or edges is a key point of unfolding in such case.
2.There are three methods for construction of prism unfolding:
3.Method for normal section.
4.Method for unrolling
5.Method for triangulation
7.1.1. Method for normal section
This method for unfolding lateral surface of a prism can be applied if in the drawing:
–edges of the prism are straight lines of the level, in other words they have the true size in one of the given projections;
–there are no images of the prism bases with the true size in the projections. NB: If in the drawing edges of the prism are straight lines of general position,
then we should change the position of the prism relative to the projection planes, transforming the edges into straight lines of the level, for example, by replacement of the projection planes.
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The construction of an unfolding of the la teral surface of a p rism while using a method f or normal section is performed according to the following g raphical algorithm:
1.D rawing a plane of normal section which is perpendicular to the edges in the projection where t hey have the true size.
2.D etermination of the true size of a poly gon of the normal section plane by any
method (for examp le using method for replace |
ent of projection planes) |
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3. |
Expansion of a natural polygon of the section int o a straigh t line in a |
free area |
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of the |
drawing a |
d drawin g perpendicular s raight lin es throu gh the po int of its |
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vertices – direction s of the edges. |
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4. |
Setting asi de true length of th e corresponding e ges on t he directi |
ns of the |
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edges on both sides of the n ormal section line. |
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5. |
Connection |
of endp oints of t he edges |
as straight line segments an d comple- |
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tion of a flat figure |
as the sid e surface of the prism. |
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6. |
Completio n of the drawing in accordance with G OSTs – fold lines should be |
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drawn by a long d shed double dotted thin line |
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Fig. 7.1. Constructio n a surface of a triangular prism w hile using m ethod for ormal sect on
7.1.2. M ethod for unrolling
This unfolding method is used when in a drawing:
– edges of a prism are s traight lin es of the level;
– bases of a prism (or one of the bases) are plan s of the level and have the true size.
The essence of this m ethod is t hat by "c utting" th e prism s urface at one of its edges, the nearest prism face is com bined with the pl ane of unfolding (this plane should b e parallel to the ed ges). The n, by rotating the prism sequ entially around the followin edges, all other faces of the prism are also aligned with t he plane of unfolding, one by one (fig. 7.2).
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ig. 7.2. Rolling a pris matic surfa e
7 .1.3. Method of triangulation
Unrolling of Pyramid S urface
The developm ent of th e side surf ace of the pyramid accordin g to the tr ue size of
its edges is carried out using the follo wing graphical algorithm. |
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1. Specify the true size for all pyramid ed es by an |
method (method |
f rotation |
around projecting line, met od of rep lacement planes of |
projectio n and etc.) and for |
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its base. (If the ba se is the plane of th e level, t hen it’s tr ue size is given in |
one of the |
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projections).
2. D raw the faces of t he pyrami d success ively in the free field of th e drawing accordin g to their true sizes with co pass by some additional tick arcs, so that they have a circular vertex S and adjoin ea ch other.
3. Finish the drawing in accordance with GOSTs – fold lines have to be drawn by Long dashed do uble dott ed thin line.
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Fig. 7.3.The unrolling of a triangular pyram id surface
7.2. Approximate unfolding of cylin drical and conical surfaces
App roximate unfolds cylindrical and coni al surfac s
Cyli ndrical a d conical surfaces are developed in the same way as prismatic and pyramida l surface . In this case, a cy lindrical surface is replaced (approxi mated) by an inline polygona l prismatic surface (usually 12-angle/faces), and a conical surface is replace d by an i nscribed polygonal pyramidal surface.
Circ ular Cyli nder surfaces.
The surface of a right c rcular cylinder can be unrolle d by the following methods:
– no rmal section meth od in the free field of the drawing, if a genera rixes is a straight line of the level, and the bases are not perpendicu lar to the generatrixes;
– method of r olling under the same conditions (unroll ng follows after projection). The rolling method is used for unrolling of an ellipti al cylind er (norma section –
ellipse) i a generatrixes is a straight l ine of the level and there is a circular b ase in the projections. Graphical algorithms for constructing surface develop ment are similar to the abov graphical algorithms for co nstructing prism unrolling.
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8. AXONOMETRIC PROJECTIONS
8.1. Method for axonometric projection
A complex drawing as a two-dimensional image is rather simple and easily measured, but it is not representable as a three-dimensional object in space. It is often necessary to have in addition to it a drawing of pictorial view, which may be obtained by projecting an object and its coordinate axes onto one plane. Then, the one projection will provide a visual and metrically distinguished image of the object. Such types of an object representation are called axonometric projections.
Depending on the direction of projecting relative to the axonometric projection plane Q, axonometric projections are classified as rectangular (projection angle is
equal to 90 degrees 90 ) and oblique ( 90 ).
Method for axonometric projection consists in the following: a given figure and axes of rectangular co-ordinates to which the figure is related in space are projected in some axonometric projection planes.
Axonometric projections as parallel projections have some properties:
–axonometric projection of a straight line is also straight;
–if lines are parallel on the object, they are also parallel in its axonometric projection;
–axonometric projection of a circle in axonometric projection is an ellipse (in general case).
Axonometric axes are projections of coordinate axes x, y and z. Therefore the projection direction is not parallel to any axes of the orthogonal coordinate system, true dimensions of the segments in the axonometric projection are distorted.
There are coefficients of distortion for determination of distortion degree for each axis. If we take some equal x, y and z coordinates, we shall obtain some distorted coordinates. The ratio of length of the axonometric projection segment to its true size is referred to as distortion coefficients for axes, as reflected in formulas (8.1)–(8.3).
Kx ex1 ; ex
Kx ex1 ; ex
Kx ex1 , ex
where Kx , K y , Kz – a distortion coefficients for axes x, y and z; ex , ey , ez – x, y and z coordinates;
ex1 , ey1 , ez1 – distortion x, y and z coordinates.
(8.1)
(8.2)
(8.3)
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