Descriptive geometry
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As the given line is parallel to the horizontal plane, it is necessary to tr ansform it into a projecting line by replacement of the frontal plane V with a new V1, which should be perpendicular to A’B’. As a result f it the given line is projected in the plane V1 as a point (B’’1 ≡A1’’).
In order to tra nsform th e general position line AB into a projecting one, it is necessary to make two replace ments, i. e. solve both probl ms, the first and the second ones, successively Firstly w e transfo m a general positio n line into a line of the level and then it is trans formed into a proje cting one.
3. Transform tion of a plane of the gener al positio α (∆AB C) into a projecting one (fig. 4.4), i. e. positione d perpendicular to o ne of the projectio n planes.
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. 4.4. Transformation of a plane o f general position into a frontal projecting plane |
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It is |
necessary to repl ace the plane of projection with a ne w plane V1 or H1, |
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which is perpendicular to the plane α. An additional plane is perpendicular to the plane α if it is drawn perpendicular to one of the main lines of t he plane (frontal or
horizonta l line). S |
firstly o ne of those lines has to be dr awn (h or f). The second step |
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is to draw a new axis x which is p |
rpendicular to the chosen main line. If it is a |
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frontal li ne f, x1 |
f’’ and it gives us a |
new horizontal pl ane of pro jection H1, if it is a |
horizonta l line h, x 1 h’ and it gives us a new frontal plane of projection V1. By the same wa y we specify all po nts of the given pl ane. All of them sh ould lie on one line
(the plane had to be transformed into a line). Angle between new |
projection, which is |
the line n ow and an axis x1 is the ang le of inclination of the given |
plane to the plane |
of projection H or V (which depends o n previou s chosen steps). |
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4. Transform the plane α (∆ABC) from a p rojecting plane int o a level p lane. The |
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true size of the plane is dete mined in this case (fig. 4.5). |
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The first step is to set an additional axis x1 parall el to the |
projection which is |
generate in a line. Then we draw an ther proj ection of ll points by measuring coor-
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dinates i previous plane of projectio n. These a ctions gi e us the t rue size o f the given plane α at the e nd.
Fig. 4.5. Transformation of a front al projecting plane into a horizontal level plane
4.2. Method for r otation a round pr ojecting a xis
According to this meth od objec has to be rotated around so me projecting axis for changing its roperty from gen eral into particular. In othe r words the object changes its position and planes of projection a re static n contrast with the previous method. As it is known line s of the level are arallel to one plane of proje ction and they are projected in this plane in true size. Acc ording to the prope rty of lin s of level one projection is inclined to he axis a d the oth er two pri nciple projections are parallel.
Therefor e line of |
the gener al positio n, which has all projections as an oblique line, |
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should ro tate in th |
following way: on e of its p ojection |
hould be come para llel to the |
axis x. Firstly, a projecting axis i sh ould be s et for this |
rotation. Axis i h as to pass |
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through any end point of the given s egment. T hen it is necessary to rotate the given projection to the horizontal position ( parallel to the axis x) by a c ompass. We finish
the task by specifying another proje ction of that segme nt by co nnection |
ines. The |
particula r case of he method of rotation around a proje ting axis , which is |
called as |
the meth od of plane-parallel transfer, and it is not consid ered here. |
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ig. 4.6. Determining the true leng th of a line by rotation around pro jecting axis: |
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a – a 3-dimantio nal model; b – a drawi g of it |
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4.3. Method for rotation ar ound main line (fr ontal or h orizontal) |
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So |
as it is obvious from the na me of the method, firstly the main lin e (f or h) |
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should b |
drawn (fig. 4.7). |
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Fig. 4.7. Dete rmining a lane true s ize by rotation around its main line (horizontal): a – a 3-dimantio nal model; b – a drawi g of it
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This line will be used as an axis line of rotation. Then we draw throug h vertexes
of the gi ven plane traces of auxiliary planes β1 and β2 which are p erpendic |
lar to the |
main line . These planes are planes w here vertexes will be rotated. The third |
vertex is |
fixed because it lies on the ain line ( axis of rotation). P oints of intersection of those |
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perpendi culars with the mai n line are centres o rotation. So, as it is shown |
n a draw- |
ing, a rad ius of rotation is the line of g eneral position. It means that, firstly, we should find its t rue size by any kn wn method (three methods for deter ination o f the true size have been already discussed ab ove). In order to finish solution of the task we have to use a comp ass as it is shown in an exam ple.
5. SURFAC ES
A s urface is a set of all successiv e positio ns of the movement of a line This line |
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is called a generatr ix; the movement may retai |
or chang e a shape. The mo vement of |
the gener atrix can be subjec ted to a law or it |
can have an arbitrary character. In the |
first case, the surf ace will be legitima te, and in |
the second one it will be random (ir- |
regularit ). The law of the generatrix movement is us ually determined by another line, which is call ed a guid e line on which the generatr x slides in its movement. In some cas es, one of the gui e lines can be con verted to a point (vertex conical surface), or in the infinity (cyli ndrical su rface).
5.1. Faceted surfaces. Prism and pyramid
A fa ceted surface is th surface, which is formed b the movement of a rectilinear generatrix on a broken line, for example, pyramid al (with top) and prismatic (without top) surfaces (fig. 5 .1).
Fig. 5.1. For ation of faceted surfaces
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Pol |
hedron i s a solid shape with four or more flat surfaces (prism (4 – faces), |
pentagon (5 – face s), hexagon (6 – fa ces), etc.). Polyhe ron outli ne is draw n as pro- |
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jections |
f its face s and ribs in a drawing |
A prismatic surface is the surface, which is formed by the movement of a recti- |
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linear ge neratrix by a broken line, du e to this w e obtain faces with the help of paral- |
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lel move ment in ac cordance with the given direction.
A prism is th e polyhedron that has two bases that a e the sam e and they are par-
allel, and some faces are tetragons ( quadrangles). Prism |
is called |
straight if its ribs |
(lines of intersection of the adjacent faces) ar perpendicular to |
the base, and is in- |
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clined if it is not [ 7]. Prism is called a s a regul ar prism i |
its bases are a regular poly- |
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gon (pol gon which is inscr bed into the circle and has e |
ual sides ) (fig. 5.2 . |
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ig. 5.2. A drawing of a prism with a regular triangle bas e
A pyramidal surface is the surfa ce, which is formed by the movement of a rectilinear ge neratrix b y a broke line with one fixe d end of t e generat rix.
Pyr mid is the polyhedron having one base, top, faces and ribs (lines of intersection of the adjacent faces) which intersect at one point (to p of pyramid) in g eneral [7]. Pyramid is called truncated if its top is cut an as a res ult of it th e pyramid has two
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bases. Py ramid is called as regular p yramid if its base is the regu lar polygon, which is inscribed into the circle an d has equ al sides (fig. 5.3).
Fig. 5.3. A drawing of a pyramid w ith a regula r square base
5.2. Surf aces of revolution. Cylinder and con e
Surf ace of revolution is the sur face which is for med by m ovement of a line (generatrix) aroun d another fixed straight line. This fixed straight line is called an axis line of rotation. The surface is called ruled if the generatrix is a straight li ne and the surface i called c rved when the generatrix is not a straight line. All points of generatrix rotate around the axis on circle s with relevant rad uses whi ch are called parallels. Parallel planes are always perpen dicular to the axis of rotatio n. The pa allel with the smallest diameter is call ed the neck, and with the larg est as the equator. A meridian is an outline which is formed by intersectio n of surf ace with the plane of level –
frontal or |
profile. The meridian, which is paral el to the rontal pla ne of projection, is |
called th |
main meridian (fi g. 5.4). |
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Fig. 5.4. A surface of revolution |
nd its special element s |
Cyli ndrical su rface of revolution is a ruled |
surface formed by parallel movement |
of a straight line ( generatrix) on circl e or ellipse line. Cy linder is a body bordered by cylindrical surface of revolution (side surface) with two parallel bases which are perpendicul ar to the axis of rota tion.
Cyli nder is c alled circ ular if its bases are circles, a nd elliptical if the bases are ellipses.
Cyli nder is called strai ht if the a xis of rotation is p rpendicu lar to its bases. There are thre e types o f cutting cylinder.
1. If cutting plane is pa rallel to the bases it cuts cuts a shape which is similar to the base (fig. 5.5, ).
2. If cutting plane is par llel to the axis of rotation it cuts some rectangle (fig. 5.5, b). 3. If cutting p lane is inclined to the axis f rotation (angle is not 90o) it cuts an
ellipse (fig. 5.5, c)
Fig. 5.5. A straight circular cylinde r:
a – by a parallel plane to its base; b – by a perpendicu lar plane to its base; c – by an incline d plane
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Fig. 5.6. Possible cylin der sections
Cyli nder has two equal rectang ular projections and the third projection as the shape of the base (circle or ellipse) in general c ase.
A conical sur face is p oduced by rotatio of a lin (generatrix) roun d the axis (directrix) intersecting it. In other w ords one end of the generat rix is fixed and referred to as a vert ex (top). Cone is a body bordered by conical su rface of evolution (side sur ace) with one base which is perpendicular to t e axis of rotation. The point of interse ction of g eneratrix with axis is fixed and is refe red to as a vertex.
Cone is called as a straight cone if the axis of rotation is perpendicu lar to the base and the cone is called as an inclined wh n the axis is not perpendicu lar to the base (fig. 5.7).
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Fig. 5.7. A |
straight circular cone |
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Tru ncated cone is form ed by a |
cutting plane parallel to the base of c one and it |
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has two b ases. |
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There are five types of conical sections which are sub-divided in accordance |
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with direction of a cutting plane. |
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1. |
A s a result of intersection of a cone wit h a plane perpendicular to th |
cone ax- |
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is, a circle is obtai ned(fig. 5.8, a) (In c ase if con e has a circle as a b ase). |
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2. |
If a cutting plane pa sses through the vertex of a |
one, it cu ts a pair |
of genera- |
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trix lines, which form a trian gle (fig. 5 .8, b). |
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3. |
If the cutti ng plane is inclined to the ro ation axi |
of a con e and does not pass |
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through its vertex then it is p ossible to |
obtain three types of lines in |
the section. |
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4. |
If a cutting plane is not parallel to any generatri x, then an |
ellipse is |
obtained |
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in its sec ion (fig. 5.8, c). |
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5. |
If a cutting plane is parallel t o one generatrix, th en a para bola is o btained in |
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its section (fig. 5.8, d). |
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6. |
If a cutting plane is parallel to |
two generatrixes, then a hyperbola i |
obtained |
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in its sec ion (fig. 5.8, e). |
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Fig. 5.8. P ossible con e sections:
a – by a p arallel plan e to its base; b – by a plane passed through the cone top; c – by an inc lined plane to all generatrixes; d – by a parallel plane to a generatrix;
e – by a perp endicular plane to its base
5.3. Points a nd lines on the surface
In order to fi nd missing projections that b elong to some surf ace, it is necessary to build a n auxilia y line on a given s urface, passing through a given point o f the projection. irstly, it is necess ry to con struct the projecti n of this auxiliary line, and then to draw the required projection of the point on it. For this p urpose we can use various li nes: lines of genera trix, parallels, mer dians, etc.
In some cases, if the surface is the proje cting surface (fig. 5.2, 5.5), (i. e. perpendicul ar to the plane of p rojection ) missing projectio n of the p oints can be found without a dditional constructions, because this kind of surfaces has collected property.
A Prism and a cylinder can be such kind of surfa es. There fore, all the points on the surfaces of the prism and the cylinder, being in a horizonta l projectio n are distributed on the lines of the base (polyhedron r circle, or ellipse) accordin g to their visibility.
We consider the examples of co nstructio n of points on a pr ism and a cylinder. The points and lin es lying on the surface canno t enter into the sur face and g o beyond it. The visible part of the fro ntal proje ction of the surfac is located below the axis of symmetr y in the h orizontal plane of p rojection, the invisible part is located above the axis of sy mmetry.
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