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Ho$e(o") no* ., THE STRAIGHT LINE

1. 4s given the straight line /) defined by points and ,. 4s required to determine(- straight line traces 6 h' h >' h>)' v' v >' v>) and < w' w >' w>);- dihedral regions that are passed by the straight line.

*' 3*' +); ,!*' 1*' !+).

2. 4s given the straight line /) defined by points and , and eternal point :m' m?)' notlying on this line. 4s as$ing to construct through the point :m' m?) a hori@ontal line A)and a frontal line 0) in intersection with /).

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Ho$e(o") no* /, THE PLANE

1. re given three points not lying on a line #' #' ); ,33' 1#' 1); C2' !' 1). 4sas$ing to determine the traces D' D?' D of the plane defined by the previous points andthe maimum angle between the plane 5D7 and the hori@ontal pro&ection plane 567.

2. /etermine the point of intersection between the straight line /d' d?) defined by points :and = and the triangular plate 5,C7. 8tudy the visibility of the straight line afterintersection.

11*' #*' #*); ,3*' 1*' *); C!*' *' 2*);:1**' +*' *); =2+' +' 1*).

!. /etermine the line of intersection between a trianghular plate 5,C7' situated in ahori@ontal plane 567 parallel) having the coordinate +* mm and a triangular obliqueplate 5:=D7. 8tudy the visibility of the intersection.

1**' 2*' +*); ,+*' *' +*); C1*' !*' +*);

:+*' +' *); =1*' *' 2*); D1*+' 3+' +).

#. /etermine the line intersection between the plates 5,C/7 and 5DF%7' using auiliarypro&ecting planes. 8tudy the visibility of the intersection.

1+*' *' 3*); ,1**' 2*' 1*); C2*' 3*' +*); /*' 11*' @);:1+*' 11*' 2*); =1**' +' 1**); D#*' *' 2+).

+. /etermine intersection point pro&ections between three planes 5%7' 5D7 and 5F7' given bytheir traces( 5%7 G vertical pro&ecting plane' 5D7 G oblique plane' 5F7 G H parallel plane.

%1**' *' *); angle H%%? I #+J;D1!*' *' *); angle HDD I 3*J; angle HDD? I 3*J;Fy I 1** mm; F@ I #* mm.

NOTE( ll plates are opaque.

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Ho$e(o") no* 0, D* G* METHODS* T1e !23!tit2tion o4 p"o5ection plane!* T1e"e#ol2tion

1. Ksing the substitution of pro&ection planes and revolution methods' determine the truelength of the line segment , and the angles L and M between this line segment and

the pro&ection planes 567 and 57. Aive the solution in different orthographicrepresentations..

+' 2*' 1+); ,#+' +*' #*).

2. 4s given the plane 5D7 by traces( D*' *' *)' angle HDD I !* ' angle HDD? I 3*. 4sas$ing to determine' using substitution of pro&ection planes method' the true si@e of theangles L and M between the plane 5D7 and pro&ection planes 567' respectively 57.

!. Ksing revolution method' determine the true length of the distance from the point : tothe straight line ,).

+*' 2*' !*); ,2+' 1*' 1+); :#*' !*' #*).

#. Ksing substitution of pro&ection planes and revolution methods' determine the truelength of the distance from the point : to the plane 5D7. Aive the solution in differentorthographic representations.

:11+' #*' !+); D1*' *' *); angle HDD I 1!+; angle HDD? I 1#* .

+. Ksing substitution of pro&ection planes and revolution methods' determine the truelength of the distance between two parallel lines /) and B). Aive the solution indifferent orthographic representations.

/) defined by points *' 1+' 2*) and ,!*' !+' ++);B) the straight line parallel to /) containing the point C+*' 1+' 2*).

3. /etermine' using revolution method' the true length of the distance between parallelplanes 5D7 and 5F7' given by their traces(

D+' *' *); angle HDD I #+; angle HDD? I !*; F#*' *' *).

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Ho$e(o") no* 6, THE METHODS OF D*G* THE COINCIDENCE* THERESTORATION

1. Ksing the coincidence with a frontal plane' determine the true si@e of the triangle ,C.*'2+'++); ,++'++'+); C1+'+'1+).

2. 4t is given the straigth line /)' defined by the points and ,. 9he point : is not lying onthis straight line. /etermine the distance from the point : to the straight line ,).

#*'2*'!*); ,1**'3*'*); :2+'++'+*).

!. /etermine the true length of the distance between two oblique parallel lines /) and B)'given by their pro&ections.

#. %epresent a square with the sides !* mm contained in a vertical pro&ecting plane 5%7

$nowing that two sides of the square are perpendicular to 57. %*'*'*); angleH%%I*' angle H%%?I#+.

+. %epresent the pro&ections of a circle with the radius r I 2+ mm' contained in a vertictalplane 5F7.

3. Ksing the coincidence method' determine the true si@e of the triangle ,C contained inthe oblique plane 5D7.

*'1*'@); ,++'*'@,); C#+'yC'*);

D12*'*'*); unghiul HDDI#+' unghiul HDD?I!*.

5

>d>

d

H

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Ho$e(o") no* :, POLYHEDRONS* Rep"e!entation* Plane !ection!

1. %epresent in orthographic representation the net polyhedrons $nowing that the lengthof the polyhedron side is equal with !* mm(- a tetrahedron'

- a cube'- an octahedron.

4s as$ing(- determine the intersection of each one of the three polyhedrons with one of the sevenparticular planes;- the true si@e of the sections;- the development of the three regular polyhedrons.

(Each student will present original solutions!!! Two identical problems are not accepted!!!)

2. 4s given a right pyramid having the base situated in the plane 57' a regular penthagon

inscribed in a circle with the radius equal with !* mm' the centre N3*' *' #*) and theheight + mm. 4s as$ing to determine the section with a vertical plane 5F7 in pyramidand the true si@e of the section.

F1+' *' *)' angle HFF I 1+*.

!. 4s given an oblique pyramid ,C/8( 1!*' #+' *); ,1**' 1+' *); C*' #+' *); /11*'+' *); 8!*' !+' +*). 4s as$ing to determine the section with an oblique plan 5D7 inpyramid and the true si@e of the section.

D#*' *' *); angle HDD I 12*; angle HDD? I 1!+.

#. 4s given the heagonal right prism having the base ,C/E0 contained in plane 567'having the height * mm. 4s as$ing to determine the section with the oblique plan 5D7 inprism and the true si@e of the section.

N1**' +*' *); r I !* mm;D1*' *' *); angle HDD I 12*; angle HDD? I 1+*.

+. 4s given the triangular oblique prism ,C/E0 having the base ,C contained in 567. 4sas$ing to determine the section with the vertical pro&ecting plane 5%7 in prism and thetrue si@e of the section.

12*' 1*' *); ,1+*' !*' *); C1*+' +*' *); E#*' 2+' *);%!*' *' *); angle H%%? I 1!+.

3. %epresent in isometric oblique frontal cavalier perspective) aonometric pro&ection thesolid given in orthographic representation(

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Ho$e(o") no* ;, POLYHEDRONSE* Plane !ection! an& &e#elop$ent!

1. 4s given an oblique pyramid 8,C/' with the base ,C/ contained in the plane 567(11*' !*' *); ,*' 3*' *); C#2' 2*' *); /*' 1*' *); 81*' *' 3*). /etermine thepoints of intersection with the straight line /)' defined by points 4+' *' 2) and 62+'

1+' 11). 8tudy the visibility of the straight line and represent the development of thepyramid.

2. 4s given the right prism with the base contained in plane 567' a regular pentagoninscribed in a circle with the radius !* mm and the height * mm. /etermine the pointsof intersection with a straight line /)' defined by points :!1' 22' 21) and =11#' 31'#2). %epresent the prism developmentand on the development the points ofintersection with the straight line.

!. 4s given an oblique frontal prism ,C1,1C1' with the base situated in 567. /eterminethe plane section in prism with a plane 5D7 perpendicular on prism edges' the true si@e

of the section' the development of the prism including bases)' respectively of thetruncated prism between the cutting plane and the base.

1#*' !' *); ,1*' 2#' *); C11' ' *); 13*' !' +); D1+' *' *).

#. 4s given the right pyramid 8,C/' with the base contained in plane 567. /etermine thesection with the oblique plane 5D7' the true si@e of the section and the development ofthe truncated pyramid between the cutting plane and the base.

13*' !' *); ,!' +!' *); C32' 1' *); /1!1' +' *); 811' !*' 3);D1*' *' *); angle HDD I 1!+; angle HDD? I 13*.

+. 4s given the right prism ,C1,1C1' with triangular base' situated in plane 57./etermine the plane section in prism with a vertical plane 5D7' the true si@e of thesection' the development of the prism including bases)' respectively of the truncatedprism between the cutting plane and the base.

3*' *' #+); ,!*' *' 2); C+' *' 1+); 13*' 3*' #+);D1+' *' *); angle HDD I 1+*.

3. 4s given the oblique pyramid 8,C/' with the base ,C/ contained in plane 567.determine the section with a vertical pro&ection plane 5D7' the true si@e of the sectionand represent the development of the truncated pyramid between the cutting plane and

the base.113' 2' *); ,2' 3*' *); C3+' 12' *); /1**' ' *); 81' #!' *);D1*' *' *); angle HDD? I 13*.

. %epresent in isometric oblique hori@ontal military perspective) aonometric pro&ectionthe solid given in orthographic representation(

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H

@

y

1* 1+ 12.+ 12.+ 1+ 1*

+

1+

+*

+

2+

+

1* 1+ 1+ 1*+ 1+ +

+

1

*

+

2*

+

#+

+

12

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Ho$e(o") no* +

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Ho$e(o") no* ++, THE CONE* THE CYLINDER

1. 4s considered a cone of revolution having the base a circle with radius r I !+ mm'contained in the plane 57' the verte 8s' s?) and the altitude * mm as perpendicularline to 57). Ksing a vertical plane 5F7 determine an elliptical section in cone' the true

si@e of the section and the development of the truncated cone between the cuttingplane and the base including the bases).

2. 4s considered a cone of revolution having the base a circle with radius r I !+ mm'contained in the plane 567' the verte v' v?) and the altitude * mm. Ksing a verticalpro&ecting plane 5D7 determine a parabolical section in cone' the true si@e of the sectionand the development of the truncated cone between the cutting plane and the baseincluding the bases).

!. 4s considered a cone of revolution having the base a circle with radius r I !+ mm'contained in the plane 567' the verte v' v?) and the altitude * mm. Ksing a vertical

pro&ecting plane 5D7 determine a hiperbolical section in cone' the true si@e of the sectionand the development of the truncated cone between the cutting plane and the baseincluding the bases).

#. /etermine the true si@e of the section with a vertical pro&ecting plane 5D7 that ma$es theangle of !*with the plane 567' in a cylinder of revolution having the base a circle withradius r I !+ mm' situated in plane 567 and the altitude + mm. %epresent thedevelopment of the truncated cylinder under the cutting plane including bases).

+. 4s considered the right cylinder having the base a circle lying in hori@ontal plane of

pro&ection' with the center in

+' #*' *)' the radius r I !* mm and the altitude + mm./etermine the section with an oblique plane 5D7' the true si@e of the section and thedevelopment of the truncated cylinder between the cutting plane and the lower base.

D1+' *' *); angle HDD I 12*; angle HDD? I 1+*.

3. 4s considered the oblique cone with the base a circle having the radius r I 2+ mm'

situated in plane 567' with the center in +*' *' *) and the verte in 81*' 1*' 3*).

/etermine the points of intersection between the cone and a hori@ontal line having thecoordinate 2*mm. 8tudy the visibility of the straight line.

. 4s considered the right cone' having the directri a circle with radius r I !* mm' lying in

plane 567' the centre in *' +*' *) and the verte in 8*' +*' 3*). /etermine thepoints of intersection between the cone and the oblique straight line /). 8tudy thevisibility of the straight line.

/)( : #*' 1*+' 1+) and = 1**' 1*' +*).

. %epresent in isometric oblique frontal cavalier perspective) aonometric pro&ection thesolid given in orthographic representation(

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Ho$e(o") no* +-, PRO9ECTIONS WITH ELEVATIONS* Sol#in= t1e "oo4!

1. /etermine the intersection edges of the roof sides given in figures.- the slope of the roof sides is !*roof no.1)' #+roof no.2)' 3*roof no.!) and !*roof no.#);

- the drawing scale is 1(1**.

16

+

-

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17

.

/

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18

0

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2. %epresent the developments of three roof sides arbitrary selected by students) for theroof no.# and two views for the roof no.! from problem no.1.

!. %epresent in isometric orthogonal aonometric pro&ection the solid given in orthographicrepresentation(

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Ho$e(o") no* +., PRO9ECTIONS WITH ELEVATIONS* Sol#in= t1e plat4o"$!

1. /etermine the boundary lines of the emban$ments for a hori@ontal platform.- the slope for the cut emban$ment I 1O1;- the slope for the fill emban$ment I 2O!;

- the scale for the drawing 1(2**.

3.**

3.

**

E+**.**

+**

+*1

+*2+*!

+*#

+*+

#11

#1.

#12

#13

#1+

#1#

#1!

.**

.

**

2. /etermine the boundary lines of the emban$ments for a hori@ontal platform and for theacces road of the construction site.- the slope for the cut emban$ment I 1O1;- the slope for the fill emban$ment I 2O!;- the slope for the acces road I 1O3;- the scale for the drawing 1(2**.

1#.

**

3.

**

.**

1#.

**

2*

*

1

1

1

1

3

1.

1

2**

2*22*!2*

#2*

2*.

13

12

2*1

2*+

2*3

3.*

*

2**.**

.**

21

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!. /etermine the boundary lines of the emban$ments for a hori@ontal platform and for theacces road of the construction site.- the slope for the cut emban$ment I 1O1;- the slope for the fill emban$ment I 2O!;- the slope for the acces road I 1O+;- the scale for the drawing 1(2**.

%epresent the longitudinal section with the plane 507 through the platform and acces road'respectively the cross section with the plane 5E7 through the platform.

.

**

1**

1**

1*1

1*2

%I1*

.*

0 0

E

E

1**

1*1

.

3

1*2

1*!

+

#

!

1*#

1*+

1*3

3.

**

3.

**

1**.**

#. %epresent in isometric oblique frontal cavalier perspective) aonometric pro&ection thesolid given in orthographic representation(

22

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H

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1+ 1++

#*

1*

!*

+

!+

+

1++

+

+ + + 1* + + +

#*

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