Descriptive geometry
.pdfA point in space has three coordinates – x, y, z. So it is possible to use axis lines x, y and z by taking some imagine base line when they are even absent. The base line enables draftsmen to draw the profile projection more comfortable.
If the horizontal projection of a surface has an axis of symmetry (circle, square and other), a base line should be drawn through this axis. If the horizontal projection of a surface has no axis of symmetry (triangle, pentagon and other), base line should be drawn through the widest place of the polygon. The widest place can be a point or a side (line).
If points belong to ribs (specific elements of surface), their projections can be found by connection line. These connection lines have to be perpendicular to base line from the frontal projection of points to the profile projection of necessary ribs. If points belong to faces, distance between the base line and the projection of a point in the horizontal projection should be measured and put aside in the profile plane of the projection.
In order to finish projection all points should be connected in the given or taken order according to their visibility. Pay attention, when two planes intersect, they always organize line of intersection. This kind of lines have to be drawn according to their visibility.
In order to solve projection points on the surface of a pyramid or a cone, one of the methods described below can be used.
5.4. Generatrix Method
This method is used on the basis of the theorem described above: if a point belongs to the line, projection of this point belongs to the projection of this line.
Line m’’ as the one of the uncountable numbers of generatrixes is drawn from the top of a pyramid or a cone up to its base through the given point K’’ (fig. 5.9, 5.10). A line is the shortest distance between two points. In order to draw any line we should draw its two end points. Both these points of the line m belong to some specific elements of the surface and their projections can be found by connection lines in accordance with their belonging and without any auxiliary actions. One of them is on the top (vertex) and another one is on the base of pyramid or cone). Two projections of a point are always connected between each other by a connection line. Connection lines have to be perpendicular to the axes lines of the planes of projection according to the properties of orthogonal parallel projection. So, for solution of the horizontal projection of the ends of the line m’ it is necessary to do the following actions. The first point is the top of the pyramid or the cone. The second point 2′ can be found by the intersection of the base outline of a pyramid or a cone with a connection line in the horizontal projection. Then the connection line is omitted from frontal projection of the given point K’’ to the intersection with horizontal projection of the line m’. The point of this intersection is the required point.
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Fi g. 5.9. Solution a point on a pyra |
id by Gen ratrix Met hod |
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and Method o f auxiliary cutting planes |
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5.5. M ethod of auxiliar |
cutting planes |
Wh en a plane |
of the le vel interse cts with |
urface, it formes a polygon or a circle |
in crosssection (i |
case if b ase of surface is a polygon or a circle and it is the plane |
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of level too). It de |
ends on the type o f surfaces |
The projection of necessary point lies |
on the si e/sides of the poly gon or on the circle.
Firstly, an imaging cut ing plane of the le vel should pass through the projection of the point P’’. This imagi ng plane α has to b e parallel to the base of a pyramid or a cone. As it has be en mentioned abov e if a cutting plane is parallel to the b ase it cuts the same shape w th the base but smaller. So the next tep is to draw this auxiliary section plane.
If it is a polygon (pyramid case), an additional poi t on any rib (exce pt vertical projection of a rib, because in this case a connection line from the additi nal point and rib line will be overlapped) has to be taken This poi nt is the itntersection point of the additional plan e with the rib of p yramid. The conn ction lin e has to be omitted
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from the projection of additional poi nt to the rojection of the relevant rib. Then an auxiliary figure is drawn, according t o all information which was given earlier, it fol-
lows that each line of the auxiliary figure has to be parallel to each relevant |
ine of the |
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base. Th |
last acti on is to draw a connection li ne from t |
e given p oint to it |
intersec- |
tion with |
the lines of the a uxiliary fi gure. This point of |
intersection is th |
required |
point. If a surface has a line of symme try, the s olution can have two symmetry points. If it is a circle (cone and torus cases), w e can just measure a radius (distance between the center line (axis line) and the gener trix line (surface bo undary li ne)). Then we draw a circle in the horizontal projection by a compas and omit the conn ction line from the projection of the gi ven point P’’ to the intersection with the horizontal projec-
tion of ou r additional circle. The point of their intersection is the req uired point.
Fig. 5.10. So lution a point on a con e by Generatrix Meth od and Method o f auxiliary cutting planes
5.6. Surfaces of re volution Sphere nd torus
Spherical surface is a urface formed by rotation of a circle round its diameter. This surface is a set of equally-space d points from the center. Sph ere is the only one
surface w |
hich has an infinit y number s of axis p assed thr ough the |
center. This proper- |
ty is used |
in solving points o n a surfac e and many other d escriptive |
tasks. |
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Sphere is drawn as a c ircle in all three projections with the equal di meter. In other words all its outlines are circle s. Every p oint on t he sphere is rotated on some circles which are called parallels.
A frontal outline is the main fro ntal merid ian. It is projected as a horiz ontal line on the ho rizontal a xis line i n the horizontal pro jection a d as a vertical line the same with the vertical ax is line in the profile projecti on.
A horizontal outline is |
the equator of sp here. It takes the s ame place with the |
horizonta l axis in the frontal and the profile projections. |
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A p ofile outline is the |
ain profile meridia and it is combined with the vertical ax- |
is line in he horizontal projection and w ith the vertical axis line in the frontal projection.
A ball is a body bord ered by a spherical |
surface. Method of auxiliary cutting |
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planes is used for s olving a point on t he ball. |
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There is only one type of section for a b all it is a circle. Bu t this cir le can be |
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projected as three different images i n the dra |
ing. It depends on the direction of a |
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cutting plane. |
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1. |
If a cutting plane is a horizon tal plane |
of level, t his sectio n is draw |
as a line |
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in the |
frontal and |
profile projections and as |
a circle in the ho rizontal |
projection |
(fig. 5.11, a). |
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2. |
If a cutting plane is a profile lane of t |
e level, this sectio n is draw |
as a line |
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in the |
frontal and |
horizontal projec tions and |
as a cir le in the profile |
projection |
(fig. 5.11, a). |
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3. |
If a cutting |
plane is a projecting plane, this section is dra wn as a line in this |
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plane of projection and as ellipses in two other planes (fi g. 5.11, b). |
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Fig. 5.11. Possible ca ses of proj ection for ball section: a – by level plan es; b – by a projecting plane
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Taking more additional points is recommended for rawing of the ellipse.
A s phere has unique property, it has only one possible type o f section and it has simple sh ape – circle, whic needs o nly specifying its c nter and knowing its radius. So accor ding to this all points on its surface ca n be solved by the method of auxiliary cutting planes in all planes of projections. See the follow ng examp le (fig. 5.12).
Fig. 5.12. Solvin g a point on a circle s urface
Tor us surface is formed by rotation |
of fo rming circle on the guidex round the |
axis whi ch belongs to the same plane as |
the c ircle but it does n ot pass through the |
center of this circl . The guidex is a c ircle with a radius.
Tor us is a bo dy border ed by a torus surface. Torus s a body bordered by a torus |
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surface. Torus is open if the |
outlines of the ge neratrix c rcle do not touch e ach other |
and do not interse t (fig. 5.1 |
3, b). In other words if R is more tha n r (R > r) where R |
is the radius of guide circle nd r is the radius of the generating circle.
Tor us is close d if R is e qual to r (R = r), what means that outlines of the generat-
ing circl e touches each oth er (fig. 5.13, c). T orus is self-crossin g if outli |
es of the |
generating circle intersect. In this cas e R is le s than r R < r). This kind |
of surface |
formation can produce sever l types of surfaces uch as to oid and globoid (fi |
. 5.13, a), |
and toroid (fig. 5. 13, d). If torus has the front al project ng axis I it is dra |
n as it is |
shown in an exam le below. |
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Fig. 5.13 . Torus surface types:
a – a globoid; b – an open torus; c – a closed torus; d – a toroid
There are six types of sections for torus. Two of them are circles if a cutting plane pas ses throu gh the axis of rotation.
Method of cutting planes is used for solution of the projectio n points on a torus surface. When a g uidex circle is parallel to th e frontal projectio n, auxiliary cutting
planes are planes of frontal level which are pr |
jected as circles in the frontal projec- |
tion and as lines i n the horizontal pr jection. |
he follo ing figure (fig. 5. 4) shows |
an examp le of solving point s on torus by the Method of cutting planes. |
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1. If a cutting plane is perpendicular to the |
axis line, the section is a pl ane which |
is border d by two circles with a big radius (Rbc) and a small radius (Rsc) (fig. 5.15, a).
2. |
If a cutting |
plane combines ( passes through the axis line) with the axis line, |
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the section is two |
circles |
which have the same radius with the generatrix circle |
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(fig. 5.15 , b). Distance t is |
ero in th is case. This distance t is tak en for explanation. |
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It is a distance which descri |
e locatio n of the c tting pla e. |
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3. |
W hen R ≤ t < R2, wh ere R is the radius of the guide circle and R2 is the radius |
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of the biggest circle of a torus, the section is an ellipse with two axes of symmetry (vertical and horiz ontal).
4. |
W hen R1 < t < R, w ere R1 is the radius of the s mallest cir cle of the torus the |
section is a wavy curve. |
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5. |
W hen t = 1, the section is a double petal curve. This section is cal ed a lem- |
niscate of Bernoul i. |
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6. |
W hen t < R 1, the sec ion is tw o ellipses with one line of sy mmetry. |
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Fig. 5.14. Solving points on a torus surface
Fi g. 5.15. Possible cases torus sectio ns:
a – wo circles which are e xactly the ame with generatrix c ircle;
b – a plane border ed by two circles (with a big radius and with a small)
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All these four sections are called curves of Perseus (fig. 5.16).
Fig. 5.16. Pos sible cases torus secti ns
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6. INTERSEC TION OF SURF CES |
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The simplest body is a cone, a sphere, etc . But a c omplex b ody whic |
contains |
several bodies is used more often. |
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The line of in tersectio of two s urfaces i a set of the point s belongi |
g to both |
surfaces. So, findi ng these points is th e way to solve the line of intersection |
Shape of |
the inters ection line depend s on mutual location of intersecting surfaces and their locations according to the projection pla nes.
6.1. Possible cas es of surface intersection
There are four types of mutual location.
1. Full penetration. All generating lines of the first surface (cylinder) intersect another surface, b ut not all generatrix es of the second surface intersect the first one.
In this c ase the intersection line of t he surface s decom |
oses into two closed curves |
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(fig. 6.1, a). |
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2. Partial cu ting-in. Not all generatrixes |
of both |
urfaces i ntersect each other. |
In this case the int ersection line is one closed c |
rve (line) (fig. 6.1, b). |
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Fig. 6.1. Typ es surface intersection s: a – Full penetration; b – Partial cuttin g-in
3. Unilateral or one-si de contact. All gen erating lines of the o ne surface intersect the other surface, but not all generatrixes of th second surface intersect the first one. There is a commo n tangent plane in one point of the surfaces (fig. 6.2, a). The intersection line decomposes into two close d curves which meet in the point of contact.
4. Bilateral o r double contact. All generating lines of bot h |
surfaces |
intersect |
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each oth er. The intersecting surfaces have two common |
tangent planes. In |
this case |
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the inters ection line decomp oses into two plane curves |
meeting in |
the points of con- |
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tact (fig. 6.2, b). |
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Fig. 6.2. Typ es surface intersection s:
a – Unilateral or one-side contact; b – Bilateral or double contact
There are tw o big gro ps of int ersection surfaces a ccording to their location in space. In other words there are two v ariants ac cording to relative l ocation of surfaces
and plan s of proje ction: private cases |
and gene ral cases. |
Four private c ases: |
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1. I tersection of some surfaces, |
here side faces are the projecting faces (fig. 6.3). |
Fig. 6.3. Intersectio n of a straight circular cylinder with a hexago nal prism
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