Descriptive geometry
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three pri ncipal planes of pro jection. I t cannot appear in its true len gth in any of three principal planes of projection [5] (fig. 2.1).
Fig. 2.1. A line of gen eral positio n
The point of intersection of a str aight line with the lane of p rojection is called a trace of a straight line.
The straight line, whic h is parallel to one or two pl nes of projection, s called a
line of p rticular position. |
hen the line in sp ace is oriented in su ch a way, that it is |
parallel to the given plane o projectio n, it is pr ojected as a true len gth. |
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There is next sub-class fication: |
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The straight line, whic |
is parall el to one plane projection, is called a line of lev- |
el and all its points have |
the same distance to this plane, i. e. they have the |
same coo rdinate. In other w ords, the line, which is parallel to one plane of projection, is seen i n an adja ent view as a line parallel to the fol ing (axis) line. (OX, OY or OZ). If t e line is parallel to the horizontal plan e of proje ction it is called a horizontal level line (fig. 2.2).
The line, which is parallel to t wo plane s of proj ction, is called the projecting line. In other ords, this line is perpendicu lar only to one plane of projection. If the line is perpendicular to the horizo ntal plane of projection it will be called the horizontal projecting line. There is another name for this line, in some coun tries it is called th e frontal p rofile lin e. These straight lin es are pr jected as the point in the indicated p lane of projection and as the true len gth in other two planes of projection (fig. 2.3).
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Fig. 2.2. Lines o level:
a – a fr ontal level line; b – a horizontal level line; c a profile level line
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Fig. 2.3. Projecting lines:
a – frontal pr jecting lin ; b – a horizontal projecting line; c – a profile projectin g line
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2.3. Determ ning the true length of a lin e by the right triangle Method
Fin ing the true length of a segm ent is a basic prob em in Descriptive geometry. There ar e several different methods of its determining. The Rig ht triangl Method, Method for replacement pr ojection planes, M ethod for rotation r ound projecting or main line and etc.
The right triangle Meth od: there is a right triangle where one leg is equal to any
projection of the segment, |
nd the se cond leg is equal to the dist ance bet ween end- |
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points of the other projection of the segment. A |
hypotenuse of the |
triangle is the true |
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length of the segment. |
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In other word s, the hyp |
tenuse of the right triangle is equal to the true len gth of the |
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segment AB, if one of two |
sides is pr ojection |
f this segment, and |
another one is the |
difference between the ends of the othe r projecti on which has to be |
measured fig. 2.4). |
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Fi g. 2.4. Determining true length of a line of general position: a – a 3-di entional m odel; b – a drawing of this model
2.4 . Position s of straight lines relative to each other
Straight lines can be parallel (fig. 2.5), intersecting (fig. 2.6) or crossing (fig. 2.7).
1.If two lines are parallel in spa ce, they have parallel projections in all planes of projection. (In general case) The converse is also true.
2.If two line s intersect, they org anize a common point whic h is the p oint of intersectio . Both p rojections of this point lie on the sam e connection line. The con-
verse is also true and these lines are c alled intersecting li nes (fig. 2.6. point |
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3. If two lines |
are crossed, the place of the ir interse ction seem |
s like the |
common |
point in one plan e |
of proj ction an d like tw o points in anothe r |
projecti on. These |
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points ar e called c ompeting points. T hey belong to the s me conn ection lin but have different location in space: o ne under another or one behind anoth er. The po int which is the nea rest to observer is visible (fi g. 2.7).
Fig. 2 .5. Parallel lines:
a – a 3-dimantional model; b – a drawing of them
Fig. 2.6 . Intersecting lines:
a – a 3-dimantional model; b – a drawing of them
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Fig. 2.7. Crossin g lines:
a – a 3-dimantional model; b – a drawing of them
2.5. Projection of a straight angle – Theorem
A straight an gle is projected in a plane of projection as the tr ue size, if one side of the st aight ang le is para llel to the plane of projecti on, and th e other side is not perpendi cular to this plane of projecti on (fig. 2.8). In oth er words if one side is a level
line and another o |
e is not a projecting line a straight ang le is proj ected in its real size |
in that projection |
here the level line has a projection as a true len gth. |
F g. 2.8. Projection of a straight an le
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3. PROJE TION OF A PLANE. CLA SSIFIC |
TION O F PLANES |
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AC CORDI G TO THEIR L |
CATIO N IN SPA CE |
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3.1. Plane. Setting m ethods of plane in |
a drawi ng |
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Projection of a plane in a drawing can be set in the following ways: |
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– by |
projecti on of three points |
(points |
o not lie |
on the same straight line) |
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(fig. 3.1, a); |
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– by |
projectio n of the s traight li |
e and pr |
jections |
f the poi nt (both of them do |
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not lie on |
the same straight line) (fig. 3.1, b); |
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– by |
projection of two parallel li nes (fig. 3.1, c); |
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– by |
projection of two |
ntersectin g lines (fig. 3.1, d); |
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– by |
projection of any |
hape (tria ngle, rectangle) (fig. 3.1, e); |
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– by |
plane traces (fig. 3 .1, f). |
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Fig. 3.1. Setting metho ds of plane:
a – by pro jection of three points (А, В, С); b – by proje tion of the straight line (AB) and projections of the point; c – by rojection of two parallel lines (AB , CD); d – by projectio n of two intersecting lines ( AB, CD); e – by projection of an y shape (quadrangle ABCD); f – b y plane traces (P)
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Plan es are al |
ays drawn with li mited size. But, in principle, t he plane has indef- |
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inite extent. |
plane is th e straight line of i ntersectio n of the plane with the plane |
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Tra e of the |
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of projection. The point of intersectio n of the lane with |
axes is c alled the vanishing |
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point of traces. |
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3.2. Point belonging to a plane. L ine belon ging to a plane |
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There are two theorem |
of the descriptive geometry |
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1. If point belongs to t |
e line wh ich belongs to the plane it m eans poi t belongs |
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to this pl ane. |
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2. The line be longs to the plane, if it pass es through |
two poin ts belonging to the |
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plane or if it passes through one point of the above plane |
and is pa rallel to a line con- |
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tained in the plane
According to these th eorems the following proble ms of de scriptive geometry can be s olved: drawing a line on the plane; c onstructing a point on the p ane; constructing a lacking projecti on of the point co ntained o the plan e; checking of the point belonging to the plane (fig. 3.2, a).
Fig. 3.2. Plane lines:
a – belonging a line (m) to a plane ( ABCD), and belongin a point (K) to a line (m); b – main (principle) lines of a plane frontal – f, horizontal – h)
Mai n (principle) lines of a plane
There are a lo t of lines belongin g to the p ane. Tho e of them which h ave a special or particular position should be distinguish ed.
Main line of plane is a straight line of le vel which belongs to the plane. Frontal line is th e line of the frontal level belonging to the given plane. Horizontal line is the line of the horizontal level belonging to the giv en plane (fig. 3.2, b).
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3.3. Classification of plane s accordi ng to their location in space
A p ane can have the fo llowing p ositions relative to the plane s of proje ction:
– an inclined position to all plane s of proje ction;
–a perpendicular posit on to the plane of p rojection;
–a parallel p sition to the plane of projection.
First variant is called as the pl ane of general position and both the other are called as the plane of particu lar (specific) position.
The plane, which is no t parallel and not perpendicular to any planes of projec-
tion, is c alled the |
plane of general position. This plane i inclined to all major planes |
of projection. (We |
can also call this plane as he oblique plane) (fig. 3.3). It cannot |
appear in its true s ze in any principal planes of projection.
Fig. 3.3. Plane o f general position (AB CD): a – a 3-dimantio nal model; b – a drawi g of it
The plane, which is perpendicular to one plane of projection , is calle d the projecting p lane (fig. 3.4). If a plane is perpendicular to the frontal plane of pr ojection it is a frontal projecting plane. It has fr ntal proj ection as the straigh t line. Th is projection can be also co nsidered as a trace of the pl ane. Other projectio ns are projected in
non-true size. Th e followi g figure |
(fig. 3.4) shows |
all three passible |
projecting |
planes: a frontal projecting plane (a), |
a horizontal projecting plan e (b) and |
a profile |
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projecting plane (c). |
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There is an i portant p roperty of the projecting planes, which is calle |
as a col- |
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lective o ne: if a point, a line or a figur e are contained in |
plane, which is perpendicu- |
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lar to the projection plane, t eir proje |
tions on the above plane coi ncide wit |
the trace |
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of the pr jecting plane. |
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Fig. 3.4 . Projectin planes:
a – a frontal p ojecting plane; b – a horizontal p ojecting plane; c – a pro file projecting plane
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