2.First angle projection views of a line.ppt

A spatial line is infinite length, but we usually
adopt a definite line segment AB to represent it.
Point A and point B are called endpoints. So,
the projection of a line is actually the projection
of two endpoints on line AB.
Lesson two Views of a Line
When Line plane H, the top
∥
view, ab, appears as True
Length (TS).
When Line ⊥ plane H,
the top view, a(b), appears as
a convergent point.
When Line AB is oblique to Plane H, the top view
appear as Foreshortened ( 透视缩短 ) Line (FL).

1. Three Views of a Line
We will get three views of a line if connecting the
projections of two endpoints in the same
projection plane and brightening them.

In general, a line in space is at angle to three
projection planes, respectively. Namely,
(Line AB ^ plane H) = , (Line AB ^plane V) =,
(Line AB ^ plane W) = .

2. Characteristics of Views of Lines Located in
Different Positions in Space
According to the spatial position between a line and
projection plane, a spatial line may be categorized as:
∥H
1) parallel —— principal lines V
∥
∥W
⊥H
2) perpendicular —— lines perpendicular to principal planes V
⊥
⊥W
3) oblique —— oblique line
principal lines and lines perpendicular to principal planes are
called peculiar position lines but oblique line is called
general position line.

2.1 Principal lines ( 投影面平行线 )
1)Definition:
Line parallel to one principal projection plane and
inclined to the other two is called principal line.
2) Kinds: ∥plane H –- Horizontal line
∥plane V –- Frontal line
∥plane W –- Profile line
3) Characteristics of projection of principal lines

Take Horizontal line for example:
AB H :
∥

ab= AB (TL)
(ab ^ OX) = (AB ^ V) =β
(ab ^ OY )= (AB ^ W) =γ
ZA = ZB : a′b′ OX
∥
a″b″ OY
∥
Frontal line and Profile line are represented in Chart 2.1.
H
H
ZA = ZB

2.2 Lines perpendicular to principal projection
planes
1)Definition:
Line that is perpendicular to one principal projection
plane and parallel to the other two is called lines
perpendicular to principle plane
2) Kinks:⊥H – line perpendicular to Horizontal plane
⊥V – line perpendicular to Frontal plane
⊥W – line perpendicular to Profile plane
3) Characteristics of views of lines perpendicular to
principle projection plane

Take line perpendicular to plane V example:
Line V :
⊥ Front view 1′ (2′)
converges to a point.
Line H :
Ⅱ∥ 12 = (TL)
ⅠⅡ
Line∥ W: 1″2″= ⅠⅡ (TL)
X =X : 12 OY
∥
Z = Z : 1″2″ ∥ OYW
H
Ⅱ
Ⅰ
Ⅰ Ⅱ
Line perpendicular to plane H and line perpendicular
to plane W are represented in Chart 2.2.

2.3 Oblique line
1) Definition:
A line oblique to three projection planes is called oblique
line. Namely, there is an acute angle between oblique line AB
and three projection planes respectively. (AB ^ H) = , (AB ^
V) =, (AB ^ W) = .
2) Projection:
ab  AB (SL)
a′b′ AB (SL)
a″b″ AB (SL)
The angle ,  and 
are not represented in three views
Note:

3. How to judge the spatial position of a line
according to the given views ?
According to the coordinates, we may judge the spatial
position of a line
1) If there is a kind of coordinates value being equal, the line
represented by given views must be Principal line.
When Z coordinates value are equal, the line must be
Horizontal line.
When Y coordinates value are equal, the line must be
Frontal line.
When X coordinates value are equal, the line must be Profile
line.

Example1:
Judge the spatial position of
line according to the given
views.
Solution: AB V
∥
AB is a Frontal line.

Example 2:
views.
Solution
:
AB W
∥
AB is a Profile line.

2) If there is two kinds of coordinates value being equal or
there is a convergent point ( 积聚点 ) in certain view, the line
represented by given views must be line perpendicular to
principal plane.
When X and Y coordinates value equal or there is a
convergent point in top view, the line must be line
perpendicular to Horizontal line.
When X and Z coordinates value equal or there is a
convergent point in front view, the line must be line
perpendicular to Frontal line.
When Z and Y coordinates value equal or there is a
convergent point in left view, the line must be line
perpendicular to Horizontal line.

Example 3:
views.
Solution:
AB W
⊥
AB is a line perpendicular to
Profile plane.

Example 4:
Judge the spatial position of line AA
according to the given views.
Solution:
AA H
⊥
AA is a line perpendicular to
Horizontal plane.
1
1
1

Example 5:
Judge the spatial position of line AB according to the
given views.
Solution:
AB ∥ H
AB is a Horizontal Line.

3) If there is not coordinates value being equal to
views or two views are oblique to projection axes,
the line represented by two views must be oblique
line.

4. How supplement the third view of a line according
to the given views?
To supplement the third view of a line is to
supplement the third view of two endpoints of
a line.

5.Points in Line
There are two characteristics with respect to points
on lines.
) If a spatial point is on a spatial line, the views
f the point appear as on the corresponding view
of the line.
2) Point dividing a line segment in a given ratio
will divide any view of the line in the same
ratio.

AK: KB = ak : kb = a′k′ : k′b′´ = a″k″ : k″b″
That is to say:

Example 6:
Judge whether point C is on
the line AB.
a)
b)
Solution:
a) Point C locates on the line AB.
b) Point C dose not locate on the
line AB.

Example 7:
Judge whether point C locates
on the line AB.
Method 1: Usage the third view.
Solution:
Supplement left view of line AB
and point C. According to the
third view, we know point C dose
not locate line AB.
What is Method 2 ?
We may make use of ratio to judge.

Example 8:
Given: point K locates on the line AB.
Supplement front view of point K.
Method 1: Supplement e the third view.
Solution:
Step 1:
Supplement left view of line
AB.

Step 2:
Supplement left view of point C.
Step 3:
Supplement front view of point C.
What is Method 2 ?

For example 8:
Find point C on the line AB to make AC:CB=2:3
Solution:
With the assistant of auxiliary line aBo,
line AB can be divided into two segments in ratio
2:3.
Look at picture please.

Step 1:
In any direction, draw auxiliary
line through top view a.
Step 2:
Put 6 equidistant dividing
points in the auxiliary line
from a to Bo so that we obtain 5
equidistant segments and
obtain point Co.
Step3:
After connecting Bo, b, through point Co draw line parallel
to Bob to intersect ab at c. After drawing projection
connecting line through top view c, we obtain front view c′.
Point C (c, c′ ) is the solution.

6. Relative Position Between Two Lines
There are three situations of spatial relationship
between two lines which are intersection,
parallelism and skew.

1) Intersecting Lines
If two lines intersect then their views intersect and
intersecting point conforms to the projection rule of
a point.

2) Parallel Lines
If two lines are parallel each other then their views
are parallel each other respectively.

3) Skew Lines
Nonintersecting, nonparallel lines are called skew
lines. Skew line locate two planes respectively( 异面直线

There is not a common point on skew
lines but two pairs coincided points.
Point locates on line
Ⅰ AB and point locates
Ⅱ
on line CD. Point is above point . In top view,
Ⅰ Ⅱ
point is visible which indicated by
Ⅰ 1 but point is
Ⅱ
invisible which indicated by (2).

Point locates on line
Ⅲ CD and point Ⅳ on line AB.
Point Ⅲ is front of point . In front view, point
Ⅳ Ⅲ is
visible which indicated by 3′ but point Ⅳ is invisible
which indicated by (4′ ).

Example 9:
Judge whether line AB intersects with line CD.
Solution:
Line AB does not intersect
with line CD.
Why ?

Because projection connecting line is not
perpendicular to axis OX , there is not a intersecting
point (a common point) on line AB and line CD. So,
line AB and line CD are skew line but intersecting
lines.

Example 10:
Judge whether line AB is
parallel to line CD.
Solution:
After supplementing left
view of line AB and line
CD we know line AB is

Example 11:
Judge whether line AB is
Solution:
After supplementing left view
of line AB and line CD we know
line AB is not parallel to line
CD.

Example 12:
From Point C draw horizontal
Line CD to intersect Line AB.
Questions:
1) How many solutions ?
2) To draw start from which
views ?
Who is willing to answer ?
Answer 1): There is only one solution in the problem
as the highness of the Horizontal line is definite.
Answer 2): To draw start from front view. The reason
is the same as aforementioned ( 上述的 ).

Solution:
Step
1:
Through c′ draw line parallel to axis
OX to intersect a′b′ at
k ′and extend to d′.
Step 2:
Draw projection connecting line
k ′k.
Step 3:
Connect c,k and extend. Lastly,
draw projection connecting line
d′d.
Line CD (c′d′, cd ) is the solution.

7. Theorem of right angle
If AB BC and AB H
⊥ ∥
Then ab bc.
⊥
1)
theorem:
2) Prove: ∵AB BC
⊥ and
Bb H
⊥
∴AB Plane BbcC and AB bc
⊥ ⊥
∵AB H, AB ab
∥ ∥
∴ ab bc
⊥
If AB BC and AB V
⊥ ∥
Then a′ b ′ b′
⊥ c′ .
3) The another form of
theorem:

4) Two views of the theorem of right angle.

Example 13:
Given: Line CD and point A
Demand:
Through Point A construct
a frontal line Line CD.
⊥

Solution:
Draw frontal line AB ⊥ Line CD.
1) Draw front view: a′ b′ c′ d′
⊥
2) Draw top view: ab OX
∥
Finish drawing.

Given:
The edge AB of a rectangle is
parallel to H plane.
Complete two view of the
rectangle.
Analyse:
∵ AB AC and AB H plane
⊥ ∥
∴ ab ac
⊥
Example 14:

Drawing:
1) Through top view a, draw
line perpendicular to ab.
2) Draw projection connecting
Line c′ c.
3) Through point C draw line CD line
∥ AB and
through point B draw line BD line
∥ AC.
Finish drawing.

2.First angle projection views of a line.ppt

More Related Content

Similar to 2.First angle projection views of a line.ppt (20)

Recently uploaded (20)

2.First angle projection views of a line.ppt