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1. TA 111/D : Engineering
Graphics
2024-25 II Semester
Dr. Arnab Samanta
Department of Aerospace Engineering
Indian Institute of Technology Kanpur
Office : A-02 Low Speed Laboratory
Phone : 2182
Email : asamanta@iitk.ac.in

2. Lecture 16
Space Geometry II
2

3. Direct Formula
Rotating Line Method
Auxiliary Plane Method
Review: Finding TL of a Line
3

4. Rotating Line Method
TL
Review: Finding TL of a Line
H
F
 Rotation axis in
plane normal to ‘the
plane’ to view TL
 Rotate until line is
parallel to plane
 The length in the
other plane is TL
TL in Frontal Plane

5. TL
H
F
TL
=
𝐁𝐅
𝐓
AH
BH
AF
BF
𝐀𝐅
𝐓
𝐁𝐇
𝐓
𝐀𝐇
𝐓
α
β
Rotating Line Method
5
TL in ANY plane
True Length is same in both planes

6. X
Z
X
Y
TL
Review: Finding TL of a Line
Auxiliary Plane Method
 Plane parallel to TL of the line

7. TRUE LENGTH(TL)
LINE
OF
SIGHT
AH BH
A
B
BA
AA
AF
BF
F
P
A
H
Notice the distances from the hinge lines
d1
d2
d1
d2
TL
AA
BA
Auxiliary Plane (A)
H
F
Review: Concept of Auxiliary Plane
H

8. 8
F
H
bH
bF
aH
aF
TL
H
aA1
bA1
A
1
A2
A1
aA2, bA2
PV of AB
dA
dB
d
A
d
B
d
1
A
d
1
B
d1A
=d1B
 Find TL and PV of line AB
Concept of Auxiliary Plane: Lines
Notice the direction of Projectors
Notice the distance
measurements along Projectors

9. 9
H
F
P
BH
AH
BF
AF
AP
BP
AA1
BA1
F
A
1
AA2,BA2
TL
A
1
A
2
PV of AB
dA
dB
d
A
d
B
d
A
d
B
d
1
A
d
1
B
d
1
A
=
d
1
B
Concept of Auxiliary Plane: Lines
 Find TL and PV of line AB
Notice the direction of Projectors

10. 10
A
B
C
A B C
A
B
C
A
B
C
A B C
A B
C
A
B
C
A
B
C
A
B
C
EV
EV
TS
TS
EV
EV
TS
EV
EV
Note: TS of plane is visible if LOS is perpendicular to EV
True Shape of Planes

11. 11
A
B
C
A
B
C
A
B
C
True shape?
A
True Shape?
EV of plane ABC
True Shape of Planes
TS of plane not projected in any other principal planes

12. 12
A
B
C
A
B
C
B
C
True
Shape?
True
Shape?
A
True
Shape?
True Shape of Planes
Plane neither parallel nor perpendicular to any principal
planes: most general case

13. 13
Edge View of Plane
TL
H
F
E
V
H
A1
PV
Note:
If one sees any line of a plane in PV,
that plane will be in EV
Find EV of plane ABC
View a line contained in the plane in TL
Find PV of the line in TL

14. 14
True Shape of Plane
TL
H
F
E
V
H
A1
PV
A1 A2
TRUE
SHAPE
Recall:
TS of a plane is available if Line of
Sight is perpendicular to EV
Find TS of plane ABC

15. 15
Projecting from FP
𝐴𝑉
𝐵𝑉
𝐶𝑉
𝐴𝐻
𝐵𝐻
𝐷𝐻
TL
H
F
F A1
PV
𝐷𝑉
EV
𝐵𝐴1
𝐶 𝐴1
A1
A2
TS
𝐵𝐴 2
𝐶𝐴 2
𝐴𝐴2
True Shape of Plane
Find TS of plane ABC
𝐶𝐻

16. 16
Projecting from both HP
& FP: compare shapes
True Shape of Plane
𝐶𝐻
𝐴𝑉
𝐵𝑉
𝐶𝑉
𝐴𝐻
𝐵𝐻
𝐷𝐻
TL
H
F
F A1
PV
𝐷𝑉
EV
𝐵𝐴1
𝐶𝐴1
A1
A2
TS
𝐵𝐴 2
𝐶𝐴 2
𝐵𝐴 2
𝐴𝐴2
𝐶𝐴 2
TS
Find TS of plane ABC

17. 17
Any two lines in space can be intersecting, non-
intersecting, parallel, perpendicular or skewed to
each other
Various relationships between a line and a plane are
possible: line parallel to a plane, perpendicular to a
plane, intersecting or non-intersecting with the
plane, etc
Given orthographic views of two lines or lines and
planes: to find the relationship between them
Relationship between Lines and Planes

18. 18
Relationship between Lines and Planes
We will learn about
Conditions for intersection, parallelism, and
perpendicularity of lines
Shortest distance from a point to a line
Shortest distance from a line to a line
Conditions for intersection, parallelism, and
perpendicularity of planes
Line piercing a plane and its visibility
Angle between two planes

19. 19
 Lines that have common ‘space direction’ (i.e. slope
and bearing) are parallel lines
 Lines may appear to be parallel in one or even two
views but may not be parallel
 Parallel lines will remain parallel in ALL the views
H
F
Are the two lines parallel?
Yes !
Oblique lines are parallel if they
appear parallel in any two
principal orthographic views
P
Parallel Lines

20. 20
AH
BH
CH
DH
AF
BF
CF
DF
AP
CP
BP
DP
F
H
P
Parallelism for non-oblique lines needs to be tested in all
three principal views
Parallel Lines

21. 21
Case A: Yes! AB and CD are perpendicular to each other
Case B: Yes! AB and CD are perpendicular to each other
Case C: May or may not be perpendicular to each other, need to
check using an Auxiliary View with one of these lines in TL
Perpendicular Lines

22. 22
 Given: AB in H and F views; BC only in F view. Complete
line BC in H view such that AB and BC are perpendicular
to each other
F
H
bH
bF
aH
aF
d
TL
aA1
bA1
A1
cF
cA1
cH
If the lines are making a 90o
angle in an auxiliary view where one of
the lines in TL, the lines are perpendicular to each other
Think: If two lines (none
in TL) appears to meet at
90o
in a view, are they
perpendicular to each
other?
Perpendicular Lines
H

23. 23
 The concept of perpendicular lines can be used to
determine shortest distance from a point to a line
 Given: AB in TL and O and AB are in same vertical plane
O
F
H
A B
TL P
What is the shortest distance
between O and AB?
What is the TL of this shortest
distance?
OP! Because O and AB are
in the same vertical plane
Shortest Distance from a Point to a Line

24. 24
O
F
H
A B
P
Is OP perpendicular to AB?
Yes!
Is the TL of OP seen in F
view?
No, it needs to be found
out in an auxiliary view
Shortest Distance from a Point to a Line
Given: AB in TL and O and AB are in same vertical plane
TL

25. 25
𝐴𝐻
𝐴𝑉
𝐶𝐻
𝐷𝐻
𝐶𝑉
𝐷𝑉
𝐶1
𝐷1
𝑻𝑳
𝐴1
𝑷𝑽 (𝐶𝐷)
𝐴2
𝑻𝑳
𝐵1
𝐵𝐻
𝐵𝑉
F
H
A1
H
A1 A2
Perpendicular from a Point to a Line
Find perpendicular from A on CD

26. 26
6. O is not fixed in space,
somewhere in line AB
7. Project OP back in A1, H,
and F views (from A2)
9. How do you locate
point O in A1?
8. Notice that P is in CD
and O is in AB
5. CD is NOT in TL in A2. OP is the
required TL of the shortest distance
between AB and CD (in A2)
AH
BH
CH
DH
OH
PH
AF
CF
DF
BF
OF
PF
BA1
AA1
DA1
PA1
OA1
F
H
H
A
A
1
A
2
CA2
PV of AB (OA2)
DA2
PA2
900
1. Get TL of AB and project CD in A1
2. Get PV of AB and project CD in A2
4. In A2, which line is in TL? CD or OP?
10. OP is parallel to hinge-
line A1-A2. Why?
TL
Shortest Distance between Two Lines
TL
3. Drop perpendicular from PV of
AB to CD in A2
Find shortest distance
between AB and CD
CA1

27. 27
Thank You