TA lecture at Indian institute of technology kanpur of bachelor of technology or bachelor of science program 1st year.

1. Department of Civil Engineering
Indian Institute of Technology Kanpur
TA 111 2023 – 2024 Sem - 1 Prof. Sudhir Misra
TA 101: Engineering Graphics
2023 - 24 Sem – I
Lecture 17 – Oct 9, 2023
Space Geometry
Lines and Planes – III
Sudhir Misra
Department of Civil Engineering
IIT Kanpur, Kanpur 208 016
1
Room No.: WLE 303 C;
Phone – 7346
E-mail: sud@iitk.ac.in

2. Department of Civil Engineering
Indian Institute of Technology Kanpur
TA 111 2023 – 2024 Sem - 1 Prof. Sudhir Misra
LINE
OF
SIGHT
AH BH
A
B
BA
AA
AF
BF
F
P
A
H
Normal/Auxiliary View – Quick Review
Notice the distances from the hinge lines
d1
d2
d1
d2
TL
BA
AA
AA
BA
2

3. Department of Civil Engineering
Indian Institute of Technology Kanpur
TA 111 2023 – 2024 Sem - 1 Prof. Sudhir Misra
F
H
bH
bF
aH
aF
d aA1
bA1
Notice the direction of Projectors
Notice the distance
Measurements along Projectors
bA2
, aA2
Normal View and PV of an Oblique Line
Projected from the Top/Horizontal View
PV
d1
d1
3

4. Department of Civil Engineering
Indian Institute of Technology Kanpur
TA 111 2023 – 2024 Sem - 1 Prof. Sudhir Misra
Parallel Lines
AH
BH
CH
DH
AF
BF
CF
DF
AP
CP
BP
DP
F
H
P
Parallelism for Non-oblique lines needs to be checked
in all three principal views

5. Department of Civil Engineering
Indian Institute of Technology Kanpur
TA 111 2023 – 2024 Sem - 1 Prof. Sudhir Misra
Perpendicular Lines
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 from
an auxiliary view where one of these line is in TL

6. Department of Civil Engineering
Indian Institute of Technology Kanpur
TA 111 2023 – 2024 Sem - 1 Prof. Sudhir Misra
Review
• By now, you must have learnt to find
– Point view of a line
– True length of a line
– Edge view of a plane
– True shape of a plane
• You have also learnt that
– If one sees any line of the plane in PV, the plane will be in EV
– True Shape of a plane is available when line of sight is
perpendicular to the EV
– Correct slope of a line is visible when the line is in TL

7. Department of Civil Engineering
Indian Institute of Technology Kanpur
TA 111 2023 – 2024 Sem - 1 Prof. Sudhir Misra
Relationship between Lines and Planes
• Any two lines in space can be intersecting, non-intersecting,
parallel, perpendicular, or simply skewed to each other
• Similarly, 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 and so on
• Given the orthographic views of two lines, or lines and planes,
we will learn to find relationship between them

8. Department of Civil Engineering
Indian Institute of Technology Kanpur
TA 111 2023 – 2024 Sem - 1 Prof. Sudhir Misra
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
Relationship between Lines and Planes

9. Department of Civil Engineering
Indian Institute of Technology Kanpur
TA 111 2023 – 2024 Sem - 1 Prof. Sudhir Misra
A Line in a Plane • Is line AB in the plane 1-2-3-4?
• How will you determine this?
• Get EV of the plane & project the line
in the same view. If no part of the
line is outside of EV, then the line is
in the plane
• There is another easier way to
establish this
• Check for the ‘common point of
intersection’ between plane & line
• If the intersection of AB with lines 1-
2 and 3-4 stay aligned in both H and F
view, then line AB is in the plane
AH
BH
BF
AF
2H
3H
2F
3F
1H
4H
4F
F
H
1F
Line AB is in the
Plane 1-2-3-4

10. Department of Civil Engineering
Indian Institute of Technology Kanpur
TA 111 2023 – 2024 Sem - 1 Prof. Sudhir Misra
AH BH
CH
AF
BF
CF
DH
DF
PH
QH
PF
QF
AA1
BA1
CA1
DA1
PA1
QA1
H
A1
F
H
A plane may be extended to check this
Notice that AC is in TL in Top view
So, we see that the line PQ is contained in the plane that contains the given plane ABC
TL
Get EV of plane, if line is in plane, line will completely lie within the EV
Is there another way to check this? YES!
Once the plane is extended, common points of intersection (between line &
plane) may be checked if they are aligned or not
A line in a plane

11. Department of Civil Engineering
Indian Institute of Technology Kanpur
TA 111 2023 – 2024 Sem - 1 Prof. Sudhir Misra
𝐴𝑉
𝐵𝑉
𝐶𝑉
𝐴𝐻
𝐵𝐻
𝐶𝐻
𝐷𝐻
H
V
V A1
𝐷𝑉
𝐵𝐴1
𝐶𝐴1
𝑃𝐻
𝑃𝑉
𝐴𝐴1, 𝐷𝐴1
𝑃𝐴1
𝑄𝐴1
𝑄𝑉
𝑄𝐻
Line Parallel to A Plane
11

12. Department of Civil Engineering
Indian Institute of Technology Kanpur
TA 111 2023 – 2024 Sem - 1 Prof. Sudhir Misra
A line is perpendicular to plane when the line is in
true length at a 900 angle to an edge view of the
plane. Line OP is perpendicular to plane 1-2-3-4
pH
1H
2H 3H
4H
H
F
oH
oF
pF
4F3F
1F2F
900
How to draw a
perpendicular from a
point with this concept?
A line perpendicular to a plane
Very useful in finding
shortest distance from
a point to a plane

13. Department of Civil Engineering
Indian Institute of Technology Kanpur
TA 111 2023 – 2024 Sem - 1 Prof. Sudhir Misra
pH
1H
2H
3H
4H
pF
OH
900
2F
1F
3F
4F
OA
d
4A1A
3A2A
PA
H A
d
OF
H
F
PARALLEL
Draw a perpendicular
to a plane from point
1. Get the edge of plane
2. See if any line in TL on the plane
3. See through the line and project all points
4. Line 2-3 in top view in TL in aux view.
5. Take distance from FV and locate points
6. Draw a perpendicular from OA on EV ~ OA
PA (in TL?)
7. Project PA to Top view (see the arrows)
8. Project PH to Front view see PF
Note: Edge view of plane
and perpendicular at 900
needed

14. Department of Civil Engineering
Indian Institute of Technology Kanpur
TA 111 2023 – 2024 Sem - 1 Prof. Sudhir Misra
𝐴𝑉
𝐵𝑉
𝐶𝑉
𝐴𝐻
𝐵𝐻
𝐶𝐻
𝐷𝐻
H
V
V A1
𝐷𝑉
𝐵𝐴1
𝐶𝐴1
𝑃𝐻
𝑃𝑉
𝐴𝐴1, 𝐷𝐴1
𝑃𝐴1
𝑄𝐴1
𝑄𝑉
𝑄𝐻
Line Perpendicular to A Plane
14

15. Department of Civil Engineering
Indian Institute of Technology Kanpur
TA 111 2023 – 2024 Sem - 1 Prof. Sudhir Misra
𝐴𝑉
𝐵𝑉
𝐶𝑉
𝐴𝐻
𝐵𝐻
𝐶𝐻
𝐷𝐻
H
V
A1
𝐷𝑉
𝐵𝐴1
𝐶𝐴1
𝑃𝐻
𝑃𝑉
𝐴𝐴1, 𝐷𝐴1
𝑃𝐴1
𝑄𝐴1
𝑄𝑉
𝑄𝐻
𝐵𝐴2
𝐴𝐴2
𝐶𝐴2
TS
A1
A2
𝑃𝐴2, 𝑄𝐴2
V
Verification:
PQ in PV
15

16. Department of Civil Engineering
Indian Institute of Technology Kanpur
TA 111 2023 – 2024 Sem - 1 Prof. Sudhir Misra
Intersection of two lines
16

17. Department of Civil Engineering
Indian Institute of Technology Kanpur
TA 111 2023 – 2024 Sem - 1 Prof. Sudhir Misra
𝐴𝐻
𝐵𝐻
𝐴𝑉
𝐵𝑉
𝐶𝐻
𝐷𝐻
𝐶𝑉
𝐷𝑉
𝐶1
𝐷1
𝐴1
𝐵1
𝑃𝑉 (𝐶𝐷)
𝐴2
17
Intersection of two lines

18. Department of Civil Engineering
Indian Institute of Technology Kanpur
TA 111 2023 – 2024 Sem - 1 Prof. Sudhir Misra
𝐴𝐻
𝐴𝑉
𝐶𝐻
𝐷𝐻
𝐶𝑉
𝐷𝑉
𝐶1
𝐷1
𝐴1
𝑃𝑉 (𝐶𝐷)
𝐴2
𝐵1
𝐵𝐻
𝐵𝑉
Perpendicular to a line from a point
18

19. Department of Civil Engineering
Indian Institute of Technology Kanpur
TA 111 2023 – 2024 Sem - 1 Prof. Sudhir Misra
Perpendicular Lines : Example
Given: AB in Top and Front views; BC only in F view. Complete line
BC in Top view such that AB and BC are perpendicular to each other
F
H
bH
bF
aH
aF
aA1
bA1
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

20. Department of Civil Engineering
Indian Institute of Technology Kanpur
TA 111 2023 – 2024 Sem - 1 Prof. Sudhir Misra
𝐴𝑉
𝐶𝑉
𝐴𝐻
𝐵𝐻
𝐶𝐻
𝐷𝐻
H
V
V A1
𝐷𝑉
𝐵𝐴1
𝐶𝐴1
𝑃𝐻
𝑃𝑉
𝐴𝐴1, 𝐷𝐴1
𝑃𝐴1
𝑄𝐴1
𝑄𝑉
𝑄𝐻
A line is perpendicular to a
plane when the line in its TL is
at 90o with the EV of the plane
Perpendicular from a point to a plane

21. Department of Civil Engineering
Indian Institute of Technology Kanpur
TA 111 2023 – 2024 Sem - 1 Prof. Sudhir Misra
• 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 Between Two Lines

22. Department of Civil Engineering
Indian Institute of Technology Kanpur
TA 111 2023 – 2024 Sem - 1 Prof. Sudhir Misra
Shortest Distance from a Point to a Line
Given: AB in TL and O is not in the vertical plane that contains AB
O
F
H
A B
TL 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

23. Department of Civil Engineering
Indian Institute of Technology Kanpur
TA 111 2023 – 2024 Sem - 1 Prof. Sudhir Misra
𝐴𝐻
𝐴𝑉
𝐶𝐻
𝐷𝐻
𝐶𝑉
𝐷𝑉
𝐵𝐻
𝐵𝑉
𝐶1
𝐷1
𝐴1
𝐵1 𝑃𝑉 (𝐶𝐷)
𝐴2
𝐵2
𝑇𝐿 ?
𝑀2
𝑀1
𝑁1
𝑀𝐻
𝑁𝐻
𝑀𝑉
𝑁𝑉
Shortest Distance Between Two Lines
23

24. Department of Civil Engineering
Indian Institute of Technology Kanpur
TA 111 2023 – 2024 Sem - 1 Prof. Sudhir Misra
5. Obviously, 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
CA1
PA1
OA1
F
H
CA2
BA2,OA2,AA2
DA2
PA2
900
1. Get TL of AB and project CD in A1
2. Get PV of AB and project CD in A2
3. Drop perpendicular from PV of
AB to CD in A2
4. In A2, which line is in TL? CD or OP?
7. Project OP back in A1, H,
and F views (from A2)
8. Notice that P is in CD
and O is in AB
9. How do you locate
point O in A1?
10. Use another basic
concept, OP will be parallel
to hinge-line A1-A2, why?
In TL
6. O is not fixed in space,
somewhere in line AB

25. Department of Civil Engineering
Indian Institute of Technology Kanpur
TA 111 2023 – 2024 Sem - 1 Prof. Sudhir Misra
25
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