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Skew Lines

Topic Integration
DESIGN AND COMMUNICATION GRAPHICS
Solutions to problems should be taught with….
“an explanation as to how the construction was derived ”
Draft Guidelines for Teachers (page 14)

Two key concepts of plane and descriptive geometry

relating to skew lines are
parallel lines
the parallel plane

Understanding the Why rather than knowing the How
PROPERTIES AND PROJECTIONS OF SKEW LINES
“Define the concept of skew lines and their use in solving practical problems”
page 17 of syllabus

 Clearances between cables, pipes and braces
 Shortest connecting tunnel between two mine tunnels
 Shortest connection between two oblique sewer lines or pipelines
Skew lines are lines which are non-intersecting and non-parallel.

Skew lines are
non-coplanar

If you look at a bridge over a river, you are looking at an example of skew lines.
An overpass is an excellent example of skew lines. The roadway represents one
line and the pedestrian bridge represents another line. The “lines” do not intersect
because they are on different planes.
Which set of lines is skew?
Intersecting Lines

Skew Lines

When two lines intersect, the point of
intersection will align in elevation and
plan, and any other view.

Skew lines do not intersect. Their
apparent point of intersection will not
align in elevation, plan or any other view.

C1

A1

A1

D1

B1

A

C1

B1

D1
D

B

C
D

C
B

Find the vertical distance between the two skew lines?

A
THE CONCEPT OF A PARALLEL PLANE
PARALLEL LINES REMAIN PARALLEL IN EVERY VIEW

Lines which are parallel in space will appear parallel in all views
e D r a w i n g s C o n tr o l

Parallel Lines
PARALLEL LINES REMAIN PARALLEL IN EVERY VIEW

Lines which are parallel in space will appear parallel in all views
except
in the views in which they appear as points
or
where one line is behind the other
PARALLEL PLANE
If a line is parallel to any line in a plane, it is parallel to the plane
The projections of two skew lines AB and CD are shown.

(a)

Find a plane containing the line CD and parallel to the line AB.

(b)

Prove that the plane is parallel to the line.

C

A

D

B
C
B

A

D
e D r a w i n g s C o n tr o l

Parallel Plane 1
C

M
A

D

B

C

true length (strike)
true length (strike)

A

D

M
B
A
D

C

B
SHORTEST DISTANCE BETWEEN
TWO SKEW LINES
What types of applications of skew lines are around us?
The directions of two parachute jumpers landing are represented by the skew lines AB and CD.
(a)
(b)

Determine the shortest distance between the two skew lines.
Determine the projections of this shortest distance.
D1

B1

A1

C1

A

B

D

C
e D r a w i n g s C o n tr o l

Shortest Distance
B2
D1

D3

B1

D2
A2

B3

A1

A3

C1

A
C2

B

D

C3

datum line
C
SHORTEST HORIZONTAL DISTANCE
BETWEEN TWO SKEW LINES
The directions of two javelins are represented by the skew lines AB and CD.
Determine the projections of the shortest horizontal distance between the two skew lines.
D1

B1

A1

C1

A

B

D

C
e D r a w i n g s C o n tr o l

Shortest Horizontal
B2
D1

B1

D2
A2

A1
D3

C1

A
C2

B

D

A3

B3

C3

datum line
C
V

D1

B1

A1

T

C1

A

B

D

Where is the other
plane director?

traces of plane
director

C
H
e D r a w i n g s C o n tr o l

Hyperbolic Paraboloid
MINING GEOMETRY
headwall

Earth’s surface

footwall
Skew boreholes

Line on headwall

Line on footwall

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Skew lines

  • 2. DESIGN AND COMMUNICATION GRAPHICS Solutions to problems should be taught with…. “an explanation as to how the construction was derived ” Draft Guidelines for Teachers (page 14) Two key concepts of plane and descriptive geometry relating to skew lines are parallel lines the parallel plane Understanding the Why rather than knowing the How
  • 3. PROPERTIES AND PROJECTIONS OF SKEW LINES “Define the concept of skew lines and their use in solving practical problems” page 17 of syllabus  Clearances between cables, pipes and braces  Shortest connecting tunnel between two mine tunnels  Shortest connection between two oblique sewer lines or pipelines
  • 4. Skew lines are lines which are non-intersecting and non-parallel. Skew lines are non-coplanar If you look at a bridge over a river, you are looking at an example of skew lines.
  • 5. An overpass is an excellent example of skew lines. The roadway represents one line and the pedestrian bridge represents another line. The “lines” do not intersect because they are on different planes.
  • 6. Which set of lines is skew? Intersecting Lines Skew Lines When two lines intersect, the point of intersection will align in elevation and plan, and any other view. Skew lines do not intersect. Their apparent point of intersection will not align in elevation, plan or any other view. C1 A1 A1 D1 B1 A C1 B1 D1 D B C D C B Find the vertical distance between the two skew lines? A
  • 7. THE CONCEPT OF A PARALLEL PLANE
  • 8. PARALLEL LINES REMAIN PARALLEL IN EVERY VIEW Lines which are parallel in space will appear parallel in all views
  • 9. e D r a w i n g s C o n tr o l Parallel Lines
  • 10. PARALLEL LINES REMAIN PARALLEL IN EVERY VIEW Lines which are parallel in space will appear parallel in all views except in the views in which they appear as points or where one line is behind the other
  • 11. PARALLEL PLANE If a line is parallel to any line in a plane, it is parallel to the plane
  • 12. The projections of two skew lines AB and CD are shown. (a) Find a plane containing the line CD and parallel to the line AB. (b) Prove that the plane is parallel to the line. C A D B C B A D
  • 13. e D r a w i n g s C o n tr o l Parallel Plane 1
  • 14. C M A D B C true length (strike) true length (strike) A D M B A D C B
  • 16. What types of applications of skew lines are around us?
  • 17. The directions of two parachute jumpers landing are represented by the skew lines AB and CD. (a) (b) Determine the shortest distance between the two skew lines. Determine the projections of this shortest distance. D1 B1 A1 C1 A B D C
  • 18. e D r a w i n g s C o n tr o l Shortest Distance
  • 21. The directions of two javelins are represented by the skew lines AB and CD. Determine the projections of the shortest horizontal distance between the two skew lines. D1 B1 A1 C1 A B D C
  • 22. e D r a w i n g s C o n tr o l Shortest Horizontal
  • 24. V D1 B1 A1 T C1 A B D Where is the other plane director? traces of plane director C H
  • 25. e D r a w i n g s C o n tr o l Hyperbolic Paraboloid
  • 28. Skew boreholes Line on headwall Line on footwall