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 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.

C1 A1 C1 A1 D1 B1 B1 D1 A D C B D C B A Which set of lines is skew? Intersecting Lines When two lines intersect, the point of intersection will align in elevation and plan, and any other view. Skew Lines Skew lines do not intersect. Their apparent point of intersection will not align in elevation, plan or any other view. C1 D B A C A1 D1 B1 C1 A C D B A1 D1 B1 Find the vertical distance between the two skew lines?

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

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

C B A D 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 B A D

Parallel Plane 1

M C A D B C true length (strike) B A true length (strike) D M C D B A

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) Determine the shortest distance between the two skew lines. (b) Determine the projections of this shortest distance. A B C D A1 B1 C1 D1

Shortest Distance

A2 B2 C2 D2 A B C D A1 B1 C1 D1 A3 B3 C3 D3 datum line

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. A B C D A1 B1 C1 D1

Shortest Horizontal

A2 B2 C2 D2 A B C D A1 B1 C1 D1 A3 B3 C3 D3 datum line

traces of plane director H V A B C D A1 B1 C1 D1 Where is the other plane director? traces of plane director

Hyperbolic Paraboloid

MINING GEOMETRY

headwall Earth’s surface footwall

Skew boreholes Line on headwall Line on footwall