Example of the use of descriptive geometry to find the shortest line segment between two given skew lines PR and SU, based on exercise at http://www3.eng.cam.ac.uk/DesignOffice/1adraw/2005_06old/2005_06old/exercise2/eps/linequesimage1.eps . Given the X, Y and Z coordinates of P, R, S and U, projections 1 and 2 are drawn to scale. To get a true view (length in the projection is equal to length in 3D space) of one of the lines: SU in this example, projection 3 is drawn with hinge line H2,3 parallel to S2-U2. To get an end view of SU, projection 4 is drawn with hinge line H3,4 perpendicular to S3-U3. The perpendicular distance d gives the shortest distance between PR and SU. To get points Q and T on these lines giving this shortest distance, projection 5 is drawn with hinge line H4,5 parallel to P4-R4, making both P5-R5 and S5-U5 true views (any projection of an end view is a true view). Projecting the intersection of these lines, Q5 and T5 back to projection 1 (magenta lines and labels) allows their coordinates to be read off the X, Y and Z axes. The red, yellow and green highlights show distances which are the same for point P.
Own work
Cmglee