Descriptive geometry

Heuristics

There is heuristic value to studying descriptive geometry. It promotes visualization and spatial analytical abilities, as well as the intuitive ability to recognize the direction of viewing for best presenting a geometric problem for solution. Representative examples:

The best direction to view

• Two skew lines (pipes, perhaps) in general positions in order to determine the location of their shortest connector (common perpendicular)
• Two skew lines (pipes) in general positions such that their shortest connector is seen in full scale
• Two skew lines in general positions such the shortest connector parallel to a given plane is seen in full scale (say, to determine the position and the dimension of the shortest connector at a constant distance from a radiating surface)
• A plane surface such that a hole drilled perpendicular is seen in full scale, as if looking through the hole (say, to test for clearances with other drilled holes)
• A plane equidistant from two skew lines in general positions (say, to confirm safe radiation distance?)
• The shortest distance from a point to a plane (say, to locate the most economical position for bracing)
• The line of intersection between two surfaces, including curved surfaces (say, for the most economical sizing of sections?)
• The true size of the angle between two planes

A standard for presenting computer-modeling views analogous to orthographic, sequential projections has not yet been adopted. One candidate for such is presented in the illustrations below. The images in the illustrations were created using three-dimensional, engineering computer graphics.

Three-dimensional computer modeling produces virtual space behind the screen and may produce any view of a model from any direction within this virtual space. It does so without the need for adjacent orthographic views and therefore may seem to render the circuitous, stepping protocol of descriptive geometry obsolete. However, since descriptive geometry is the science of the legitimate or allowable imaging of three or more dimensional space, on a flat plane, it is an indispensable study, to enhance computer modeling possibilities.

Examples

Finding the shortest connector line QT between two given skew lines PR and SU

Given the X, Y and Z coordinates of P, R, S and U, projections 1 and 2 are drawn to scale on the X-Y and X-Z planes, respectively. Projections 1 and 2 are delineated by hinge line H_1,2, and aligned such that each point projects perpendicularly across the hinge line (P1:P2, R1:R2, S1:S2, U1:U2).

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, the projection 3 view is chosen perpendicular to S₂U₂ by drawing a hinge line H_2,3 parallel to S₂U₂. To get an end view of SU, the projection 4 view is chosen is perpendicular to the true view of line S₃U₃ by drawing a hinge line H_3,4 perpendicular to S₃U₃. The perpendicular QT is the true length of the connector and its distance d gives the shortest distance between PR and SU.

To locate points Q and T on these lines giving this shortest distance, projection 5 is drawn with hinge line H_4,5 perpendicular to QT and parallel to P₄R₄, making both P₅R₅ and S₅U₅ 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.

General solutions

General solutions are a class of solutions within descriptive geometry that contain all possible solutions to a problem. The general solution is represented by a single, three-dimensional object, usually a cone, the directions of the elements of which are the desired direction of viewing (projection) for any of an infinite number of solution views.

For example: To find the general solution such that two, unequal length, skew lines in general positions (say, rockets in flight?) appear:

• Equal length
• Equal length and parallel
• Equal length and perpendicular (say, for ideal targeting of at least one)
• Equal to lengths of a specified ratio
• others.

In the examples, the general solution for each desired characteristic solution is a cone, each element of which produces one of an infinite number of solution views. When two or more characteristics of, say those listed above, are desired (and for which a solution exists) projecting in the direction of either of the two elements of intersections (one element, if cones are tangent) between the two cones produces the desired solution view. If the cones do not intersect a solution does not exist. The examples below are annotated to show the descriptive geometric principles used in the solutions. TL = True-Length; EV = Edge View.

Figs. 1-3 below demonstrate (1) Descriptive geometry, general solutions and (2) simultaneously, a potential standard for presenting such solutions in orthographic, multiview, layout formats.

The potential standard employs two adjacent, standard, orthographic views (here, Front and Top) with a standard "folding line" between. As there is no subsequent need to 'circuitously step' 90° around the object, in standard, two-step sequences in order to arrive at a solution view (it is possible to go directly to the solution view), this shorter protocol is accounted for in the layout. Where the one step protocol replaces the two-step protocol, "double folding" lines are used. In other words, when one crosses the double lines he is not making a circuitous 90° turn but a non-orthodirectional turn directly to the solution view. As most engineering computer graphics packages automatically generates the six principal views of the glass box model, as well as an isometric view, these views are sometimes added out of heuristic curiosity.

Figure 1: Descriptive geometry - skew lines appearing perpendicular
Figure 2: Descriptive geometry - skew lines appear equal length
Figure 3: Descriptive geometry - skew lines appear in specified length ratio