Descriptive geometry

Descriptive geometry is the branch of geometry which allows the representation of three-dimensional objects in two dimensions, by using a specific set of procedures. The resulting techniques are important for engineering, architecture, design and in art.[1] The theoretical basis for descriptive geometry is provided by planar geometric projections. The earliest know publication on the technique was "Underweysung der Messung Mit dem Zirckel un Richtscheyt", published in Linien, Nuremberg: 1525, by Albrecht Dürer. Gaspard Monge is usually considered the "father of descriptive geometry". He first developed his techniques to solve geometric problems in 1765 while working as a draftsman for military fortifications, and later published his findings.[2]

Monge's protocols allow an imaginary object to be drawn in such a way that it may be modeled in three dimensions. All geometric aspects of the imaginary object are accounted for in true size/to-scale and shape, and can be imaged as seen from any position in space. All images are represented on a two-dimensional surface.

Descriptive geometry uses the image-creating technique of imaginary, parallel projectors emanating from an imaginary object and intersecting an imaginary plane of projection at right angles. The cumulative points of intersections create the desired image.

Protocols

Example of four different 2D representations of the same 3D object

Aside from the Orthographic, six standard principal views (Front; Right Side; Left Side; Top; Bottom; Rear), descriptive geometry strives to yield four basic solution views: the true length of a line (i.e., full size, not foreshortened), the point view (end view) of a line, the true shape of a plane (i.e., full size to scale, or not foreshortened), and the edge view of a plane (i.e., view of a plane with the line of sight perpendicular to the line of sight associated with the line of sight for producing the true shape of a plane). These often serve to determine the direction of projection for the subsequent view. By the 90° circuitous stepping process, projecting in any direction from the point view of a line yields its true length view; projecting in a direction parallel to a true length line view yields its point view, projecting the point view of any line on a plane yields the plane's edge view; projecting in a direction perpendicular to the edge view of a plane will yield the true shape (to scale) view. These various views may be called upon to help solve engineering problems posed by solid-geometry principles

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

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 tube", as it were, 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 between two given skew lines PR and SU

Example of the use of descriptive geometry to find the shortest connector between two skew lines. The red, yellow and green highlights show distances which are the same for projections of point P.

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.

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 S2U2. To get an end view of SU, projection 4 is drawn with hinge line H3,4 perpendicular to S3U3. 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 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:

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

See also

References

  1. ↑ Joseph Malkevitch (April 2003), "Mathematics and Art", Feature Column Archive, American Mathematical Society
  2. ↑ Ingrid Carlbom, Joseph Paciorek (December 1978), "Planar Geometric Projections and Viewing Transformations", ACM Computing Surveys, 10 (4): 465–502, doi:10.1145/356744.356750
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