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. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position The resulting techniques are important for engineering, architecture, design and in art. Engineering is the Discipline and Profession of applying technical and scientific Knowledge and The term architecture (from Greek αρχιτεκτονικήarchitektoniki) can be used to mean a process a profession or documentation Design is used both as a Noun and a Verb. The term is often tied to the various Applied arts and Engineering (See design disciplines Art refers to a diverse range of Human activities creations and expressions that are appealing to the Senses or Emotions of a human individual ^[1] The theoretical basis for descriptive geometry is provided by planar geometric projections. Graphical projection is a Protocol by which an image of an imaginary three-dimensional object is projected onto a planar surface without the aid of mathematical calculation Gaspard Monge is usually considered the "father of descriptive geometry". Gaspard Monge Comte de Péluse ( May 10, 1746 &ndash July 28, 1818) was a French Mathematician and inventor of 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 3-D modeled. 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 drawing 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.

Contents 1 Protocols 2 Heuristics 2.1 The best direction to view: 3 General solutions 4 References 5 See also

Protocols

• Project two images of an object into mutually perpendicular, arbitrary directions. Each image view accommodates three dimensions of space, two dimensions displayed as full-scale, mutually-perpendicular axes and one as an invisible (point view) axis receding into the image space (depth). Each of the two adjacent image views shares a full-scale view of one of the three dimensions of space.

• Either of these images may serve as the beginning point for a third projected view. The third view may begin a fourth projection, and on ad infinitum. These sequential projections each represent a circuitous, 90° turn in space in order to view the object from a different direction.

• Each new projection utilizes a dimension in full scale that appears as point-view dimension in the previous view. To achieve the full-scale view of this dimension and accommodate it within the new view requires one to ignore the previous view and proceed to the second previous view where this dimension appears in full-scale.

• Each new view may be created by projecting into any of an infinite number of directions, perpendicular to the previous direction of projection. (Envision the many directions of the spokes of a wagon wheel each perpendicular to the direction of the axle. ) The result is one of stepping circuitously about an object in 90° turns and viewing the object from each step. Each new view is added as an additional view to an orthographic projection layout display and appears in an "unfolding of the glass box model". Orthographic projection is a means of representing a three- Dimensional (3D object in two dimensions (2D

Aside from the Orthographic, six standard principal views (Front; Right Side; Left Side; Top; Bottom; Rear), descriptive geometry strives to yield three basic solution views: the true length of a line (i. e. , full size, not foreshortened), the point view (end view) of a line, and the true shape of a plane (i. e. , full size to scale, or not foreshortened). 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:

• 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. In Geometry, skew lines are two lines that do not intersect but are not Parallel. 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 renders the circuitous, stepping protocol of Descriptive Geometry obsolete.

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.

References

1. ^ Joseph Malkevitch (April 2003), “Mathematics and Art”, Feature Column Archive (American Mathematical Society), <http://www.ams.org/featurecolumn/archive/art1.html> 
2. ^ Ingrid Carlbom, Joseph Paciorek (Dec. The American Mathematical Society (AMS is an association of professional Mathematicians dedicated to the interests of mathematical research and scholarship which 1978), Planar Geometric Projections and Viewing Transformations, vol. v. 10 n. 4, ACM Computing Surveys (CSUR), pp. The Association for Computing Machinery, or ACM, was founded in 1947 as the world's first scientific and educational Computing society p. 465-502, DOI 10. 1145/356744. 356750 

See also

Projective geometry is a non- metrical form of Geometry, notable for its principle of duality. Graphical projection is a Protocol by which an image of an imaginary three-dimensional object is projected onto a planar surface without the aid of mathematical calculation Orthographic projection is a means of representing a three- Dimensional (3D object in two dimensions (2D Axonometric projection ("to measure along axes" is a technique used in orthographic pictorials Isometric projection is a form of Graphical projection —more specifically an Axonometric projection. Dimetric projection is a form of Axonometric projection, in which its direction of viewing is such that two of the three axes of space appear equally foreshortened of which Trimetric projection is a form of Axonometric projection, where the direction of viewing is such that all of the three axes of space appear unequally foreshortened This article discusses imaging of three-dimensional objects For an abstract mathematical discussion see Projection (linear algebra. Perspective (from Latin perspicere to see through in the graphic arts such as drawing is an approximate representation on a flat surface (such as paper of an image as it is perceived Perspective (from Latin perspicere to see through in the graphic arts such as drawing is an approximate representation on a flat surface (such as paper of an image as it is perceived A technical drawing is a form of graphic communication This type of Drawing is used in the transforming of an idea into physical form An engineering drawing is a type of Technical drawing, used to fully and clearly define requirements for engineered items and is usually created in accordance

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