Кеңістіктік объектілерді екі өлшемде бейнелеу геометриясы

Branch of geometry which allows the representation of three dimensional objects in two dimensions

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. The theoretical basis for descriptive geometry is provided by planar geometric projections. The earliest known publication on the technique was "Underweysung der Messung mit dem Zirckel und Richtscheyt" (Observation of the measurement with the compass and spirit level), published in Linien, Nuremberg: 1525, by Albrecht Dürer. Italian architect Guarino Guarini was also a pioneer of projective and descriptive geometry, as is clear from his Placita Philosophica (1665), Euclides Adauctus (1671) and Architettura Civile (1686—not published until 1737), anticipating the work of Gaspard Monge (1746–1818), who is usually credited with the invention of descriptive geometry. Gaspard Monge is usually considered the "father of descriptive geometry" due to his developments in geometric problem solving. His first discoveries were in 1765 while he was working as a draftsman for military fortifications, although his findings were published later on. 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.

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". 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

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:

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.