Descriptive geometry

Descriptive Geometry is the portion of the geometry, which deals with the geometric- constructive method of projections of three-dimensional objects on a two dimensional representation level. The areas of application their methods are wide-ranging and extend alongside the now well-known applications in engineering and architectural presentation on art, painting, cartography and computer graphics. The descriptive geometry is not limited to displaying of spatial objects, but also provides opportunities spatial geometric drawings to solve problems: for example, the determination of the intersection of a line and a plane or the intersection curve of two surfaces or the shadow of an object.

Screenshot of CATIA V5: Machinery

Unlike in the past (see web link Delabar ) Descriptive Geometry is no longer the only means to represent spatial objects graphically or solve spatial geometric problems. For this purpose one uses computers today (see links and Geometric Modeling ). The Importance of Descriptive Geometry is now rather in the training of the users of geometric software, so they understand what a software can and demands of inputs. For the first sketches of a ( spatial ) idea or interpretations and amendments of computer drawings drawing with ruler and compass is a great exercise.

In the representation of spatial objects in a drawing plane two competing factors play a significant role. If you want to achieve dimensional accuracy, it is usually possible only with loss of clarity. For example, make the following two pictures of a house close easily on length, width and height; but they are not very descriptive. In contrast, the next two pictures bring the spatial impression more advantage. Exact dimensions can, however, be difficult to read (especially from the right image).

Imaging method

In the Descriptive Geometry, use is essentially two imaging methods. Here, points and curves of an object with the aid of beams ( lines ) are projected onto a screen panel ( level):

A) parallel projection

The imaging rays are parallel, such as the sunlight. Here, a distinction between the two cases:

  • The rays are perpendicular to the picture board (vertical parallel projection or orthogonal or normal projection ).
  • The rays are not perpendicular to the image board (inclined or oblique parallel projection ).

Parallel projections are often used by engineers because of their loyalty division ratio ( Teiverhältnisse remain invariant straight lines ). The special case bird's-eye view is an oblique parallel projection, which is used in particular to illustrate Maps. It can be produced relatively easily by hand. Parallel projections can be produced using the Einschneideverfahrens quickly as axonometric images or, for larger objects.

For almost all constructions in Descriptive Geometry using ground plan and elevation of an object. These are vertical parallel projections onto a horizontal (plan ) and vertical plane ( elevation ) (see Table Two projection). Due to it is ( with appropriate labels ) an object spatially clearly described.

B ) central projection

All imaging beam passing through one point, the projection center or eye point. In parallel projection, the images of parallel lines ia parallel again. In the central projection, the images of parallel lines cut ia at one point, the vanishing point of the parallel bundle.

That a central projection gives the best visual impact to show the images with a row of houses. In the image in parallel projection of the rear house appears larger than the first. This is due to an optical illusion. The eye recognizes the house as a spatial object and expects an equally large, remote house is small, but this is not the case with parallel projection.

Methods of Descriptive Geometry

The most important tool in Descriptive Geometry are plan and elevation and their assignments. They provide the spatial information for specific representations and constructions. The knowledge you learn in

  • Two panel projection: Occasionally it is necessary to introduce further cracks. We also speak of Umprojektionen and multi panel projections.

With plan and elevation can then with the help of

  • Axonometric,
  • Einschneideverfahren
  • Orthogonal and Axonometric
  • Outline structure ( for curved surfaces ). See also channel surfaces.

Produce vivid images of spatial objects in parallel projection.

Images in central projection is constructed with the best

  • Architects arrangement (similar to the Einschneideverfahren for parallel projections ) and
  • Frontal perspective (perspective with an essential vanishing point, the main point).

To avoid distorted picture acting parts, one should look for the choice of the position of image panel, key point and the eye point to the first

  • Viewing circle

Draw. Because only parts of the image within the vision circle appear undistorted in the image perspectives.

An important basic task of Descriptive Geometry is the ( graphic ) Determination of the intersection of a line and a plane. The method for this is called

  • Piercing point method.

Penetration points are, for example, required in the

  • Shade structure.

For the construction of points of the curve of intersection of two surfaces (cylinder, cone, sphere, torus surface of rotation ), there are three standard procedures:

  • The auxiliary plane methods,
  • The pendulum plane methods and
  • The auxiliary sphere method.

In the manufacture of models of interpenetrating cylinders and / or cones often wound transactions of these surfaces can be used. How to become handles cylinders and cones in

  • Completion

Described.

In addition to the two- and multi- panel projections, there are special Eintafelprojektionen. Here are floor plans with additional information describing the objects spatially. In road using the

  • Listed projection to
  • Slope surfaces to construct and present.

With the construction of the ridge, fillet and ridge lines (cut line) of flat roof areas is concerned, the

  • Roof design.

Inclined lines and plane figures in parallel projections equalized with you

  • True length and true form.

Analog methods for central projections (Photos) provides the

  • Reconstruction.

Circles and ellipses play especially as boundary curves of objects such as cylinders, cones, and surfaces of revolution an important role (see photo: tower with a gate, bridge in frontal perspective). How it maps with parallel projection and the central projection and then characterized, in

  • Ellipse ( Descriptive Geometry )

Described.

The representation of a sphere is very simple in vertical parallel projection. Its outline is a circle with the radius of the sphere. In all other major projection methods, such as bird's eye view, cavalier perspective and central projection, the outline of a ball appears, except in special cases, as an ellipse. How to outline the ellipse of a sphere is constructed in

  • Ball ( Descriptive Geometry )

Explained.

Special perspectives (views)

The basis for almost all representations and constructions in Descriptive Geometry

  • Plan and elevation: Sekrechte parallel projections on a horizontal or vertical image panel.

The word perspective is often used in Descriptive Geometry to name particular ideological views of a spatial object:

  • The cavalier perspective or cabinet perspective is an oblique parallel projection (see Axonometric ) on a vertical screen panel. All plane figures that are parallel to the image panel, are mapped without distortion (see picture)
  • Bird's eye view or military perspective is
  • An engineer projection is an axonometric image with simple reductions (0.5, 1, 1). The images of the x -and y- axis close to the image of the z- axis angles of 132 and 97 degrees (see Axonometric ). Their advantages are: a) simple reductions, b ) good image effect c ) (scaled ) orthogonal, d) contours of spheres are circles.
  • An isometry is an axonometric image in which the distortions in xyz directions are all equal. In the standard isometry also applies: the images of the coordinate axes intersect at an angle of 120 degrees. A typical characteristic is: In the projection of an axis- parallel cube fall two points together.
  • Central perspective is a central projection.
  • Perspective is often used as shorthand for central perspective.
  • Frontal perspective is a central projection of an object with three substantially mutually orthogonal directions (eg square, house ), two of these directions are parallel to the image panel and thus their vanishing points are located in the " infinite". One calls such a view also perspective with one vanishing point (see picture). The special vanishing point is the main point in the rule. The advantage of Frontalpesrpektive: all plane figures in planes parallel to the image panel are only scaled but mapped undistorted. (see examples: Bridge and house in frontal perspective).
  • Perspective with two vanishing points is a central projection in which the vanishing points to two mutually perpendicular orthogonal directions (usually horizontal) play an essential role (see picture).
  • Perspective with three vanishing points is a central projection in the three vanishing points play an essential role. Here are the panels is inclined (see house in bird's eye view of the central projection ).
  • Parallel perspective is a parallel projection
  • Polar perspective is a former name of central projection (see weblink Delabar, Gangolf )

Training

Descriptive Geometry is now a subject in technical and vocational schools and a basic subject in the training of engineers at a technical university or college.

Topic is the acquisition and representation of spatial, especially technical structures ( geometric bodies, structures, representation of the terrain, etc ).

Top Resources are engineering drawings, perspective (central projection), Axonometric, Listed projection and the like. Apart from learning drawing techniques, the spatial imagination and expression are encouraged and cross connections to mathematics, to engineering and the fine arts are produced.

Until the 2000s, a purely graphical compartment ( Applied geometrical drawing ), it is today in many areas of computer graphics. In recent years, the importance of the subject had not general, but decreased in the training because the computer-aided design (CAD) other skills required as the graphical representation of hand since even schools are well equipped with computers in the classroom, the subject heard again the most important technical basis for training at all, and also includes learning related programs - generally market- leading specialist CAD applications the industry.

The actual mental work, converting the 2D representation ( whether paper or screen) in a 3D ( thinking ) model remains even with the use of CAD to the design engineer or designer will receive. In contrast, it is difficult ( eg connection profiles for inclined gates ) to detect spatial design problems when ( and because) you rely on the software.

" Descriptive Geometry is not in a superficial sense requirement to master a CAD program. You to practice, but is a primary experience by the spatial imagination, estimating and selecting solution strategies and the precision of thinking can be trained. "

The History of Descriptive Geometry

For the systematic construction of buildings plans play an important role with specific guidelines. Even in ancient times were used ground plans and elevations. The earliest written evidence of this is the book De architectura of the Roman architect Vitruvius. But it was Albrecht Dürer (1471-1528) wrote in the Middle Ages, the first real textbook of Descriptive Geometry: Underweysung with compasses and Richtscheydt (Nuremberg, 1525). On pages 34-37 of the first book also come forward to the conics ellipse, parabola and hyperbola. Gaspard Monge (1746-1818) resulted in his book, descriptive geometry for the first time a strict allocation of plan and elevation to the drawings to solve spatial problems. The basic tasks of Descriptive Geometry can be found there as early as the still used today version.

The foundations of the central projection were the Greeks and Romans already known. But until the Renaissance, this type of presentation of spatial conditions was rediscovered by the painting and brought to bloom. See De pictura of Leone Battista Alberti ( 1404). The masters of this period were Albrecht Dürer (1471-1528), Leonardo da Vinci (1452-1519) and Michelangelo ( 1475-1564 ).

Line–line intersection Depiction Integrated Authority File
125457
de