Axonometric projection

The Axonometric is a method in descriptive geometry to relatively simple presentation of spatial objects in a drawing plane. Here, using the coordinates of points and the substantial images of the three coordinate axes in a plane of the drawing. The result for any choice of image axes up to a scaling a parallel projection. In general, there is an oblique (or slant ) parallel projection. Only for special choice of the image axes and the distortion ratios results in an orthogonal (or perpendicular ) parallel projection, ie the imaging beam are perpendicular to the image plane (see orthogonal axonometric ).

Archway (circles) in cavalier projection

Principle of Axonometric

The coordinate axes is thought to be projected with the aid of parallel rays on a character level along with the points. The unit routes be distorted in the rule. The distortion relations are with, and inscribed. A point is now registered as follows in the image with the coordinate axes:

( The order can be interchanged as desired. )

As for a concrete projection direction and position of the image plane, it is very difficult to construct the image axes and the distortions are simply selects the images of the coordinate axes in the plane and adopt any appropriate distortion. The mathematical justification for this is the set of Pohlke: For ( almost) any choice of axes and image distortion is obtained up to similarity ( scaling) the image of a parallel projection. ( The images of the coordinate axes can not be on a straight line, the distortions should be greater than zero. )

Selecting the image axes and distortions

Only with an appropriate choice of the image axes and distortions of the image impression is good. The best image effect is achieved if one so chooses the image axes and the distortion ratios that the result is a perpendicular parallel projection. As one with the simplest possible distortion ratios (eg 1 or 0.5 ) would like to work you can in choosing the image axes and distortions in the following examples ( see picture) are based.

In all four examples, the image plane is parallel to the yz plane. There are thus called cavalier projection or cavalier perspectives. The visual effect is like that for a rider. A simplification occurs, even if the images of the x-axis and the y-axis intersect at right angles in the image. Such axonometries be used in Maps to achieve justice scale (horizontal) and vividness of buildings. This axonometries hot bird's eye or military projections.

If you have plaid paper available, then offers the following choice for axes and distortions: Two coordinate axes coincide with the principal directions of plaid paper, the third axis in the direction of the Karo- diagonal ( see input image ). To simplify the design, you should choose the units on the horizontal and vertical axis and two boxes on the diagonal direction of a diagonal box as a unit. Then, for example (see image above).

Axonometries with two identical distortions hot dimetric, with three equal isometric distortion, otherwise trimetric.

Engineer projection

In an engineering projection, the distortion

In the projection is the

(see protractor )

Advantages of the engineering projection are:

• Simple distortion
• Almost a vertical Axonometric ( good image effect, the scaling factor is 1.06 ),
• The contour of a sphere is a circle ( otherwise it is an ellipse ).

Isometric Axonometric

In the isometric axonometric, in short: isometry, the distortions are all equal. The angle between the axis images can be freely selected. ( Note the multiple meaning of the term isometry in mathematics. )

As a standard isometry has proven that, for the distortions all equal to 1 and the angles between the axes are all 120 degrees. The advantages of this choice of parameters are:

• The coordinates can be used unchanged,
• The axonometric image is scaled by a factor orthogonal ( perpendicular parallel projection ). This results in a good visual effect and the outline of a sphere is a circle.
• Sign systems, such as xfig, provide a triangle grid to facilitate the drawing of objects with axis- parallel edges (see image ).

A blemish because of the symmetry is that two of the eight vertices of an axis-aligned cube coincide (see image ).

Overview of the special axonometries

In a general axonometric the two angles between the axes and the distortion ( almost) can be chosen freely. Thus, all three axis images do not lie on a straight line, has to be. This restriction on the choice of the angle guarantees a view obliquely from above. The restriction provides views from obliquely below. It reversed the usual orientation of the x-axis to the y- axis. Negative distortion would alter the usual orientation of the axes.

Circles in the axonometric

Circuits can be established in parallel projection generally ellipses. An important special case: a circle, the circle plane is parallel to the image panel is imaged without distortion. This is for example the case with a trivial projection is imaged without distortion in the yz-plane ( see example). For a bird's eye view all horizontal circuits remain undistorted. If a circle is distorted into an ellipse (see picture), can be mapped some points and a tangent square and in the image of the square ( parallelogram ) an ellipse by hand or fit with a drawing program. It should be noted that the images are not the main ellipse axes of the image, but conjugated diameter of circle diameters is generally vertical. From these you can see the main axes with the axis Rytzsche construction reconstruct. Then can the ellipse with a drawing program or a trammel draw accurately. If you have only compass, ruler and a curve ruler at hand, the ellipse can be amazingly good and quick with the help of the vertex curvature circles approximately draw (see ellipse or C. Leopold, p 64). In the orthogonal axonometric it mostly comes without the elaborate Rytzkonstruktion.

Balls in the axonometric

The outline of a sphere is only at orthogonal axonometric simply a circle with the radius of the sphere. Since both the Ingenieuraxonometrie and the Standardisometrie are scaled orthogonal (see above), the outline of a sphere appears also in each case as circular, but scaled. In any Axonometric the outline of a ball appears as an ellipse, which may irritate the viewer (see ball in isometric bird's eye view ). Therefore it is better to depict scenes with balls with orthogonal axonometric or at least in engineering axonometric or Standardisometrie.

Swell

• Fucke, Kirch, Nickel: Descriptive Geometry. Textbook -Verlag, Leipzig, 1998, ISBN 3-446-00778-4
• Cornelia Leopold: Geometric foundations of architectural representation. Publisher W. Kohlhammer, Stuttgart, 2005, ISBN 3- 17-018489 -X