Axonometry

Axonometry is a graphical procedure belonging to descriptive geometry that generates a planar image of a three-dimensional object. The term "axonometry" means "to measure along axes", and indicates that the dimensions and scaling of the coordinate axes play a crucial role. The result of an axonometric procedure is a uniformly-scaled parallel projection of the object. In general, the resulting parallel projection is oblique (the rays are not perpendicular to the image plane); but in special cases the result is orthographic (the rays are perpendicular to the image plane), which in this context is called an orthogonal axonometry.

Cavalier projection of a semicircular archway.

In technical drawing and in architecture, axonometric perspective is a form of two-dimensional representation of three-dimensional objects whose goal is to preserve the impression of volume or of relief. Sometimes also called rapid perspective or artificial perspective, it differs from conical perspective and does not represent what the eye actually sees: in particuliar parallel lines remain parallel and distant objects are not reduced in size. It can be considered a conical perspective conique whose center has been pushed out to infinity, i.e. very far from the object observed.

The term axonometry is used both for the graphical procedure described below, as well as the image produced by this procedure.

Axonometry should not be confused with axonometric projection, which in English literature usually refers to orthogonal axonometry.

Principle of axonometry

The point ${\displaystyle {\overline {P}}}$ is the projection of a point ${\displaystyle P=(x,y,z)}$ onto the projection plane Π. The foreshortenings are ${\displaystyle v_{x}}$, ${\displaystyle v_{y}}$ and ${\displaystyle v_{z}}$.

Pohlke's theorem is the basis for the following procedure to construct a scaled parallel projection of a three-dimensional object:[1][2]

1. Select projections of the coordinate axes, such that all three coordinate axes are not collapsed to a single point or line. Usually the z-axis is vertical.
2. Select for these projections the foreshortenings, ${\displaystyle v_{x}}$, ${\displaystyle v_{y}}$ and ${\displaystyle v_{z}}$, where ${\displaystyle v_{x},v_{y},v_{z}>0.}$
3. The projection ${\displaystyle {\overline {P}}}$ of a point ${\displaystyle P=(x,y,z)}$ is determined in three sub-steps (the result is independent of the order of these sub-steps):
   • starting at the point ${\displaystyle {\overline {O}}}$, move by the amount ${\displaystyle v_{x}\cdot x}$ in the direction of ${\displaystyle {\overline {x}}}$, then
   • move by the amount ${\displaystyle v_{y}\cdot y}$ in the direction of ${\displaystyle {\overline {y}}}$, then
   • move by the amount ${\displaystyle v_{z}\cdot z}$ in the direction of ${\displaystyle {\overline {z}}}$ and finally
4. Mark the final position as point ${\displaystyle {\overline {P}}}$.

In order to obtain undistorted results, select the projections of the axes and foreshortenings carefully (see below). In order to produce an orthographic projection, only the projections of the coordinate axes are freely selected; the foreshortenings are fixed (see de:orthogonale Axonometrie).[3]

The choice of the images of the axes and the forshortenings

Parameters and notations

Notation:

• ${\displaystyle \alpha$ :} angle between ${\displaystyle {\overline {z}}}$-axis and ${\displaystyle {\overline {x}}}$-axis
• ${\displaystyle \beta$ :} angle between ${\displaystyle {\overline {z}}}$-axis and ${\displaystyle {\overline {y}}}$-axis
• ${\displaystyle \gamma$ :} angle between ${\displaystyle {\overline {x}}}$-axis and ${\displaystyle {\overline {y}}}$-axis.

The angles can be chosen so that ${\displaystyle 0^{\circ }<\alpha +\beta <360^{\circ }\ .}$
The forshortenings: ${\displaystyle 0<\;v_{x},\;v_{y},\;v_{z}\ .}$

Only for suitable choices of angles and forshortenings does one get undistorted images. The next diagram shows the images of the unit cube for various angles and forshortenings and gives some hints for how to make these personal choices.

Various axonometric images of a unit cube. (The image plane is parallel to the y-z-plane.)
The left and the far right images look more like prolonged cuboids instead of a cube.

Axonometry (cavalier perspective) of a house on checked pattern paper.

In order to keep the drawing simple, one should choose simple forshortenings, for example ${\displaystyle 1.0}$ or ${\displaystyle 0.5}$.

If two forshortenings are equal, the projection is called dimetric.
If the three forshortenings are equal, the projection is called isometric.
If all forshortenings are different, the projection is called trimetric.

The parameters in the diagram at right (e.g. of the house drawn on graph paper) are: ${\displaystyle \alpha =135^{\circ },\beta =90^{\circ },\ v_{y}=v_{z}=1,\;v_{x}=1/{\sqrt {2}}\ .}$ Hence it is a dimetric axonometry. The image plane is parallel to the y-z-plane and any planar figure parallel to the y-z-plane appears in its true shape.

Special axonometries

Commonly used projections and perspectives with angles between the transformed coordinate axes (α, β, γ), angles from the horizon (α_h, β_h, without γ_h = 90°) and scaling factors (v_i)

Name or property	α = ∠x̄z̄	β = ∠ȳz̄	γ = ∠x̄ȳ	αh	βh	vx	vy	vz	v
Orthogonal, orthographic, planar	90°	0°	270°	0°	270°	v		0%	any
Trimetric	90° + αh	90° + βh	360° − α − β	any	any	any	any	any	any
Dimetric							v
Isometric						v
Normal									100%
Oblique, clinographic				< 90°	< 90°	any	any	any	tan(αh)
Symmetric		α	360° − 2·α	< 90°	αh				any
Equiangular	120°			30°
Normal, 1∶1 isometric						v			100%
Standard, shortened isometric									√⅔ ≈ 81%
Pixel, 1∶2 isometric	116.6°		126.9°	arctan(v)					50%
Engineering	131.4°	97.2°	131.4°	arccos(¾)	arcsin(⅛)	50%	v		100%
Cavalier	90° + αh	90°	270° − α	any	0°	any
Cabinet, dimetric cavalier						< 100%
Standard, isometric cavalier	135°		135°	45°		v
Standard 1∶2 cabinet						50%	v
30° cabinet	116.6°		153.4°	arctan(vx)
60° cabinet	153.4°		116.6°	arccot(vx)
30° cavalier	120°		150°	30°		any
Aerial, bird's eye view	135°		90°	45°		v		any	100%
Military						v
Planometric	90° + αh	180° − αh		any	90° − αh				any
Normal planometric									100%
Shortened planometric									⅔ ≈ 67%

Parameters of special axonometries.

Engineer projection

In this case [4][5]

• the forshortenings are: ${\displaystyle v_{x}=0.5,\ v_{y}=v_{z}=1\ }$ (dimetric axonometry) and
• the angles between the axes are: ${\displaystyle \alpha =132^{\circ },\ \beta =97^{\circ }\ .}$

These angles are marked on many German set squares.

Advantages of an engineer projection:

• simple forshortenings,
• a uniformly scaled orthographic projection with scaling factor 1.06,
• the contour of a sphere is a circle (in general, an ellipse) .

For more details: see de:Axonometrie.

Cavalier perspective, cabinet perspective

• image plane parallel to y-z-plane.

In the literature the terms "cavalier perspective" and "cabinet perspective" are not uniformly defined. The above definition is the most general one. Often, further restrictions are applied.[6][7] For example:

cabinet perspective: additionally choose ${\displaystyle \alpha =135^{\circ }}$ (oblique) and ${\displaystyle v_{x}=0.5}$ (dimetric),

cavalier perspective: additionally choose ${\displaystyle \alpha =135^{\circ }}$ (oblique) and ${\displaystyle v_{x}=1}$ (isometric).

Birds eye view, military projection

• image plane parallel to x-y-plane.

military projection: additionally choose ${\displaystyle v_{z}=1}$ (isometric).

Such axonometries are often used for city maps, in order to keep horizontal figures undistorted.

Isometric axonometry

standard isometry: cube, cuboid, house and sphere

(Not to be confused with an isometry between metric spaces.)

For an isometric axonometry all forshortenings are equal. The angles can be chosen arbitrarily, but a common choice is ${\displaystyle \alpha =\beta =\gamma =120^{\circ }}$.

For the standard isometry or just isometry one chooses:

• ${\displaystyle v_{x}=v_{y}=v_{z}=1}$ (all axes undistorted)
• ${\displaystyle \alpha =\beta =\gamma =120^{\circ }\ .}$

The advantage of a standard isometry:

• the coordinates can be taken unchanged,
• the image is a scaled orthographic projection with scale factor ${\displaystyle {\sqrt {1.5}}=1.225}$. Hence the image has a good impression and the contour of a sphere is a circle.
• Some computer graphic systems (for example, xfig) provide a suitable raster (see diagram) as support.

In order to prevent scaling, one can choose the unhandy forshortenings

• ${\displaystyle v_{x}=v_{y}=v_{z}={\sqrt {2/3}}}$ (instead of 1)

and the image is an (unscaled) orthographic projection.

various axonometries of a tower
dimetric military projection: ${\displaystyle v_{z}=0.5,v_{x,y}=1}$, dimetric engineering and cavalier projections: ${\displaystyle v_{x}=0.5,v_{y,z}=1}$, isometric axonometry: ${\displaystyle v_{x,y,z}=1}$

Circles in axonometry

A parallel projection of a circle is in general an ellipse. An important special case occurs, if the circle's plane is parallel to the image plane–the image of the circle is then a congruent circle. In the diagram, the circle contained in the front face is undistorted. If the image of a circle is an ellipse, one can map four points on orthogonal diameters and the surrounding square of tangents and in the image parallelogram fill-in an ellipse by hand. A better, but more time consuming method consists of drawing the images of two perpendicular diameters of the circle, which are conjugate diameters of the image ellipse, determining the axes of the ellipse with Rytz's construction and drawing the ellipse.

• 
  Cavalier perspective: circles
• 
  Military projection: sphere

Spheres in axonometry

In a general axonometry of a sphere the image contour is an ellipse. The contour of a sphere is a circle only in an orthogonal axonometry. But, as the engineer projection and the standard isometry are scaled orthographic projections, the contour of a sphere is a circle in these cases, as well. As the diagram shows, an ellipse as the contour of a sphere might be confusing, so, if a sphere is part of an object to be mapped, one should choose an orthogonal axonometry or an engineer projection or a standard isometry.

References

• Graf, Ulrich; Barner, Martin (1961). Darstellende Geometrie. Heidelberg: Quelle & Meyer. ISBN 3-494-00488-9.
• Fucke, Kirch Nickel (1998). Darstellende Geometrie. Leipzig: Fachbuch-Verlag. ISBN 3-446-00778-4.CS1 maint: ref=harv (link)
• Leopold, Cornelie (2005). Geometrische Grundlagen der Architekturdarstellung. Stuttgart: Kohlhammer Verlag. ISBN 3-17-018489-X.CS1 maint: ref=harv (link)
• Brailov, Aleksandr Yurievich (2016). Engineering Graphics: Theoretical Foundations of Engineering Geometry for Design. Springer. ISBN 978-3-319-29717-0.CS1 maint: ref=harv (link)
• Stärk, Roland (1978). Darstellende Geometrie. Schöningh. ISBN 3-506-37443-5.CS1 maint: ref=harv (link)

Notes

1. Graf 1961, p. 144.
2. Stärk 1978, p. 156.
3. Graf 1961, p. 145.
4. Graf 1961, p. 155.
5. Stärk 1978, p. 168.
6. Graf 1961, p. 95.
7. Stärk 1978, p. 159.

External links

• Orthogonal axonometry