Axes and foreshortening
SŠ9 minWhy one image is enough – three axes, three ratios and Pohlke’s theorem.
System
Orthogonal
Oblique
Ratios and angles
- p (x)
- 0.8165
- q (y)
- 0.8165
- r (z)
- 0.8165
- ∠(x, y)
- 120°
- ∠(y, z)
- 120°
- ∠(z, x)
- 120°
Every orthogonal axonometry gives exactly 2 – that is its fundamental theorem. Try moving the view direction: the ratios change, the sum does not.
Show
Spatial view
Theory
Axonometry shows a body with a single image instead of two. The three coordinate axes project to three axes on the drawing, and the point [x, y, z] is laid off as x·(image of x) + y·(image of y) + z·(image of z). Each axis carries its own foreshortening ratio p, q, r.
A ratio says how much of its true length a unit on that axis keeps in the drawing. That is why the axonometric image of a cube is never a cube: edges in different directions are shortened differently, and that difference is the entire spatial information a single image carries.
Pohlke’s theorem says something surprising: any three segments from one point (as long as they are not collinear) can be taken as the parallel projection of three equal, mutually perpendicular segments. So you may choose the axes almost freely and the image is still a valid projection of a cube – which is precisely why oblique axonometry exists alongside orthogonal.
Step-by-step construction
Practice
Build the construction yourself
What is the next step?
1 / 5Choose the step that comes next. The drawing only advances on a correct choice.
Check your understanding
What does the ratio p on the x-axis tell you?
Why does the axonometric image of a cube never look like a cube?
What does Pohlke’s theorem state?
Downloadable material
A PDF for this lesson is in preparation.
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