Isometry, dimetry, trimetry
SŠ9 minEqual foreshortening on all three axes – and what happens when they are not equal.
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
Orthogonal axonometry is classified by how many ratios agree: isometry (p = q = r), dimetry (exactly two agree) and trimetry (all three differ). Isometry is the most used because it needs only one scale.
In isometry the axis images meet at 120° and all three ratios equal p = q = r = √(2/3) ≈ 0.8165. That number is not a convention – it follows from the view direction (1, 1, 1), which leans equally toward all three axes.
In technical practice one usually draws at 1 : 1 : 1 rather than 0.8165. That enlarges the image by about 22 %, but the shape is identical – what matters are the ratios between the axes, not their absolute values. Likewise dimetry is drawn as 1 : ½ : 1 although the exact values are 0.9428 : 0.4714 : 0.9428.
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 angle do the axis images make in isometry?
What is the exact foreshortening on each axis in orthogonal isometry?
What distinguishes dimetry from trimetry?
Downloadable material
A PDF for this lesson is in preparation.
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