The axonometric triangle
SŠ11 minThe image plane’s traces on the axes – and the origin as the orthocentre.
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
The origin O is where the triangle’s altitudes meet – its orthocentre. The dashed lines are those altitudes.
Spatial view
Theory
The image plane ρ cuts the coordinate axes at three points X, Y, Z. Joining them gives the axonometric triangle – the figure most constructions start from, because it holds everything you need to know about that axonometry.
The key theorem: in an orthogonal axonometry the origin O projects to the point where the triangle’s altitudes meet – its orthocentre. So once the triangle is drawn you find the origin by constructing two altitudes, not by estimating.
The second property that characterises orthogonal axonometry is the relation p² + q² + r² = 2. It holds for every view direction without exception – it follows from the image plane’s normal being a unit vector. If a triple of ratios does not sum to 2 this way, your axonometry is not 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
How does the axonometric triangle arise?
Where does the origin O project to in an orthogonal axonometry?
An axonometry has ratios 1; 0.5; 1. Is it orthogonal?
Downloadable material
A PDF for this lesson is in preparation.
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