Isometric projection is a graphical method used to represent three-dimensional objects in two dimensions. In this type of projection, the angle between any two of the three coordinate axes (X, Y, Z) appears to be the same, typically $120^\circ$.

Calculating Isometric Scale

A key characteristic of true isometric projection is that lines parallel to the isometric axes are not shown at their true length; they are scaled down by a constant factor known as the foreshortening factor.

• The theoretical foreshortening factor for lines parallel to the isometric axes is calculated using geometric principles.
• This factor is mathematically represented as the ratio of the projected length to the true length.
• Calculation: The foreshortening factor is $ \frac{\sqrt{2}}{\sqrt{3}} $.
• Approximate Value: $ \frac{\sqrt{2}}{\sqrt{3}} \approx 0.816496... $

Isometric Axis Representation

To express this factor as a percentage of the true length:

• Percentage $= \left( \frac{\sqrt{2}}{\sqrt{3}} \right) \times 100\% $
• Percentage $\approx 0.8165 \times 100\% $
• Percentage $\approx 81.65\% $

Therefore, features depicted along the isometric axes in a true isometric projection are shown at approximately $81.6\%$ of their actual length.