It is possible to find the axial directions in order to obtain the axonometric axial directions if the numbers for the scales are given.

Given: Length Ratios( 1:X:Y)

1. Draw an arbitrarily sized semi-circle on the line B’C’.
2. Draw a line O’C’ such that it is at an angle arcCos(X) with the horizontal and intersects the semi-circle at point O’.
3. Draw a line (O’B’1) at angle arcCos(Y) to the horizontal.
4. Transfer points B’1, O’, and C’ down to the bottom view.
5. Draw a circle centered at O through B.
6. Transfer point B’ down such that it lies on the constructed circle, creating point B. Then use point O and point B to construct a line.
7. Construct a perpendicular line OB through point C until it intersects vertically with point B at a new point: A. Connect A to point O.
8. The line segments BO, AO, and CO form the angles of the graphic scale.

An example problem is given below.

Construction of a graphic directional scale using given scale factors.