Axonometric Projection by Rotation and Tilt

Rotation and Tilt angles applied within a sphere.

The three axes of an axonometric projection are three mutually perpendicular edges in space. If the angle of rotation and the angle of tilt of the object are known or decided upon, the three axes for the pictorial drawing and the location of the orthographic views for projection to the picotiral can easily be found.
The three orthographic views of a cube are shown. The three mutually perpendicular edges OA, OB, and OC will be foreshortened differently when the cube is rotated in space for some axonometric position, but the ends of the axes a, b and c will always lie on the surface of a sphere whose radius is OA = OB = OC as illustrated by figure 1.

Fig. 1 — Radius of Sphere = OA = OB = OC

Fig. 2 — Completed Axonometric Projection

At any particular angle of tilt of the cube, the axes ends a¹ and c¹ will describe an ellipse, as shown, if the cube is rotated about the axes OB. The axes OB will appear foreshortened to Ob¹ as shown in figure 3 below.

Fig. 3 — Axes Ends a¹ and c¹ Describe an Ellipse

Thus, for any particular position of the cube in space, representing some desired axonometric position, the axes can be located and their relative amounts of foreshortening found.

Moreover, if a face of the cube is revolved about an axis at right angles to the axis which is perpendicular to the face, an orthographic view of the face in projection with the axonometric view, will result. Thus, the plan and elevation views may be located and the axonometric projection can be obtained by direct projection from any two of the three revolved views.

The following diagrams illustate the practical use of this theory. The actual size of the sphere is unimportant, as it is used only to establish the direction of the axes.

First, the desired angle of rotation, R and the angle of tilt, T are decided upon and laid out as already shown in figure 3 above. The minor diameter for the ellipse upon which A and C will lie is found by projecting vertically from b and drawing the circle as shown in figure 4. A and C on the major-diameter circle of the ellipse will be at a and b. On the minor-diameter circle, they will be at am and cm; and they are found in the axonometric position by projecting, as in the concentric-circle ellipse method, to a¹ and c¹ . The foreshortened position of B is found by projecting horizontally across from b to b¹.

Fig. 4 — Foreshortening of Axes

The revolved plan view will be parallel to oa and oc, and projection from this othographic view to the axonometric will be vertical (parallel to aa¹ and cc¹). Projection from an orthographic left-side view will be as follows. The left side containing axes OA and OB, is found by projecting from a¹ and b¹ parallel to oc¹, to locate a² and b² on the circle representing the sphere. The sides of the object are parallel to oa² and ob² as shown in figure 5. Projection from the left side view to the axonometric view is in the dirction of oc¹.

Fig. 5 — Revolved Left Side View

Projection from an orthographic right-side view will be as follows. The right side containing axes OB and OC, is found by projecting from b¹ and c¹ parallel to oa¹, to locate b³ and c³ on the circle representing the sphere. The sides of the object are parallel to ob³ and oc³ as shown in figure 6. Projection from the right side view to the axonometric view is in the direction of oa¹.

Fig. 6 — Revolved Right Side View

For convenience and clarity the the orthographic views are laid out in positions removed from the sphere in figure 7 below.

Fig. 7 — Laying out the Orthographic Views

Once the positions for the orthograhic views has been established, the views can be completed as shown if figure 8 below.

Fig. 8 — Completion of Orthographic Views

The axonometric view can now be completed by projection from the orthographic views as follows:-

• Projection from the plan view to the axonometric will be vertical (parallel to aa¹ and cc¹).
• Projection from the left side view to the axonometric view is in the dirction of oc¹.
• Projection from the right side view to the axonometric view is in the direction of oa¹.
The completed Axonometric View using the method of Rotation and Tilt is now shown in figure 9.

Fig. 9 — The Completed Construction