**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*

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

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*

*Fig. 5 — Revolved Left
Side View*

*Fig. 6 — Revolved Right
Side View*

*Fig. 7 — Laying out
the Orthographic Views*

*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.

*Fig. 9 — The Completed
Construction*