Solids with the Axis Inclined to One or Both Reference Planes

By the end you should be able to: Apply the change-of-position method to draw projections of a solid whose axis is inclined to one reference plane, and extend it to an axis inclined to both.

Draw as you read. This topic is a procedure, and procedures are learned by executing them.

The whole method in one sentence

Draw the solid in the easy position, then tilt it — and when it is tilted, copy lengths rather than re-measuring them.

Every problem in this topic is that sentence applied once (axis inclined to one plane) or twice (inclined to both). There is nothing else to it.

Why we do not draw the tilted solid directly

Because the base is then inclined to both planes, so neither view shows its true shape, and there is no way to construct a regular pentagon that is foreshortened by an unknown amount. Instead we draw the base in true shape while it is still parallel to a plane, and carry that construction into the tilted position by projection.

The change-of-position method — axis inclined to one plane

Three stages.

Stage 1 — simple position. Draw the solid with its axis perpendicular to the plane it will eventually be inclined to. If the axis is to end up at θ to the HP, start with the axis perpendicular to the HP — the solid resting on its base. Draw the top view (true base shape) and project the front view, as in the previous topic.

Stage 2 — tilt. Redraw the front view of Stage 1, tilted so that the axis makes the required angle θ with XY. Do not construct it afresh: copy it. The shape of the front view is unchanged by the tilt; only its orientation changes. Trace or transfer every point.

Stage 3 — new top view. Project each point of the tilted front view downward, and each corresponding point of the Stage 1 top view horizontally across. Where a point's vertical projector meets its own horizontal line is its new top view position. Join the points in the same order as before.

The principle that makes Stage 3 work: tilting about an axis perpendicular to the VP moves points vertically and along XY, but does not change their distance from the VP. Distance from the VP is measured perpendicular to XY in the top view, so those distances carry straight across from the Stage 1 top view.

That is the rule to memorise: when the tilt is shown in the front view, the top view keeps its distances from XY. And symmetrically: when the tilt is shown in the top view, the front view keeps its heights.

Axis inclined to both planes

Four stages — the same trick twice.

Stage 1 — simple position, axis perpendicular to HP. Stage 2 — tilt the front view so the axis is at θ to the HP. Stage 3 — project the new top view (as above). Stage 4 — now redraw the Stage 3 top view, rotated so that the axis makes the required angle φ with XY, and project the final front view upward from it, taking heights from the Stage 2 front view.

Note the switch: the second tilt is applied to the top view, and the quantity carried across is now height above XY rather than distance from XY.

The commonest mistake in the whole module is applying both tilts to the front view. If you do, the solid has been rotated twice about the same axis and the second condition is not satisfied at all.

Reading what the question asks for

The wording specifies the tilt, and small differences matter:

"axis inclined at 30° to the HP" — tilt the front view so the axis is at 30° to XY.
"axis inclined at 30° to the HP and 45° to the VP" — two tilts, in the order above.
"resting on an edge of the base" or "on a corner of the base" — this fixes where the tilt pivots, and the stated edge or corner stays on the HP throughout.
"the axis appears inclined at 45°" — an apparent angle, measured in the finished view, not a true angle. Such problems are solved by tilting until the drawn angle matches, rather than by setting the angle directly.

That last distinction — true angle versus apparent angle — is a favourite examiner's trap, exactly as it was for lines in Module 1.