Projection of Solids in Simple Position
Draw as you read. Every procedure below is written as numbered construction steps. Follow them with a pencil.
The solids in this syllabus
Only these appear, and the list is worth knowing because it bounds what can be asked:
Prisms and pyramids with triangular, square, rectangular, pentagonal and hexagonal bases; cones; cylinders.
- A prism has two identical parallel polygonal faces joined by rectangles. Its cross-section is constant.
- A pyramid has one polygonal base and triangular faces meeting at an apex. Its cross-section shrinks with height.
- A cylinder and a cone are the circular limiting cases.
Terminology that the questions rely on:
- Axis — the line joining the centres of the two ends, or base centre to apex.
- Base edge and lateral edge — for a pyramid the lateral edges are the slant edges to the apex.
- Slant height of a pyramid or cone — apex to the midpoint of a base edge (pyramid) or to a point on the base circle (cone).
Slant height and lateral edge are different for a pyramid, and confusing them is a common and expensive error. For a pyramid of base edge and height , the slant height is measured to the midpoint of a base edge, whereas the lateral edge runs to a corner and is longer.
The governing rule
Draw the view that shows the base as its true shape first.
If the axis is perpendicular to the HP — the solid resting on its base — then the base is parallel to the HP, so the top view shows the true shape of the base, a regular polygon or a circle. Start there and project the front view up from it.
If the axis is perpendicular to the VP, the front view shows the true base shape, and you start there.
This one rule removes most of the difficulty. Students who start with the wrong view spend the whole question fighting the construction.
Procedure — solid resting on HP, axis perpendicular to HP
- Draw XY.
- Top view first. Draw the true shape of the base — the regular polygon or circle — below XY, positioned as the question requires (often with one edge parallel or perpendicular to the VP, which is specified and matters).
- Mark the centre; this is the top view of the axis, which appears as a point because the axis is perpendicular to the HP.
- Project every corner of the base upward to XY.
- The base of the solid lies on the HP, so the base line of the front view lies on XY.
- Mark the height along the axis. For a prism or cylinder, draw the top face parallel to XY at the given height and join corresponding corners with vertical lateral edges. For a pyramid or cone, mark the apex on the axis at the given height and join it to every base corner.
- Apply visibility: edges hidden behind the solid are dashed.
Where the marks are lost: in step 2, the orientation of the base polygon. "A hexagonal prism with an edge of the base parallel to the VP" and "...perpendicular to the VP" give different drawings, and the question always specifies which.
Visibility
The rule is mechanical once stated: in the front view, an edge is hidden if it lies behind another face as seen from the front — that is, if it is further from the VP. In the top view, an edge is hidden if it lies below another face.
For a pyramid resting on its base with the apex above, all lateral edges are visible in the top view; in the front view, the lateral edge running to the rearmost base corner is hidden.
Cylinders and cones have no edges to hide — only the two extreme generators appear as outlines, and the base circle appears as a line.
The profile (side) view
A third view, projected onto a plane perpendicular to both HP and VP.
Procedure: draw a 45° mitre line from the intersection of XY and the vertical reference line. Project horizontally from the front view and vertically from the top view; bounce the top-view projectors off the 45° line to convert depth into horizontal distance. Their intersections give the side view.
The syllabus asks for the profile view of solids in simple position, so it is worth practising this bouncing construction until it is automatic.