Question 5 of 10: Constructive Solid Geometry — Boolean Primitive Construction

Question 5: Constructive Solid Geometry — Boolean Primitive Construction (10 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Object A is a rectangular block with a smaller, stepped-up rectangular boss at one end and a horizontal cylindrical through-hole running the full length (its axis shown by two parallel hidden lines threading through both the block and the boss in the given views). Object B is a hub/flange: a ring-shaped body (constant wall thickness — a torus of rectangular, not circular, cross-section, i.e. a tube) with a smaller cylindrical boss unioned onto its outside diameter, a central bore, and a smaller bore through the boss.

Find. An ordered sequence of primitive solids and Boolean operations (union ∪, difference ∖, intersection ∩) that reproduces each object exactly.

Approach. Constructive Solid Geometry (CSG) builds any polyhedral/rotational part from a short tree of simple primitives combined by Boolean set operations; the standard method is to first union together every "added material" primitive (bosses, ribs, pads) onto a base primitive, then subtract every "removed material" primitive (holes, slots, counterbores) last, since subtraction must always act on the already-unioned solid to guarantee the hole passes cleanly through every layer it should.

1. Object A — base primitive. Start with a right rectilinear prism sized to the main block's length × width × height.
2. Object A — union the boss. Union a second, smaller right rectilinear prism, positioned flush with one end and offset upward, to form the stepped boss seen in the given views.
3. Object A — difference the hole. Subtract a cylinder, axis horizontal and coincident with the hidden-line pair, long enough to pass fully through both the main prism and the boss, producing the through-hole.
4. Object B — base primitive. Start with a torus of rectangular cross-section (equivalently, a large cylinder with a smaller, shorter, coaxial cylinder subtracted — but since a torus/ring is offered as an explicit primitive in this exam's approved list, model the main flange body directly as one ring primitive) sized to the flange's outside diameter and axial thickness.
5. Object B — union the boss. Union a smaller cylinder onto the outside diameter of the ring, positioned so its axis is parallel to the ring's axis, forming the side boss.
6. Object B — difference the bores. Subtract a cylinder along the main axis (the central bore) and a second, smaller coaxial cylinder through the boss (the boss hole) — both subtractions performed last, after the union, so each bore passes cleanly through the already-combined solid.