Solid Geometry, Sections and Development of Surfaces
This domain covers true shapes of sections and the flat development (net) of solid surfaces - the two skills that turn a 3D object into manufacturable information.
Sections
A cutting plane cuts the solid; the true shape of the cut face is obtained by an auxiliary view taken perpendicular to the cutting plane. Project the points where the plane crosses the edges from the view in which the plane appears as a straight line, then transfer their distances from a datum using compasses or a mitre line. The section appears in every view, so a cut on a cone shows as a line on the elevation and as a closed curve on the plan.
Development methods
- Parallel-line - prisms and cylinders: stretch the perimeter into a straight line, divide it accurately, and project each height.
- Radial-line - pyramids and cones: true slant lengths must first be found by rotating the slant edge parallel to the vertical plane.
- Triangulation - oblique solids and transition pieces.
For a right cone: sector radius = true slant height L; arc length = πD; sector angle = 360° × D ÷ (2L).
Worked example
A cone has base diameter 60 mm and vertical height 80 mm. L = √(30² + 80²) = 85.4 mm. Sector angle = 360 × 60 ÷ (2 × 85.4) = 126.4°. Set the compasses to 85.4 mm, strike a 126.4° sector, divide the arc into twelve equal parts, and step the generators out to the true lengths taken from the elevation.
Traps
Developing an apparent slant length instead of the true one; omitting the section from one view; forgetting that a truncated cone's development uses radii measured to the cut, not the apex only; no allowance for the seam.