Solids in dihedral projection: polyhedra, surfaces of revolution, sections and developments
Here the exam gives a solid by position and size - an edge on a plane, an axis, a face in a projecting plane - and asks for its projections, a section, or its development. The mark depends on hidden edges and on true sizes obtained before developing.
Polyhedra
Regular polyhedra, prisms and pyramids are placed by an edge, a face, an axis or a diagonal. Build the base in true size by a change of plane or by rotation, then take heights perpendicular to the horizontal plane. A cube resting on a face gives a square plus vertical edges; a cube standing on a diagonal needs the diagonal measured in true length and the outline rebuilt from it. Always draw hidden edges dashed.
Surfaces of revolution
Cylinder given by axis and radius; cone given by axis, apex and generatrix or by base radius and height, with g = √(r² + h²); sphere given by centre and radius, whose apparent contour is a circle of the same radius in both projections; torus given by axis and generating circle. The apparent contour is not the same as the equator or the silhouette of a section.
Sections
A plane cuts a polyhedron along a polygon whose vertices lie on the edges: pass auxiliary projecting planes through the edges and join the piercing points. For a cone, the section is an ellipse when the plane cuts all generatrices, a parabola when it is parallel to one generatrix, a hyperbola when it is parallel to two (cutting both nappes), a circle when perpendicular to the axis, and a point or two lines through the apex. For a cylinder: ellipse, rectangle, circle or two lines. For a sphere the section is always a circle, found by a change of plane or by auxiliary planes.
Developments
Cylinder: a rectangle 2πr by h. Cone: a circular sector of radius g and angle α = 360°·r/g; a truncated cone is the sector minus an inner sector, and the cut curve is transferred point by point with the true distance from the apex. Pyramid and prism: true lengths of the edges by right triangle, then the faces laid out in order.
Worked example
A right cone has r = 20 mm and h = 48 mm, so g = √(400 + 2304) = 52 mm. Its development is a sector of radius 52 mm and angle α = 360·20/52 = 138.5°. Any section point is transferred by its true distance from the apex, measured on the corresponding generatrix.
Traps
Developing with the apparent generatrix instead of the true one; placing section vertices inside a face rather than on an edge; omitting hidden edges; forgetting that a section of a sphere is a circle of radius √(R² − d²).