Development of Surfaces and Intersections
Development methods
Parallel-line development suits prisms and cylinders: stretch the perimeter or circumference into a straight line and erect the true heights. Radial-line development suits pyramids and cones: obtain the true slant edge or slant height and lay true lengths along an arc. Triangulation divides an oblique surface into triangles, each developed from true lengths. Where the solid is cut by a plane, the cut points are projected onto the development and the true shape of the cut is attached.
Intersections
The line of intersection of two solids is found by passing a series of auxiliary cutting planes through both, drawing the section each plane makes with each solid, and marking where those sections cross. Planes parallel to the axis suit cylinders and prisms; concentric spheres suit two solids of revolution with a common axis. Whether each part of the curve is visible or hidden depends on which solid lies in front.
Worked example
A cone has base radius 30 mm and height 80 mm. Slant height L = √(30² + 80²) = 85.4 mm. Its development is a sector of radius 85.4 mm with angle 360 × 30/85.4 = 126.4°, plus the base circle. For a cylinder of the same radius the development is a rectangle 2π × 30 = 188.5 mm wide; dividing into 12 elements gives 15.7 mm spacing.
Traps
Using the height instead of the slant height for a cone; omitting the base circle attached to the cone development; missing the points where one solid's outline crosses the other, so the curve breaks; drawing an intersection as a straight line; failing to number the cutting planes 1, 2, 3 … so points cannot be transferred consistently.