Development of surfaces and intersections
Development of surfaces and intersections
This domain tests the flat pattern of a solid's surface and the curve formed where two solids meet. You need true lengths, an accurate stretch-out and a clean intersection curve.
Core rules
- Use the parallel-line method for prisms and cylinders: draw a stretch-out line equal to the perimeter of the base, divide it into equal parts, and project heights from the front view. For a cylinder, stretch-out length is πD, not D.
- Use the radial-line method for pyramids and cones: find the true slant edge, draw an arc with radius equal to the slant height, and step off the base edges around it. For a truncated cone, develop the full cone first and cut off the top portion.
- Use triangulation for oblique or transition pieces. Divide the surface into triangles, find the true length of each side, and build the triangles in sequence.
- For intersections, locate common points by cutting planes or by projecting visible points from one view to the other. Join the points in order and decide visibility; the curve is the line of interpenetration.
Worked example
Develop a cylinder of diameter 40 mm and height 60 mm. The stretch-out length is π × 40 = 125.7 mm. Draw a rectangle 125.7 mm wide and 60 mm high. Divide the width into 12 equal parts (each about 10.5 mm) to match the 12 divisions of the top view, then draw vertical generators. Reason: rolling the cylinder on a plane unrolls the circumference into a straight line while the height stays unchanged.
Traps
Using the diameter instead of the circumference is the most common error. Others are failing to use the true slant height for a cone, using too few divisions for a smooth curve, and drawing an intersection curve through points that are not common to both solids.