Technical Drawing · Lesson 10

Intersections of prisms, cylinders, cones and spheres; common-sphere method

WAECNECO

What the syllabus asks

WAEC: WAEC/NECO Technical Drawing, Solid Geometry item 18 (interpenetration of prisms, cylinders, cones and pyramids; lines of intersection; developments of intersecting solids). (Not a JAMB UTME subject: Technical Drawing is absent from JAMB’s official IBASS e-syllabus list.)

NECO: WAEC/NECO Technical Drawing, Solid Geometry item 18 (interpenetration of prisms, cylinders, cones and pyramids; lines of intersection; developments of intersecting solids). (Not a JAMB UTME subject: Technical Drawing is absent from JAMB’s official IBASS e-syllabus list.)

What you will learn

Explain what a line of intersection is and how visibility is decided.
Draw intersections of prisms with prisms by the edge method.
Draw intersections of cylinders (equal, unequal, offset, inclined) by generators and cutting planes.
Draw intersections involving cones and pyramids by cutting planes.
Apply the common-sphere method and recognise Monge cases.
Develop the parts of intersecting solids.

Key points

Points of intersection lie on both surfaces.
Prisms: straight lines, edge by edge.
Cylinders: generators read from the view where the other is a circle.
Equal cylinders: straight lines in elevation.
Unequal tee: height √(R² − y²); worked dip 6.7 mm.
Cutting planes: lines and circles crossing.
Common spheres for solids of revolution with intersecting axes.
Number generators and join in order, smoothly.
Show hidden portions where the back differs.
Develop parts after the intersection is found.

Start here: What interpenetration is

When two solids join, the line where their surfaces meet is called the line (or curve) of intersection, and the drawing of the joined solids is called interpenetration. Every pipe tee, branch duct, chimney through a roof, bolt through a plate, and dome on a cylindrical tank has one. Before a fabricator can develop the parts (td9), the line of intersection must be drawn accurately.

Combinations in the syllabus

Prism with prism (square, hexagonal, triangular), axes intersecting or offset.
Cylinder with cylinder, equal or unequal diameters, at right angles or inclined, axes intersecting or offset.
Cylinder with prism.
Cylinder or prism with cone or pyramid.
Cone with cylinder (common-sphere method).

Key idea

A point on the line of intersection lies on both surfaces. So we look for lines on one surface (edges of a prism, generators of a cylinder or cone) and find where each meets the other surface. Joined in the correct order, these points form the line of intersection.

Visibility

A point of the intersection is visible in a view only if it is on a visible part of both surfaces. Parts of the curve at the back are drawn as hidden lines if needed, but in symmetrical problems the back half usually coincides with the front half in the elevation.

Reading the question

Interpenetration questions give two views (often the plan and an incomplete elevation) and ask you to complete the elevation with the line of intersection, sometimes adding a development of one part. Before plotting, identify: (1) which solid passes into which; (2) whether the axes intersect, are offset or inclined; (3) which view shows each solid as an edge. The answers decide the method. Spend two minutes on this analysis; it saves twenty minutes of plotting points that do not help.

Two families of method

Line (generator) method: take an element of one solid and find where it pierces the other, usually from the view in which the other solid appears as an edge (a circle or a line).
Cutting-plane method: pass a series of planes through both solids. Each plane cuts simple lines or circles from both; where these cross are points on the intersection. The common-sphere method is a special version using spheres.

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Practice questions

1. Two cylinders of equal diameter meeting at right angles with axes intersecting have a line of intersection that appears in the front view as

two straight lines
a circle
a parabola
a sine curve

Show answer

A: two straight lines
Equal tee: z = ±x.

2. The common-sphere method can be used when two solids of revolution have

parallel axes only
skew axes
no axes
intersecting axes

Show answer

D: intersecting axes
A sphere centred on the common point cuts both in circles.

3. Two solids of revolution on a common axis meet in

straight lines
parabolas
circles
ellipses

Show answer

C: circles
Coaxial solids meet in circles perpendicular to the axis.

4. A vertical cylinder of radius 20 mm passes centrally through a sphere of radius 50 mm. The circles of intersection are from the centre

20 mm
45.8 mm
30 mm
50 mm

Show answer

B: 45.8 mm
√(50² − 20²) ≈ 45.8 mm.

5. The usual method for finding the intersection of two cylinders is

the cutting-plane (generator) method
the trammel method
the radial-line method
oblique projection

Show answer

A: the cutting-plane (generator) method
Pass planes or generators through both solids.

Key words

Interpenetration: Two solids passing into each other.
Line of intersection: Line common to both surfaces.
Generator: Straight surface line of a cylinder or cone.
Generator method: Finding where each generator meets the other solid.
Cutting-plane method: Using planes to cut simple lines from both solids.
Common-sphere method: Using spheres centred on the axis intersection.
Monge’s theorem: Solids tangent to one sphere intersect in plane curves.
Tee: Branch at right angles to a main pipe.
Lateral: Branch at an angle other than 90°.
Offset branch: Branch whose axis misses the main axis.
Solid of revolution: Solid formed by rotating a shape about an axis.
Outline generator: Generator forming the visible outline in a view.

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