Last reviewed 16 Sept 2026 · 7 min read
Intersection of solids
When one solid penetrates another, their surfaces meet along lines or curves of intersection (interpenetration). These curves are needed for:
- Pipe joints — tees, branches, elbows, ducts.
- Sheet metal work — accurate cutting before developing surfaces.
- Structural and machine parts — domes with openings, boilers with nozzles, chimneys through roofs, junctions of columns and slabs.
Principle
A point on the curve of intersection lies on the surfaces of both solids. Such points are found systematically and joined in the proper order.
Methods
| Method | Procedure | Suitable for |
|---|---|---|
| Line (generator) method | Draw lines (edges/generators) on the surface of one solid; find where they pierce the other solid's surface (using a view in which that surface appears as an edge/line) | Prisms and cylinders — when one solid's surface appears as a line or curve in some view |
| Cutting plane method | Pass a series of cutting planes through both solids; each plane cuts simple sections (lines or circles) from both; their intersections are points on the curve | Cones, spheres, and general cases |
Choice of cutting planes: planes giving simplest sections — e.g. for a cone with vertical axis, horizontal planes give circles; planes through the apex give straight generators.
Typical intersections
| Case | Curve of intersection (in view where both axes are parallel to the plane) |
|---|---|
| Two cylinders of equal diameter, axes intersecting at right angles | Straight lines (two lines at 45° forming an X for a full cross; a V for a tee) |
| Two cylinders of unequal diameter, axes intersecting at right angles | Curves bending towards the axis of the larger cylinder |
| Cylinder penetrating a prism (or vice versa) | Curves on each flat face (portions of ellipses) |
| Prism penetrating a prism | Straight lines (polygonal lines) joining points on edges |
| Cylinder penetrating a cone | Space curve found by horizontal cutting planes (circles on cone and lines on cylinder) |
| Axes offset (not intersecting) | Asymmetric curves |
Visibility
- A segment of the intersection curve is visible only if it lies on visible surfaces of both solids in that view.
- The order of joining points follows their positions on the generators/edges; the extreme (critical) points — on contour generators, outermost or innermost points — must be located first.
Auxiliary views
Principal views (front, top, side) show surfaces parallel to principal planes in true shape. Surfaces inclined to the principal planes appear foreshortened. An auxiliary view is a view on an auxiliary plane parallel to the inclined surface, showing its true shape.
Types of auxiliary planes
| Auxiliary plane | Position | View obtained | Measurements transferred |
|---|---|---|---|
| Auxiliary vertical plane (AVP) | Perpendicular to HP, inclined to VP | Auxiliary front view (auxiliary elevation) | Heights (distances above HP) taken from the front view |
| Auxiliary inclined plane (AIP) | Perpendicular to VP, inclined to HP | Auxiliary top view (auxiliary plan) | Distances from VP taken from the top view |
Procedure (primary auxiliary view)
- Identify the inclined surface appearing as an edge (line) in one principal view.
- Draw a new reference line (X₁Y₁) parallel to that edge at a convenient distance.
- Draw projectors perpendicular to X₁Y₁ from all points of the edge view.
- Transfer distances from the other principal view (measured from XY) along the projectors from X₁Y₁.
- Join the points — the true shape of the inclined surface.
Usually only the inclined surface is drawn in the auxiliary view (a partial auxiliary view), with break lines, to avoid confusing foreshortened features.
Secondary auxiliary view
For an oblique surface (inclined to all principal planes), which does not appear as an edge in any principal view:
- Draw a primary auxiliary view in which the surface appears as an edge (view along a true-length line of the surface).
- Draw a secondary auxiliary view on a plane parallel to that edge — showing the true shape.
Other applications of auxiliary projection
- True length of a line — auxiliary plane parallel to the line.
- Point view of a line — auxiliary plane perpendicular to the true-length view.
- Edge view of a plane — view along a true-length line lying in the plane.
- Dihedral angle between two planes — view along their line of intersection (point view).
- Shortest distance between skew lines.