Steinmetz Solid

Steinmetz Solid

Mathematical Definition

Let r ≥ 0 be the common radius. The first cylinder is parallel to the x-axis, the second to the y-axis, and the optional third to the z-axis. They are infinite cylinders; their axes meet at the origin.

The finite wireframe guides in the visualization illustrate these infinite cylinders. Their displayed end rings are not solid boundaries. The filled mesh is built from the cylindrical surface patches that satisfy the remaining inequalities. The amber section is drawn through the shell so its complete outline stays visible.

Key Concepts

Two cylinders: square slices

At fixed height z, with |z| ≤ r, both |x| and |y| are at most a = √(r² − z²). The section is therefore a square of side 2a. Integrating its area gives an exact volume without π.

For r = 1, the central section has area 4 and the volume is 16/3. For r = 2 and z = 1, the section has area 12 and the volume is 128/3. The solid occupies two thirds of its bounding cube, which has volume 8r³.

Three cylinders: a disk-clipped square

The third cylinder adds the disk condition x² + y² ≤ r² to each square slice. If a ≤ r/√2, all square corners lie inside the disk, so the section remains a square. Otherwise, remove four disjoint circular caps from the disk. Each cap has area r² arccos(a/r) − a√(r² − a²).

At z = 0, the square contains the entire disk and A₃ = πr². At z = r/2, the area is r²(π/3 + √3). The two formulas join at |z| = r/√2 with area 2r²; the Critical slice button selects this transition exactly. At z = ±r, either solid has a point section of area zero.

An independent tricylinder volume calculation

Use eight octants and the six possible orderings of nonnegative coordinates. In the wedge x ≥ y ≥ z ≥ 0, the strongest cylinder constraint is x² + y² ≤ r². Thus 0 ≤ y ≤ r/√2, y ≤ x ≤ √(r² − y²), and 0 ≤ z ≤ y. Symmetry boundaries have zero volume, so multiplying this wedge by 48 gives the full volume.

The unit-radius tricylinder volume is approximately 4.686292, smaller than the bicylinder volume 5.333333. It still contains the unit ball, whose volume is approximately 4.188790: every point of that ball satisfies each cylinder inequality.

Midpoint integration and scaling

Divide [−r, r] into N equal layers and sample the area at each layer midpoint. The bars in the area plot have width Δz and height A(z_i); their summed areas represent a volume approximation. The displayed absolute error compares this sum with the exact formula.

For two cylinders with r = 1, N = 4 gives 5.5, N = 20 gives 5.34, and N = 100 gives 5.3336. The exact quadratic error identity follows from summing the midpoint samples of z². Both solids have concave area profiles, so midpoint integration overestimates their volumes; increasing the resolution leads to convergence. Do not apply the quadratic error identity to the three-cylinder profile.

Multiplying r by a factor k multiplies lengths by k, section areas by k², and volumes by k³. The 3D panel keeps a fixed physical scale at each viewport size; the section plan and area plot fit their own panels. Radius zero is a valid limiting degeneration: the common solid is the origin and all dimensional areas, volumes, and errors are zero. The normalized slice height has no physical effect in that case.

Historical Context

Cylinder intersections connect classical solid geometry to modern integral methods. Hubbell's 1965 National Bureau of Standards paper treats common volumes of intersecting cylinders, including different radii and non-right angles. Ramsay's three-cylinder investigation uses the orthogonal intersection to develop three-dimensional visualization and multivariable integration. This module restricts the model to equal radii and perpendicular intersecting axes.

Real-world Applications

• Beam overlap: Intersecting cylindrical beam models help estimate a common interaction volume. Hubbell discusses crossed-beam and radiation experiments.
• Solid modeling: Boolean intersections in CAD retain exactly the points common to the input solids. A tessellated boundary can display the result while analytic sections verify its volume.
• Numerical integration: Known exact volumes provide benchmarks for slice methods and a clear way to separate rendering approximation from numerical quadrature error.

Related Concepts

• Double & Triple Integrals — express volume as an integral over a region or its cross sections.
• Numerical Integration — compare midpoint approximations with exact values.
• Arc Length & Surface Area — distinguish boundary measurements from enclosed volume.

References

• J. H. Hubbell, Common Volume of Two Intersecting Cylinders (1965), Journal of Research of the National Bureau of Standards, 69C, 139–143; section 3.
• John Ramsay, Three Cylinder Intersection Problem, in Problems for Student Investigation, MAA Notes 30 (1993), pp. 137–142.