Steinmetz solid
E61291
The Steinmetz solid is a three-dimensional geometric shape formed by the intersection of two or more cylinders at right angles, often studied in calculus and solid geometry for its interesting volume and symmetry properties.
Statements (37)
| Predicate | Object |
|---|---|
| instanceOf |
geometric solid
ⓘ
mathematical object ⓘ solid of intersection ⓘ |
| appearsIn |
problems on triple integrals
ⓘ
undergraduate calculus textbooks ⓘ |
| boundary | formed by cylindrical surfaces ⓘ |
| category | solid of revolution applications ⓘ |
| context | Euclidean 3-dimensional space ⓘ |
| coordinateSystem |
often described in Cartesian coordinates
ⓘ
often described in cylindrical coordinates ⓘ |
| definedAs | intersection of two or more cylinders at right angles ⓘ |
| dimension | 3 ⓘ |
| field |
calculus
ⓘ
geometry ⓘ solid geometry ⓘ |
| generalizationOf | intersection of more than two mutually perpendicular cylinders ⓘ |
| hasAxisConfiguration | cylinder axes intersect at right angles ⓘ |
| hasExample | intersection of two perpendicular unit cylinders ⓘ |
| hasProperty |
high degree of symmetry
ⓘ
non-convex surface ⓘ smooth surface ⓘ |
| namedAfter | Charles Proteus Steinmetz ⓘ |
| relatedTo |
cylinder
ⓘ
multiple integrals ⓘ volume of intersection ⓘ |
| specialCase | intersection of two equal cylinders of radius r with perpendicular axes ⓘ |
| studiedFor |
applications in analytic geometry
ⓘ
integration over complex regions ⓘ symmetry properties ⓘ |
| symmetry |
invariant under 90-degree rotations exchanging cylinder axes
ⓘ
symmetric with respect to coordinate planes containing cylinder axes ⓘ |
| topology | simply connected ⓘ |
| usedIn |
examples of change of variables in integration
ⓘ
multiple integral exercises ⓘ visualization of solid intersections ⓘ volume computation problems ⓘ |
| visualization | often illustrated as a shape with four rounded lobes ⓘ |