Slab (geometry)

In geometry, a slab is a region between two parallel lines in the Euclidean plane, ^[1] or between two parallel planes in three-dimensional Euclidean space or between two hyperplanes in higher dimensions. ^[2]

Set definition

A slab can also be defined as a set of points: ^[3] $${\displaystyle \{x\in \mathbb {R} ^{n}\mid \alpha \leq n\cdot x\leq \beta \},}$$ where ${\displaystyle n}$ is the normal vector of the planes ${\displaystyle n\cdot x=\alpha }$ and ${\displaystyle n\cdot x=\beta }$.

Or, if the slab is centered around the origin: ^[4] $${\displaystyle \{x\in \mathbb {R} ^{n}\mid |n\cdot x|\leq \theta /2\},}$$ where ${\displaystyle \theta =|\alpha -\beta |}$ is the thickness of the slab.

See also

• Bounding slab
• Convex polytope
• Half-plane
• Hyperplane
• Prismatoid
• Slab decomposition
• Spherical shell

References

1. Preparata, Franco P.; Shamos, Michael Ian (1985). "2.2.2.1 The slab method". Computational Geometry: An Introduction. New York: Springer. pp. 45–48. doi:10.1007/978-1-4612-1098-6. ISBN   978-1-4612-7010-2. S2CID   206656565.
2. Jacob, Goodman. "Handbook of Discrete and Computational Geometry". CRC Press LLC. Retrieved 24 July 2022.
3. S., Boyd. "Convex Optimization". Cambridge University Press. Retrieved 14 March 2022.
4. Jean-Luc, Marichal; Mossinghoff, Michael J. (2008). "Slices, slabs, and sections of the unit hypercube" (PDF). Online Journal of Analytic Combinatorics. 3 (1). arXiv: math/0607715 .