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] [math]\displaystyle{ \{x \in \mathbb{R}^n \mid \alpha \le n \cdot x \le \beta \}, }[/math] where [math]\displaystyle{ n }[/math] is the normal vector of the planes [math]\displaystyle{ n \cdot x = \alpha }[/math] and [math]\displaystyle{ n \cdot x = \beta }[/math].
Or, if the slab is centered around the origin:[4] [math]\displaystyle{ \{x \in \mathbb{R}^n \mid |n \cdot x| \le \theta / 2 \}, }[/math] where [math]\displaystyle{ \theta = |\alpha - \beta| }[/math] is the thickness of the slab.
See also
References
- ↑ "2.2.2.1 The slab method". Computational Geometry: An Introduction. New York: Springer. 1985. pp. 45–48. doi:10.1007/978-1-4612-1098-6. ISBN 978-1-4612-7010-2.
- ↑ Jacob, Goodman. "Handbook of Discrete and Computational Geometry" (in en). CRC Press LLC. http://www.csun.edu/~ctoth/Handbook/HDCG3.html. Retrieved 24 July 2022.
- ↑ S., Boyd. "Convex Optimization" (in en). Cambridge University Press. https://web.stanford.edu/~boyd/cvxbook/. Retrieved 14 March 2022.
- ↑ Jean-Luc, Marichal; Mossinghoff, Michael J. (2008). "Slices, slabs, and sections of the unit hypercube". Online Journal of Analytic Combinatorics 3 (1). https://hosted.math.rochester.edu/ojac/vol3/Marichal_2008.pdf.
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