Applications of Mathematics, Vol. 68, No. 3, pp. 259-268, 2023
Tight bounds for the dihedral angle sums of a pyramid
Sergey Korotov, Lars Fredrik Lund, Jon Eivind Vatne
Received January 15, 2022. Published online June 21, 2022. OPEN ACCESS
Abstract: We prove that eight dihedral angles in a pyramid with an arbitrary quadrilateral base always sum up to a number in the interval $(3\pi,5\pi)$. Moreover, for any number in $(3\pi,5\pi)$ there exists a pyramid whose dihedral angle sum is equal to this number, which means that the lower and upper bounds are tight. Furthermore, the improved (and tight) upper bound $4\pi$ is derived for the class of pyramids with parallelogramic bases. This includes pyramids with rectangular bases, often used in finite element mesh generation and analysis.
References: [1] J. Brandts, A. Cihangir, M. Křížek: Tight bounds on angle sums of nonobtuse simplices. Appl. Math. Comput. 267 (2015), 397-408. DOI 10.1016/j.amc.2015.02.035 | MR 3399056 | Zbl 1410.51026
[2] J. Brandts, S. Korotov, M. Křížek: The discrete maximum principle for linear simplicial finite element approximations of a reaction-diffusion problem. Linear Algebra Appl. 429 (2008), 2344-2357. DOI 10.1016/j.laa.2008.06.011 | MR 2456782 | Zbl 1154.65086
[3] P. G. Ciarlet: The Finite Element Method for Elliptic Problems. Studies in Mathematics and Its Applications 4. North-Holland, Amsterdam (1978). DOI 10.1016/s0168-2024(08)x7014-6 | MR 0520174 | Zbl 0383.65058
[4] J. W. Gaddum: The sums of the dihedral and trihedral angles in a tetrahedron. Am. Math. Mon. 59 (1952), 370-371. DOI 10.1080/00029890.1952.11988143 | MR 0048034 | Zbl 0046.37903
[5] J. W. Gaddum: Distance sums on a sphere and angle sums in a simplex. Am. Math. Mon. 63 (1956), 91-96. DOI 10.1080/00029890.1956.11988764 | MR 0081488 | Zbl 0071.36304
[6] H. Katsuura: Solid angle sum of a tetrahedron. J. Geom. Graph. 24 (2020), 29-34. MR 4128185 | Zbl 1447.51020
[7] A. Khademi, S. Korotov, J. E. Vatne: On interpolation error on degenerating prismatic elements. Appl. Math., Praha 63 (2018), 237-257. DOI 10.21136/AM.2018.0357-17 | MR 3833659 | Zbl 06945731
[8] A. Khademi, S. Korotov, J. E. Vatne: On the generalization of the Synge-Křížek maximum angle condition for $d$-simplices. J. Comput. Appl. Math. 358 (2019), 29-33. DOI 10.1016/j.cam.2019.03.003 | MR 3926696 | Zbl 1426.65177
[9] S. Korotov, J. E. Vatne: The minimum angle condition for $d$-simplices. Comput. Math. Appl. 80 (2020), 367-370. DOI 10.1016/j.camwa.2019.05.020 | MR 4099857 | Zbl 1446.65100
[10] L. Liu, K. B. Davies, K. Yuan, M. Křížek: On symmetric pyramidal finite elements. Dyn. Contin. Discrete Impuls. Syst., Ser. B, Appl. Algorithms 11 (2004), 213-227. MR 2049777 | Zbl 1041.65098
[11] C. Wieners: Conforming discretizations on tetrahedrons, pyramids, prisms and hexahedrons. University Stuttgart, Bericht 97/15 (1997), 1-9.
Affiliations: Sergey Korotov (corresponding author), Division of Mathematics and Physics, UKK, Mälardalen University, Box 883, 721 23 Västerås, Sweden, e-mail: sergey.korotov@mdu.se; Lars Fredrik Lund, Department of Computer Science, Electrical Engineering and Mathematical Sciences, Faculty of Engineering and Science, Western Norway University of Applied Sciences, Postbox 7030, 5020 Bergen, Norway, e-mail: lars.fredrik.lund@hvl.no, Jon Eivind Vatne, Department of Economics, Norwegian Business School (BI), Kong Christian Frederiks plass 5, 5006 Bergen, Norway, e-mail: jon.e.vatne@bi.no
