TY - CHAP
T1 - Checkerboard Incircular Nets
AU - Bobenko, Alexander I.
AU - Lutz, Carl O.R.
AU - Pottmann, Helmut
AU - Techter, Jan
N1 - Publisher Copyright:
© 2021, The Author(s), under exclusive license to Springer Nature Switzerland AG.
PY - 2021
Y1 - 2021
N2 - In this section, as an application of two-dimensional Lie and Laguerre geometry, we present new research results. While incircular nets and their Laguerre geometric generalization to checkerboard incircular nets have been studied in great detail [Böh1970, AB2018, BST2018], we introduce their generalization to Lie geometry, and show that they may be classified in terms of checkerboard incircular nets in hyperbolic/elliptic/Euclidean Laguerre geometry. We prove incidence theorems of Miquel type, show that all lines of a checkerboard incircular net are tangent to a hypercycle, and give explicit formulas in terms of Jacobi elliptic functions. This generalizes the results from [BST2018] and leads to a unified treatment of checkerboard incircular nets in all space forms.
AB - In this section, as an application of two-dimensional Lie and Laguerre geometry, we present new research results. While incircular nets and their Laguerre geometric generalization to checkerboard incircular nets have been studied in great detail [Böh1970, AB2018, BST2018], we introduce their generalization to Lie geometry, and show that they may be classified in terms of checkerboard incircular nets in hyperbolic/elliptic/Euclidean Laguerre geometry. We prove incidence theorems of Miquel type, show that all lines of a checkerboard incircular net are tangent to a hypercycle, and give explicit formulas in terms of Jacobi elliptic functions. This generalizes the results from [BST2018] and leads to a unified treatment of checkerboard incircular nets in all space forms.
UR - http://www.scopus.com/inward/record.url?scp=85118591327&partnerID=8YFLogxK
U2 - 10.1007/978-3-030-81847-0_8
DO - 10.1007/978-3-030-81847-0_8
M3 - Chapter
AN - SCOPUS:85118591327
T3 - SpringerBriefs in Mathematics
SP - 81
EP - 115
BT - SpringerBriefs in Mathematics
PB - Springer Science and Business Media B.V.
ER -