Invariant rectification of non-smooth planar curves
Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry - Trang 1-23 - 2023
Tóm tắt
We consider the problem of defining arc length for a plane curve invariant under a group action. Initially one partitions the curve and sums a distance function applied to consecutive support elements. Arc length then is defined as a limit of approximating distance function sums. Alternatively, arc length is defined using the integral that arises when applying the method to smooth curves. That these definitions agree in the general case for the Euclidean group was an early victory of the Lebesgue integral; the equivalence also is known for the equi-affine group. Here we present a unified treatment for equi-affine, Laguerre, inversive, and Minkowski (pseudo-arc) geometries. These are alike in that arc length corresponds with a geometric average of finite Borel measures.
Tài liệu tham khảo
Balestro, V., Martini, H., Shonoda, E.: Concepts of curvatures in normed planes. Expo. Math. 37(4), 347–381 (2019)
Barrett, D.E., Bolt, M.: Cauchy integrals and Möbius geometry of curves. Asian J. Math. 11(1), 47–53 (2007)
Barrett, D.E., Bolt, M.: Laguerre arc length from distance functions. Asian J. Math. 14(2), 213–233 (2010)
Benz, W.: Vorlesungen über Geometrie der Algebren. Springer, Berlin, Geometrien von Möbius, Laguerre-Lie, Minkowski in einheitlicher und grundlagengeometrischer Behandlung, Die Grundlehren der mathematischen Wissenschaften, Band 197 (1973)
Bini, D., Geralico, A., Jantzen, R.T.: Frenet–Serret formalism for null world lines. Class. Quantum Gravity 23(11), 3963–3981 (2006)
Blaschke, Wilhelm: Vorlesungen über Differentialgeometrie, II (Affine Differentialgeometrie). Springer, Berlin (1923)
Bobenko, A.I., Suris, Y.B.: Discrete Differential Geometry: Integrable Structure. Graduate Studies in Mathematics, vol. 98. American Mathematical Society, Providence (2008)
Bonnor, W.B.: Null curves in a Minkowski space-time. Tensor (N.S.) 20, 229–242 (1969)
Brown, R., Marsland, S., McLachlan, R.: Differential invariant signatures for planar Lie group transformations with application to images. J. Lie Theory 32(3), 709–736 (2022)
Cairns, G., Sharpe, R.W.: On the inversive differential geometry of plane curves. Enseign. Math. (2) 36(1–2), 175–196 (1990)
Calabi, E., Olver, P.J., Tannenbaum, A.: Affine geometry, curve flows, and invariant numerical approximations. Adv. Math. 124(1), 154–196 (1996)
Coxeter, H.S.M.: Inversive distance. Ann. Mat. Pura Appl. 4(71), 73–83 (1966)
Crane, K., Wardetzky, M.: A glimpse into discrete differential geometry. Not. Am. Math. Soc. 64(10), 1153–1159 (2017)
Dymara, J.: Duality for curves in affine plane geometry. Geom. Dedicata 79(2), 189–204 (2000)
Ferrández, A., Giménez, A., Lucas, P.: Null helices in Lorentzian space forms. Int. J. Mod. Phys. A 16(30), 4845–4863 (2001)
Folland, G.B.: Real Analysis: Modern Techniques and Their Applications. Pure and Applied Mathematics. A Wiley-Interscience Publication, 2nd edn. Wiley, New York (1999)
Guven, J., Manrique, G.: Conformal mechanics of planar curves (2019). arXiv:1905.00488
Inoguchi, J., Lee, S.: Null curves in Minkowski 3-space. Int. Electron. J. Geom. 1(2), 40–83 (2008)
Kerzman, N., Stein, E.M.: The Cauchy kernel, the Szegö kernel, and the Riemann mapping function. Math. Ann. 236(1), 85–93 (1978)
Kisil, V.V.: Starting with the group \({{\rm SL}}_2({ R})\). Not. Am. Math. Soc. 54(11), 1458–1465 (2007)
Leichtweiß, K.: Affine Geometry of Convex Bodies. Johann Ambrosius Barth Verlag, Heidelberg (1998)
Leichtweiß, K.: On the affine rectification of convex curves. Beiträge Algebra Geom. 40(1), 185–193 (1999)
Liebmann, H.: Beiträge zur Inversionsgeometrie der Kurven. Münch. Ber. (Akademie der Wissenschaften, Munich, Sitzungsberichte), pp. 79–94 (1923)
López, R.: Differential geometry of curves and surfaces in Lorentz–Minkowski space. Int. Electron. J. Geom. 7(1), 44–107 (2014)
Maeda, J.: On the Laguerre-geometry of plane curves. Jpn. J. Math. 17, 13–25 (1940)
Maeda, J.: Differential Laguerre geometry of plane curves. Jpn. J. Math. 18, 385–581 (1942a)
Maeda, J.: Differential Möbius geometry of plane curves. Jpn. J. Math. 18, 67–260 (1942b)
Nolasco, B., Pacheco, R.: Evolutes of plane curves and null curves in Minkowski 3-space. J. Geom. 108(1), 195–214 (2017)
Olver, P.J.: Equivalence, Invariants, and Symmetry. Cambridge University Press, Cambridge (1995)
Olver, P.J.: Moving frames and differential invariants in centro-affine geometry. Lobachevskii J. Math. 31(2), 77–89 (2010)
Patterson, B.: The differential invariants of inversive geometry. Am. J. Math. 50(4), 553–568 (1928)
Stein, E.M., Shakarchi, R.: Real Analysis: Measure Theory, Integration, and Hilbert Spaces. Princeton Lectures in Analysis, vol. 3. Princeton University Press, Princeton (2005)
Struve, H., Struve, R.: Non-Euclidean geometries: the Cayley–Klein approach. J. Geom. 98(1–2), 151–170 (2010)
Su, B.C.: Affine Differential Geometry. Science Press Beijing/Gordon & Breach Science Publishers, Beijing/New York (1983)
Synge, J.L.: An action principle for photons. Nature 317, 675 (1985)
Tao, T.: An Introduction to Measure Theory. Graduate Studies in Mathematics, vol. 126. American Mathematical Society, Providence (2011)
Vessiot, E.: Sur les courbes minima. C. R. Acad. Sci. Paris 140, 1381–1384 (1905)
Yaglom, I.M.: Complex Numbers in Geometry. Translated from the Russian by Eric J. F. Primrose. Academic Press, New York (1968)
Yaglom, I.M.: A Simple Non-Euclidean Geometry and Its Physical Basis. Springer, New York (1979). An elementary account of Galilean geometry and the Galilean principle of relativity, Heidelberg Science Library, Translated from the Russian by Abe Shenitzer, With the editorial assistance of Basil Gordon