Documents del Fons Lluís Santalóhttp://hdl.handle.net/10256.2/80802017-12-18T18:56:55Z2017-12-18T18:56:55ZIntegral Geometry on Surfaces of Constant Negative CurvatureSantaló, Lluíshttp://hdl.handle.net/10256.2/108602011-04-11T09:37:07Z1943-01-01T00:00:00ZIntegral Geometry on Surfaces of Constant Negative Curvature
Santaló, Lluís
We use the expression Integral geometry in the sense given it by Blaschke in “Vorlesungen über Integrtüçeomelrie”. In a previous paper (“Integral formulas in Croflon style on the sphere and some inequalities referring to spherical curve” (1942), Duke Mathematical Journal, vol. 9, pp. 707) we generalized to the sphere many formulas of plane integral geometry and at the same time applied these to the demonstration of certain inequalities referring to spherical curves. The present paper considers analogous qüestions for surfaces of constant negative curvature and consequently for hyperbolic geometry
1943-01-01T00:00:00ZIntegral formulas in Crofton's style on the sphere and some inequalities referring to spherical curvesSantaló, Lluíshttp://hdl.handle.net/10256.2/108592011-04-11T07:44:47Z1942-01-01T00:00:00ZIntegral formulas in Crofton's style on the sphere and some inequalities referring to spherical curves
Santaló, Lluís
Several integral formulas referring to convex plane curves, notable for their great generality, were obtained by W. Crofton in 1868 and successive years from the theory of geometrical probability [6], [7], [8], [9], and [10]. A direct and rigorous exposition of Crofton's principal results, adding some (new formulas, was made in 1912 by H. Lebesgue [12]. Another systematic exposition of Crofton's most interesting formulas, together with the generalization of many of them to space, is found in the two volumes on integral geometry by Blaschke [2]. The purpose of the present paper is to give a generalization of Crofton's formulas to the surface of the sphere. This is what we do in part I. We find further integral formulas on the sphere (for instance, (16), (17), (20), (21)) which have no equivalent in the plane. Other formulas, if we consider the plane as the limit of a sphere whose radius increases indefinitely, give integral formulas referring to plane convex curves (e. g., (34), (35)) which we think are new. In part II, with simple methods of integral geometry [2], we obtain three inequalities referring to spherical curves. Inequality (38) is the generalization to the sphere of an inequality that Hornich [11] obtained for plane curves. (52) and (58) contain the classical isoperimetric inequality on the sphere. Finally, inequality (61) gives a superior limitation for the "isoperimetric deficit" of convex curves on the sphere
1942-01-01T00:00:00ZAffine Invariants of Certain Pairs of Curves and SurfacesSantaló, Lluíshttp://hdl.handle.net/10256.2/108582011-04-05T07:57:41Z1947-01-01T00:00:00ZAffine Invariants of Certain Pairs of Curves and Surfaces
Santaló, Lluís
For two curves in a plane or two surfaces in ordinary space various projective invariants have been given by different authors. Obviously each projectiva invariant is also an affine invariant, that is, an invariant with respect to the group of affine transformations. However in
certain cases there are affine invariants which are not projective invariants. The purpose of the present paper is to study these cases giving affine invariants, as well as their affine and metrical characterization, for the following cases:
(a) two curves in a plane having a common tangent at two ordinary points
(b) two curves in a plane intersecting at an ordinary point
(c) two surfaces in ordinary space having a common tangent plane at two ordinaiy points
(d) two surfaces in ordinary space having a common tangent line but distinct tangent plane at two ordinary points.
For the cases (a),(b) of plane curves we shall consider the neighborhoods of the second and the third order of the curves at the considered points. For the cases (c), (d) of two surfaces in ordinary space we shall consider only the neighborhoods of the second order of the surfaces at the considered points
1947-01-01T00:00:00ZUna Demostración de la propiedad isoperimétrica del círculoSantaló, Lluíshttp://hdl.handle.net/10256.2/107462011-04-05T07:33:37Z1940-01-01T00:00:00ZUna Demostración de la propiedad isoperimétrica del círculo
Santaló, Lluís
Una nueva demostración de la propiedad isoperimétrica del círculo, en el plano y sobre la esfera, deducida de una fórmula de Crofton de la teoría de Probabilidades geométricas y de otra fórmula de Geometría integral
1940-01-01T00:00:00Z