On surfaces of constant negative curvature and their deformation

Abstract

We have shown that the pseudosphere is applicable to itself in an infinity of ways. Therefore these surfaces that are applicable to it can, after they are folded on the pseudosphere, be made to pass through the same deformations that the pseudosphere undergoes to reveal its applicability to itself. Hence they are applicable to the pseudosphere in an infinity of ways; and since in being applied to the pseudosphere, they are applied to each other and to themselves folded on the pseudosphere, they are therefore applicable to each other and to themselves in an infinity of ways. To illustrate the applicability of these surfaces we have taken two casts, one from each of our pseudospheres. These casts we formed into molds, and papers pressed in the smaller one which has the same Gaussian curvature as our surfaces of the elliptic and hyperbolie types, can be applied to any portion of these two surfaces or to itself. Papers pressed in the larger mold can be applied to any portion of the large pseudosphere without stretching, tearing or crumpling the paper. us first define what is meant by curvature. If w represents the angle between the positive directions of two tangents M T, and M' T', at two points M and M' which are infinitely near and on a curve c, at the point M, equal to the limit of the quotient of (arc M M' / w), as M approaches M' indefinitely; that is, R = ds/dw which is the reciprocal of the curvature. Therefore the curvature K = dw/ds. The radius of curvature lies in the osculating plane. The radius of the curvature of a normal section is measured along the normal to the surface. The radius of curvature of a normal section is obtained when the osculating plane coincides with a normal section of the surface. The principal radii of curvature of a point P on a surface are the radii of curvature of the principal normal sections, which sections pass through the axes of the indicatrix.