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Published online by Cambridge University Press: 05 June 2012
But if the particle is not forced to move upon a determinate curve, the curve which it describes possesses a singular property, which had been discovered by metaphysical considerations; but which is in fact nothing more than a remarkable result of the preceding differential equations. It consists in this, that the integral ∫ vds, comprised between the two extreme points of the described curve, is less than on every other curve.
P. S. Laplace, Mécanique Céleste (1799)In the last few chapters we developed some of the analytic and geometric properties of surfaces in ℝ3 that are consequences of the properties of the first and second fundamental forms. The important geometric properties are intrinsic, that is, preserved by an isometry of the surface. To develop further elementary geometric notions we next introduce a notion of “line” on a surface.
A “line” on a surface should enjoy some of the familiar properties of lines in the plane: For example, lines are
(1) The curves of shortest length joining two points (Archimedes).
(2) The curves of plane curvature identically zero (Huygens, Leibniz, Newton).
(3) The curves whose tangent and its derivative are linearly dependent.
Furthermore, to have geometric significance, the defining notion for a “line” on a surface should be intrinsic. We first try to adapt the condition of zero plane curvature. Let α: (–∈, ∈) → S be a curve on S, parametrized by arc length.
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