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DIFFERENTIAL) it follows, that -an infinity of geodesics may be drawn at any point of a face, each determined by its direction at the The expression for ds' takes a notable form when the parameter lines, v —constant, are geodesics and the parameter lines, u =constant, are their orthogonal trajectories. Two cases are of interest: (1) The geodesic parallel system, in which the geodesics are drawn perpendicular to an arbitrary curve c as shown in Fig. 6, and t,• . . ., are the orthogonal trajectories. Any two trajectories intercept equal lengths on the geodesics, and it is from this fact that the trajectories derive their name of geodesic parallels. If u represents the distance from c to a geodesic parallel measured on the geodesics, into the form ds' ,3dAdp, where ,3 is a function of A and p. The parame ters X, p are conjugate imaginaries and the two series of corresponding curves on the surface are imaginary. They are called the minimal lines of the surface, as it is •evident that along any one of them ds'-•=0. The tan gents of a minimal line are the minimal right lines (see CURVES OF DOUBLE CURVATURE). By integrating d"--=0, one can determine the minimal parameters and thence a pair of iso thermal parameters. Just as there is but one system of lines of curvature and one system of asymptotic lines, so there is but one system of minimal lines on a surface.

A geodesic line is a curve such that at every point P of it, the principal normal of the curve and the normal of the surface coincide. The shortest curve traced between any two points of the surface is a geodesic. A given geodesic does not always represent the shortest curve between any two of its points, but the property of shortest distance does hold for sufficiently small segments of the geodesic. On a surface of negativ e Gauss curvature (K negative at all points) a geodesic does not cease to be a curve of shortest length. On a surface of positive Gauss curvature (K positive), e.g., on a sphere, (2) The geodesic polar system, in which the curve c reduces to a point 0 as shown in Fig. 7, and the geodesics proceed from O. The orthogonal trajectories are geodesic circles. If u represents the distance from 0 to a geodesic circle, and v the angle from a fixed geodesic taken as a base, to any geodesic, there follows: ds' •-= du + Gde, where G is a function of u and v satisfying two conditions at the pole point: [ {aV G =o N=0 We mention, finally, a system of references in which the curves are geodesic ellipses and hyperbolas. If a point P moves so that the sum

(difference) of its geodesic distances from two arbitrary curves c, ci of the surface is constant, it describes what is called a geodesic ellipse (hyperbola) • c and ci must not be geodesic parallels. Weingarten has shown that the system of geodesic ellipses and hyperbolas is orthogonal. A special case is when c and ci reduce to points. Lionville has investigated a class of surfaces on which there is an isother mal system of geodesic ellipses and hyperbolas. The form of the arc element is If(u) g(v) (dui + dvi), and the differential equation of the geodesics can be integrated and brought to the form dv v f(a) + a — f v and b are constants of integration. Surfaces of the second order and surfaces of rotation belong to the Lionville surfaces. Gauss established the theorem that the sum of the angles of a geodesic triangle (the sides are geodesic lines) is greater than, less than, or equal to, ir, accord ing as the triangle lies in an elliptic, hyperbolic or parabolic region. The only surface that can contain an area of parabolic points is a develop able surface (see 16), i.e., a surface developable upon a plane.

15. Representation of One Surface upon Another Surface. Conformal Representation. In map drawing one has an in stance of the depiction or ,representation of one surface upon another. To each point of one surface corresponds a definite point of the other surface. The character of the depiction is a matter of the law of relation of the corre sponding points of the two surfaces. If the equations of a surface A are expressed in parameters is, v and those of surface B in parameters, u, vi, any equations so, = g (u, v), = h(u, v) will give a law of correspondence of points, provided that to a pair of values of v there corresponds a pair of values in, and conversely.

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