A Treatise on the Differential Geometry of Curves and by Luther Pfahler Eisenhart

By Luther Pfahler Eisenhart

Created particularly for graduate scholars, this introductory treatise on differential geometry has been a hugely profitable textbook for a few years. Its strangely unique and urban method incorporates a thorough clarification of the geometry of curves and surfaces, targeting difficulties that would be so much priceless to scholars. 1909 edition.

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Extra resources for A Treatise on the Differential Geometry of Curves and Surfaces (Dover Phoenix Editions)

Example text

W, y = sin w, z <f> so that the principal normals to the (u) are parallel to the yz-plane. Find the osculating plane and radius of x a cos u -f 6 sin w, y = a sin u first + 10. Torsion. Frenet-Serret formulas. less a curvature of 6 cos w, z It has = c sin 2 u. been seen that, un curve be plane, the osculating plane varies as the point moves along the curve. The change in the direction depends evidently upon the form of the curve. The ratio of the angle A^ between the binormals at two points of the curve and their curvi linear distance As expresses our idea of the mean change in the direction of the osculating plane.

Curve, approaches point l along M M We M consider now the circle in this plane which has the same tan M as the curve, and passes through M The limiting posi M approaches called the osculating gent at { tion of this circle, as to the curve at M. 7I/, l circle It is evident that its center normal at M. Hence, with reference F the coordinates of (7 denoted by Q cipal , X =x + Y rl, where the absolute value of y M when l X , -f rm, , to Z Q, C is on the prin any fixed axes in space, are of the form Z^=z + rn, r is the radius of the osculating circle.

Instead of considering only the points whose locus is the curve, we may look upon the moving point as the intersection of three mutually perpendicular lines which move along with the point, the whole figure rotating so that in each position the lines coin cide with the tangent, principal normal, and binormal at the point. We shall refer to such a configuration as the moving trihedral. In the solution of certain problems it is of advantage to refer the curve to this moving trihedral as axes. proceed to the con We sideration of this idea.

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